diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgcgd" "b/data_all_eng_slimpj/shuffled/split2/finalzzgcgd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgcgd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nKey agreement is one of the most important \nproblems in the cryptography, and it has been\nextensively studied in the information theory \nfor discrete sources (e.g.~\\cite{ahlswede:93,csiszar:00,csiszar:04})\nsince the problem formulation\nby Maurer \\cite{maurer:93}.\nRecently, the confidential message \ntransmission \\cite{wyner:75,csiszar:78} in \nthe MIMO wireless communication has attracted \nconsiderable attention as a\npractical problem setting \n(e.g.~\\cite{liang:08,liu:09,bustin:09,ly:09,liu:09b,ekrem:09,ekrem:10,liu:10,khisti:10}). \nAlthough the key agreement in the context of the\nwireless communication has also attracted\nconsiderable attention recently \\cite{bloch:08},\nthe key agreement from analog sources has not\nbeen studied sufficiently compared to\nthe confidential message transmission.\nAs a fundamental case of the key agreement\nfrom analog sources, we consider the key\nagreement from correlated vector Gaussian\nsources in this paper. More specifically,\nwe consider the problem in which the legitimate \nparties, Alice and Bob, and an eavesdropper, Eve,\nhave correlated vector Gaussian sources respectively,\nand Alice and Bob share a secret key from their sources\nby using the public communication.\nRecently, the key agreement from Gaussian sources\nhas attracted considerable attention in the context\nof the quantum key distribution \\cite{grosshans:03},\nwhich is also a motivation to investigate\nthe present problem.\nFig.~\\ref{Fig:scenario} illustrates a scenario\nwe are considering.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig3.eps}\n\\caption{An example of the scenarios treated in this paper.\nThe legitimate parties, Alice and Bob, and an eavesdropper, Eve,\nreceive (vector) signals from the base station respectively.\nAlice and Bob generate a secret key from their received signals\n$X$ and $Y$ by using the\npublic communication.}\n\\label{Fig:scenario}\n\\end{figure}\n\nTypically, the first step of the key agreement protocol\nfrom analog sources is the quantization of the sources.\nIn literatures (e.g.~see \\cite{bloch:08,assche:04,assche:book}), \nthe authors used the scalar quantizer, \ni.e., the observed source is quantized in each time instant.\nUsing the finer quantization, we can expect the higher key\nrate in the protocol, where the key rate is the ratio between the length of\nthe shared key and the block length of\nthe sources that are used in the protocol.\nHowever, there is a problem such that the finer quantization\nmight increase the rate of the public communication in the protocol.\nAlthough the public communication is usually regarded as a cheap resource\nin the context of the key agreement problem, \nit is limited by a certain amount in practice.\nTherefore, we consider the key agreement protocols with \nthe rate limited public communication in this paper.\nThe purpose of this paper is to clarify the trade-off\nbetween the key rate and the public communication rate\nof the key agreement protocol from vector\nGaussian sources.\n\nThe key agreement by rate limited public communication\nwas first considered by Csisz\\'ar and \nNarayan for discrete sources \\cite{csiszar:00}.\nFor the class of protocols with one-way public communication,\nthey characterized the optimal trade-off between the key rate and\nthe public communication rate in terms of the information\ntheoretic quantities, i.e., they derived the so-called\nsingle letter characterization.\nHowever, there are two difficulties to extend their \nresult to the vector Gaussian sources.\n\nFirst, the direct part of the proof in \\cite{csiszar:00}\nheavily relies on the finiteness of the alphabets\nof the sources, and cannot be applied to continuous sources.\nThis difficulty was solved by the authors in\n\\cite{watanabe:09c}, and this result will be\nalso used in this paper.\n\nSecond, although\nthe converse part of Csisz\\'ar and Narayan's characterization\ncan be easily extended to continuous sources,\nthe characterization is not computable because\nthe characterization involves auxiliary random variables\nand the ranges of those random variables are unbounded\nfor continuous sources.\n\nIn \\cite{watanabe:09c} for scalar Gaussian sources,\nthe authors showed that Gaussian auxiliary random\nvariables suffice, and derived a closed form expression\nof the optimal trade-off.\nIn the problem for scalar Gaussian sources,\nwe first solved the problem in which the sources are\ndegraded, i.e., Alice's source, Bob's source, and\nEve's source form a Markov chain in this order.\nThen, we reduced the general case to the degraded\ncase by using the fact that scalar Gaussian correlated\nsources are stochastically degraded \\cite{cover}.\n\nIn this paper for vector Gaussian sources,\nwe show that Gaussian auxiliary random\nvariables suffice, and characterize the\noptimal trade-off in terms of the\n(covariance) matrix optimization problem.\nOne of difficulties to show our result is that\nvector Gaussian sources are not\nstochastically degraded in general, and cannot\nbe reduced to the degraded case in the same manner\nas scalar Gaussian sources.\nTo circumvent this difficulty, we utilize the \nenhancement technique introduced by\nWeingarten {\\em et al}.~\\cite{weingarten:06}.\n\n\nThe rest of the paper is organized as follows:\nIn Section \\ref{sec:preliminaries}, we explain\nour problem formulation.\nIn Section \\ref{sec:main}, we show\nour main results and some numerical examples.\nIn Sections \\ref{sec:aligned-case} and\n\\ref{sec:proof-general}, our main results\nare proved.\nFinally, in Section \\ref{sec:conclusion},\nthe conclusion and the future research agenda\nare discussed.\n\n\\section{Problem Formulation}\n\\label{sec:preliminaries}\n\nLet $X$, $Y$, and $Z$ be correlated \nvector Gaussian sources on $\\mathbb{R}^{m_x}$,\n$\\mathbb{R}^{m_y}$, and $\\mathbb{R}^{m_z}$\nrespectively,\nwhere $\\mathbb{R}$ is the set of real\nnumbers.\nThen, let $X^n$, $Y^n$, and $Z^n$ be i.i.d.\ncopies of $X$, $Y$, and $Z$ respectively.\nThroughout the paper, upper case letters indicate\nrandom variables, and the corresponding lower case\nletters indicate their realizations.\nWe also use the following notations throughout the paper:\n$\\Sigma$ designates the covariance matrix of $(X,Y,Z)$.\n$\\Sigma_x$, $\\Sigma_{xy}$, and $\\Sigma_{y|x}$ designate\n$\\mathbb{E}[X^TX]$, $\\mathbb{E}[X^T Y]$, and \nthe conditional covariance of $Y$ given $X$ etc..\n$N \\sim {\\cal N}(0, A)$ means that the random variable\n$N$ is a Gaussian vector with zero mean and covariance\nmatrix $A$. We use $|A|$ to denote the determinant\nof the matrix $A$,\n$\\left|\\frac{A}{B}\\right|$ to denote \n$\\frac{|A|}{|B|}$, and we denote\n$A \\preceq B$ ($A \\prec B$) if the matrix $B-A$ is\npositive semidefinite (definite). \nThroughout the paper, we assume that $\\Sigma \\succ 0$.\n\nAlthough Alice and Bob can use public communication\ninteractively in general, we concentrate on the\nclass of key agreement protocols in which only Alice \nsends a message to Bob over the public channel.\nFirst, Alice computes the message\n$C_n$ from $X^n$ and sends the\nmessage to Bob\nover the public channel.\nThen, she also compute the key $S_n$.\nBob compute the key $S_n^\\prime$\nfrom $Y^n$ and $C_n$.\nFig.~\\ref{Fig:protocol} illustrates the protocol\nwith one-way public communication.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig4.eps}\n\\caption{The key agreement protocol with one-way public communication.\nFirst, Alice sends the message $C_n$ to Bob over the public channel, which \nmight be eavesdropped by Eve. Then, Alice compute the key\n$S_n$ and Bob compute the key $S_n^\\prime$. The rate of the public\ncommunication is limited by $R_p$.}\n\\label{Fig:protocol}\n\\end{figure}\n\nThe error probability of the protocol\nis defined by \n\\begin{eqnarray*}\n\\varepsilon_n := \\Pr\\{ S_n \\neq S_n^\\prime \\}.\n\\end{eqnarray*}\nThe security of the protocol is measured by\nthe quantity\n\\begin{eqnarray*}\n\\nu_n := \\log|{\\cal S}_n| - \n H(S_n|C_n,Z^n),\n\\end{eqnarray*}\nwhere ${\\cal S}_n$ is the range of\nthe key $S_n$.\n\nIn this paper, we are interested in the trade-off\nbetween the public communication rate $R_p$\nand the key rate $R_k$.\nThe rate pair $(R_p,R_k)$ is defined to \nbe achievable if there exists a sequence of\nprotocols satisfying\n\\begin{eqnarray*}\n\\lim_{n \\to \\infty} \\varepsilon_n &=& 0, \\\\\n\\lim_{n \\to \\infty} \\nu_n &=& 0, \\\\\n\\limsup_{n \\to \\infty} \\frac{1}{n} \\log|{\\cal C}_n| &\\le& R_p,\\\\\n\\liminf_{n \\to \\infty} \\frac{1}{n} \\log|{\\cal S}_n| &\\ge& R_k,\n\\end{eqnarray*}\nwhere ${\\cal C}_n$ is the range of the message $C_n$\ntransmitted over the public channel.\nThen, the achievable rate region is defined as\n\\begin{eqnarray*}\n{\\cal R}(X,Y,Z) :=\n\\{(R_p,R_k) : \\mbox{$(R_p,R_k)$ is achievable} \\}.\n\\end{eqnarray*}\n\nIn \\cite{watanabe:09c}, the authors showed a closed \nform expression of ${\\cal R}(X,Y,Z)$ for the scalar problem, i.e.,\n$m_x = m_y = m_z = 1$. In the next section, we show that\nthe achievable rate region for the vector problem can be characterized as a\n(covariance) matrix optimization problem.\n\n\n\\section{Main Result}\n\\label{sec:main}\n\n\\subsection{Main Theorems}\n\nIn this section, we show our main results.\nSince the security quantities $\\varepsilon_n$ and \n$\\nu_n$ only depend on the marginal distributions of\n$(X,Y)$ and $(X,Z)$ respectively, it suffice to consider\n$(X,Y,Z)$ of the form\n\\begin{eqnarray*}\nY &=& B X + W_y, \\\\\nZ &=& E X + W_z,\n\\end{eqnarray*}\nwhere $B \\in \\mathbb{R}^{m_y \\times m_x}$,\n$E \\in \\mathbb{R}^{m_z \\times m_x}$, \n$W_y \\sim {\\cal N}(0, I_{m_y})$ and\n$W_z \\sim {\\cal N}(0, I_{m_z})$.\nIn the rest of this paper, we omit the subscript\nof the identity matrix if the dimension is obvious\nfrom the context. \n\nOne of the main results of this paper is \nthe following.\n\n\\begin{theorem}\n\\label{theorem:general-case}\nLet ${\\cal R}_G(X,Y,Z)$ be the set\nof all rate pairs $(R_p,R_k)$ satisfying\n\\begin{eqnarray*}\nR_p\n&\\ge& \\frac{1}{2}\\log \\left|\\frac{\\Sigma_x }{\\Sigma_{x|u} }\\right|\n - \\frac{1}{2}\\log \\left|\\frac{B \\Sigma_x B^T + I}{\n B \\Sigma_{x|u} B^T + I}\\right|, \\\\\nR_k\n&\\le& \\frac{1}{2}\\log \\left|\\frac{ B \\Sigma_x B^T + I}{\n B \\Sigma_{x|u} B^T + I}\\right|\n- \\frac{1}{2}\\log \\left|\\frac{ E \\Sigma_x E^T + I}{\n E \\Sigma_{x|u} E^T + I}\\right|\n\\end{eqnarray*}\nfor some $0 \\prec \\Sigma_{x|u} \\preceq \\Sigma_x$.\nThen, we have\n\\begin{eqnarray*}\n{\\cal R}(X,Y,Z) = {\\cal R}_G(X,Y,Z).\n\\end{eqnarray*}\n\\end{theorem}\n\nWe are also interested in the asymptotic behavior\nof the function\n\\begin{eqnarray}\nR_k(R_p) := \\sup\\{ R_k:~(R_p,R_k) \\in {\\cal R}(X,Y,Z) \\}.\n\\label{asymptotic-bahavior}\n\\end{eqnarray}\nFollowing the approach in \\cite{liu:09b}, we can obtain\na closed form expression of \n$\\lim_{R_p \\to \\infty}R_k(R_p)$ as follows.\nLet $\\phi_i,i=1,\\ldots,m_x$ be the generalized\neigenvalues \\cite[Chapter 6.3]{strang} of the matrices\n\\begin{eqnarray*}\n\\left(\n\\Sigma_x^{\\frac{1}{2}} B^T B \\Sigma_x^{\\frac{1}{2}} + I_{m_x},\n\\Sigma_x^{\\frac{1}{2}} E^T E \\Sigma_x^{\\frac{1}{2}} + I_{m_x}\n\\right).\n\\end{eqnarray*}\nWithout loss of generality, we may assume that\nthese generalized eigenvalues are ordered as\n\\begin{eqnarray}\n\\label{eq:order-of-eigenvalues}\n\\phi_1 \\ge \\cdots \\ge \\phi_\\rho > 1 \n\\ge \\phi_{\\rho+1} \\ge \\cdots \\ge \\phi_{m_x},\n\\end{eqnarray} \ni.e., a total of $\\rho$ of them are assumed to be greater than $1$.\nThen, we have\n\\begin{eqnarray}\n\\lim_{R_p \\to \\infty}R_k(R_p) \n&=& \\max_{0 \\preceq \\Sigma_{x|u} \\preceq \\Sigma_x}\n \\left[\n \\frac{1}{2}\\log \\left|\\frac{ B \\Sigma_x B^T + I}{\n B \\Sigma_{x|u} B^T + I}\\right| \\right. \\nonumber \\\\\n&& \\left. ~~~~~~~~~~~~~- \\frac{1}{2}\\log \\left|\\frac{ E \\Sigma_x E^T + I}{\n E \\Sigma_{x|u} E^T + I}\\right|\n \\right] \\nonumber \\\\\n&=& \\frac{1}{2}\\sum_{i =1}^\\rho \\log \\phi_i.\n\\label{eq:closed-form-upper}\n\\end{eqnarray}\nSince Eq.~(\\ref{eq:closed-form-upper}) can be proved\nalmost in the same manner as \\cite[Theorem 3]{liu:09b},\nwe omit a proof.\n\nWhen $m_x=m_y=m_z$ and both $B$ and $E$\nare invertible, it suffice to consider \nthe case in which\n\\begin{eqnarray}\n\\label{eq:aligned-y}\nY &=& X + W_y, \\\\\n\\label{eq:aligned-z}\nZ &=& X + W_z,\n\\end{eqnarray}\nwhere the covariance matrices\n$\\Sigma_{W_y}$ and $\\Sigma_{W_z}$ are\nnot necessarily identity but are invertible.\nFollowing \\cite{weingarten:06}, we call this\ncase the aligned case.\nAs is usual with the vector Gaussian \nproblems (e.g.~\\cite{weingarten:06}),\nthe general statement (Theorem \\ref{theorem:general-case})\nis shown by detouring the statement for\nthe aligned case.\n\n\\begin{theorem}\n\\label{theorem:aligned-case}\nLet \n${\\cal R}^*_G(X,Y,Z)$ be the set\nof all rate pairs $(R_p,R_k)$ satisfying\n\\begin{eqnarray*}\nR_p &\\ge&\nI_p(\\Sigma_{x|u}) \\\\\n&:=& \\frac{1}{2}\\log \\left|\\frac{\\Sigma_x }{\\Sigma_{x|u} }\\right|\n - \\frac{1}{2}\\log \\left|\\frac{\\Sigma_x + \\Sigma_{W_y}}{\n \\Sigma_{x|u} + \\Sigma_{W_y}}\\right|, \\\\\nR_k &\\le&\nI_k(\\Sigma_{x|u}) \\\\\n&:=& \\frac{1}{2}\\log \\left|\\frac{ \\Sigma_x + \\Sigma_{W_y}}{\n \\Sigma_{x|u} + \\Sigma_{W_y}}\\right|\n- \\frac{1}{2}\\log \\left|\\frac{ \\Sigma_x + \\Sigma_{W_z}}{\n \\Sigma_{x|u} + \\Sigma_{W_z}}\\right|\n\\end{eqnarray*}\nfor some $0 \\prec \\Sigma_{x|u} \\preceq \\Sigma_x$.\nThen, we have\n\\begin{eqnarray*}\n{\\cal R}(X,Y,Z) = {\\cal R}^*_G(X,Y,Z).\n\\end{eqnarray*}\n\\end{theorem}\n\nTheorem \\ref{theorem:aligned-case} is shown \nin Section \\ref{sec:aligned-case}\nand Theorem \\ref{theorem:general-case} \nis shown in Section \\ref{sec:proof-general}\nby using Theorem \\ref{theorem:aligned-case}.\n\n\\subsection{Numerical Examples}\n\nIn this section, we show some numerical\nexample to illustrate Theorem \\ref{theorem:general-case}.\nIn general,\ncalculation of ${\\cal R}_G(X,Y,Z)$ involves a\nnonconvex optimization problem and is not tractable.\nHowever for $m_x \\ge 2$ and $m_y=m_z=1$, \nfollowing the method in \\cite{weingarten:06b} \n(see also \\cite{ly:09}), we can transform the calculation of\n${\\cal R}_G(X,Y,Z)$ into tractable form.\n\nFor $m_x \\ge 2$ and $m_y=m_z=1$, we have\n\\begin{eqnarray*}\nI_p(\\Sigma_{x|u}) &=& \\frac{1}{2} \\log \\left|\n \\frac{\\Sigma_x}{\\Sigma_{x|u}} \\right|\n - \\frac{1}{2} \\log \\frac{b \\Sigma_x b^T +1}{b \\Sigma_{x|u} b^T + 1},\\\\\nI_k(\\Sigma_{x|u}) &=& \\frac{1}{2} \\log\n \\frac{b \\Sigma_x b^T + 1}{b \\Sigma_{x|u} b^T + 1}\n - \\frac{1}{2} \\log \\frac{e \\Sigma_x e^T + 1}{e \\Sigma_{x|u} e^T + 1},\n\\end{eqnarray*}\nwhere $b, e \\in \\mathbb{R}^{m_x}$.\nNoting the relation\n\\begin{eqnarray*}\n\\frac{e \\Sigma_{x|u} e^T + 1}{b \\Sigma_{x|u} b^T + 1}\n = 1 + \\frac{e \\Sigma_{x|u} e^T - b \\Sigma_{x|u} b^T + 1}{ b \\Sigma_{x|u} b^T + 1},\n\\end{eqnarray*}\nwe set\n\\begin{eqnarray*}\ns &=& b \\Sigma_{x|u} b^T, \\\\\nt &=& \\frac{e \\Sigma_{x|u} e^T - b \\Sigma_{x|u} b^T + 1}{ b \\Sigma_{x|u} b^T + 1}.\n\\end{eqnarray*}\nLet \n\\begin{eqnarray*}\nI_p(\\Sigma_{x|u},s) &:=& \\frac{1}{2} \\log \\left| \\frac{\\Sigma_x}{\\Sigma_{x|u}}\\right|\n - \\frac{1}{2} \\log (b \\Sigma_x b^T + 1) \\\\\n&&~~~~~~~~~~~~~~~~ + \\frac{1}{2} \\log(1+s), \\\\\nI_k(t) &=& \\frac{1}{2}\\log \\frac{b\\Sigma_x b^T + 1}{e \n \\Sigma_x e^T + 1} + \\frac{1}{2}\\log (1 + t).\n\\end{eqnarray*}\nThen we can easily find that\n\\begin{eqnarray*}\n\\lefteqn{\n\\hspace{-35mm} {\\cal R}_G(X,Y,Z) } \\\\\n= \\{ (R_p,R_k):~R_p &\\ge& I_p(\\Sigma_{x|u},s), \\\\\n R_k &\\le& I_k(t), \\\\\n0 \\prec \\Sigma_{x|u} &\\preceq& \\Sigma_x, \\\\\nt(b \\Sigma_{x|u}b^T + 1) &\\le& e\\Sigma_{x|u}e^T - b \\Sigma_{x|u} b^T ,\\\\\nb \\Sigma_{x|u} b^T &\\le& s \\}.\n\\end{eqnarray*}\nFor fixed $(s,t)$, the optimization problem\n\\begin{eqnarray*}\n\\mbox{minimize} && I_p(\\Sigma_{x|u},s) \\\\\n\\mbox{subject to} && t(b \\Sigma_{x|u} b^T +1) \n \\le e \\Sigma_{x|u} e^T - b \\Sigma_{x|u} b^T \\\\\n&& b \\Sigma_{x|u} b^T \\le s \\\\\n&& 0 \\prec \\Sigma_{x|u} \\preceq \\Sigma_x\n\\end{eqnarray*}\nis a convex problem.\nBy sweeping $(s,t)$, we can calculate\nthe region ${\\cal R}_G(X,Y,Z)$.\n\nFor \n\\begin{eqnarray}\n\\label{eq:example-1}\n\\Sigma_x = \\left[\n\\begin{array}{cc}\n2 & 0 \\\\ 0 & 2\n\\end{array} \\right],\n~ b = \\left[\n\\begin{array}{cc}\n1 & 0.5\n\\end{array} \\right],\n~ e = \\left[\n\\begin{array}{cc}\n0.7 & 0.35\n\\end{array} \\right],\n\\end{eqnarray}\nthe region ${\\cal R}_G(X,Y,Z)$ is plotted \nin Fig.~\\ref{Fig:degraded}.\nNote that this case is degraded in the \nsense of \\cite[Definition 1]{weingarten:09},\ni.e., $X \\leftrightarrow Y \\leftrightarrow Z$ by appropriately\nchoosing the correlation between $(Y,Z)$.\nIn this case, the function $R_k(R_p)$\nconverges to $I(X;Y) - I(X;Z)$ as \n$R_p$ increases.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig_degraded.eps}\n\\caption{''region`` is \n$R_k(R_p)$ defined in Eq.~(\\ref{asymptotic-bahavior}) for the sources given\nby Eq.~(\\ref{eq:example-1}).\n''upper bound'' is the quantity $I(X;Y) - I(X;Z)$.\n}\n\\label{Fig:degraded}\n\\end{figure}\n\nFor \n\\begin{eqnarray}\n\\label{eq:example-2}\n\\Sigma_x = \\left[\n\\begin{array}{cc}\n2 & 0 \\\\ 0 & 2\n\\end{array} \\right],\n~ b = \\left[\n\\begin{array}{cc}\n1 & 0.5\n\\end{array} \\right],\n~ e = \\left[\n\\begin{array}{cc}\n0.5 & 1\n\\end{array} \\right],\n\\end{eqnarray}\nthe region ${\\cal R}_G(X,Y,Z)$ is plotted \nin Fig.~\\ref{Fig:non_degraded}.\nNote that $(X,Y,Z)$ in this example is not degraded.\nAlthough $I(X;Y) - I(X;Z) = 0$ in this example, \nFig.~\\ref{Fig:non_degraded} clarifies that appropriate\nquantization enables Alice and Bob to share a secret\nkey at positive key rate. \n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{fig_non_degraded.eps}\n\\caption{''region`` is \n$R_k(R_p)$ defined in Eq.~(\\ref{asymptotic-bahavior}) for the sources given\nby Eq.~(\\ref{eq:example-2}). ``upper bound\nis $\\lim_{R_p \\to \\infty} R_k(R_p)$ which is\nexplicitly given by Eq.~(\\ref{eq:closed-form-upper}).\n}\n\\label{Fig:non_degraded}\n\\end{figure}\n\nFor non-degraded case, \n$R_k(R_p)$ converges to the quantity\ngiven by Eq.~(\\ref{eq:closed-form-upper}) instead of\n$I(X;Y) - I(X;Z)$ as $R_p$ increases,\nand it is also plotted in Fig.~\\ref{Fig:non_degraded}.\n\n\n\\section{Proof of Theorem \\ref{theorem:aligned-case}}\n\\label{sec:aligned-case}\n\n\\subsection{Direct Part}\n\\label{subsec:direct-aligned}\n\nIn \\cite{watanabe:09c}, the present authors proved\nthe following proposition, which is an extension\nof \\cite[Theorem 2.6]{csiszar:00} to continuous sources.\n\n\\begin{proposition}\n\\label{proposition:direct-part}\nFor an auxiliary random variable $U$ satisfying the Markov chain\n\\begin{eqnarray*}\nU \\leftrightarrow X \\leftrightarrow (Y,Z),\n\\end{eqnarray*}\nlet $(R_p,R_k)$ be a rate pair such that\n\\begin{eqnarray*}\nR_p &\\ge& I(U;X) - I(U;Y), \\\\\nR_k &\\le& I(U;Y) - I(U;Z).\n\\end{eqnarray*}\nThen, we have $(R_p,R_k) \\in {\\cal R}(X,Y,Z)$.\n\\end{proposition}\n\nThe direct part of Theorem \\ref{theorem:aligned-case} is shown by\ntaking Gaussian auxiliary random variable $U$ \nsuch that the conditional covariance matrix\nof $X$ given $U$ is $\\Sigma_{x|u}$\nin Proposition \\ref{proposition:direct-part}.\n\\hfill$\\square$\n\n\\subsection{Converse Part}\n\nIn the converse proof, we will use the following\nProposition and Corollary. \nThe proposition was shown for discrete\nsources in \\cite[Theorem 2.6]{csiszar:00},\nand it can be shown almost in the same manner\nfor continuous sources.\n\\begin{proposition}\n(\\cite{csiszar:00}) \n\\label{prop:cn-bound}\nSuppose that a \nrate pair $(R_p,R_k)$ is included in\n${\\cal R}(X,Y,Z)$. Then, \nthere exist\nauxiliary random variables $U$ and $V$ satisfying\n\\begin{eqnarray}\n\\label{public-lower}\nR_p &\\ge& I(U;X|Y), \\\\\nR_k &\\le& I(U;Y|V) - I(U;Z|V),\n\\end{eqnarray}\nand the Markov chain\n\\begin{eqnarray}\n\\label{markov-condition}\nV \\leftrightarrow U \\leftrightarrow X \\leftrightarrow (Y,Z).\n\\end{eqnarray}\n\\end{proposition}\n\nFor degraded sources, we can simplify \nthe above proposition\n(see \\cite[Appendix B]{watanabe:09c} for a proof).\n\\begin{corollary}\n\\label{coro:cn-bound}\nSuppose that $(X,Y,Z)$ is degraded, i.e., $X \\leftrightarrow Y \\leftrightarrow Z$.\nIf $(R_p,R_k) \\in {\\cal R}(X,Y,Z)$, then\nthere exists an auxiliary random variable $U$ satisfying\n\\begin{eqnarray}\n\\label{eq:coro-public-upper}\nR_p &\\ge& I(U;X|Y)\n= I(U;X) - I(U;Y), \\\\\n\\label{eq:coro-key-upper}\nR_k &\\le& I(U;Y|Z)\n = I(U;Y) - I(U;Z),\n\\end{eqnarray}\nand the Markov chain\n\\begin{eqnarray}\n\\label{eq:markov-2}\nU \\leftrightarrow X \\leftrightarrow Y \\leftrightarrow Z.\n\\end{eqnarray}\n\\end{corollary}\n\nWe show a converse proof of\nTheorem \\ref{theorem:aligned-case}\nby contradiction.\nSuppose that there exists a rate pair such that\n$(R_p^o,R_k^o) \\in {\\cal R}(X,Y,Z)$\nand $(R_p^o,R_k^o) \\notin {\\cal R}_G^*(X,Y,Z)$,\nwhere we assume $R_k^o > 0$ to avoid\nthe trivial case.\nThen, there exists $0 \\prec \\Sigma_{x|u}^o \\preceq \\Sigma_x$\nsuch that $I_p(\\Sigma_{x|u}^o) \\le R_p^o$.\nTherefore, we can write \n\\begin{eqnarray}\n\\label{eq:key-rate-equality}\nR_k^o = R_k^* + \\delta\n\\end{eqnarray} \nfor some $\\delta > 0$, where $R_k^*$ is given by\nthe optimal value of\n\\begin{eqnarray}\n\\mbox{maximize} && I_k(\\Sigma_{x|u}) \\nonumber\\\\\n\\mbox{subject to} && I_p(\\Sigma_{x|u}) \\le R_p^o, \n \\label{eq:opt1} \\\\\n && 0 \\prec \\Sigma_{x|u} \\preceq \\Sigma_x.\n \\nonumber\n\\end{eqnarray}\nAn optimal solution $\\Sigma_{x|u}^*$ of\nthis optimization problem satisfies\nthe Karash-Kuhn-Tucker (KKT) \ncondition (see Appendix \\ref{appendix:kkt} for the\nderivation)\n\\begin{eqnarray}\n\\mu (\\Sigma_{x|u}^*)^{-1}\n &+& (\\Sigma_{x|u}^* + \\Sigma_{W_z})^{-1} \\nonumber \\\\\n &=& \n(1+\\mu) (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1} + M, \\nonumber \\\\\n \\label{eq:kkt-1} \\\\\n\\label{eq:kkt-2}\nM(\\Sigma_x - \\Sigma_{x|u}^*) &=& 0, \\\\\n\\label{eq:kkt-3}\n\\mu(R_p^o - I_p(\\Sigma_{x|u}^*)) &=& 0,\n\\end{eqnarray}\nwhere $\\mu \\ge 0$ and $M \\succeq 0$.\nFrom Eqs.~(\\ref{eq:key-rate-equality}) \nand (\\ref{eq:kkt-3}), we have\n\\begin{eqnarray}\n\\label{eq:contradiction}\nR_k^o - \\mu R_p^o =\n I_k(\\Sigma_{x|u}^*) - \\mu I_p(\\Sigma_{x|u}^*)\n + \\delta.\n\\end{eqnarray}\nWe shall find a contradiction to Eq.~(\\ref{eq:contradiction})\nby showing that for any $(R_p,R_k) \\in {\\cal R}(X,Y,Z)$\n\\begin{eqnarray}\n\\label{eq:goal}\nR_k - \\mu R_p \\le I_k(\\Sigma_{x|u}^*)\n - \\mu I_p(\\Sigma_{x|u}^*).\n\\end{eqnarray}\n\nThe proof of Eq.~(\\ref{eq:goal}) roughly consists\nof three steps: In the first step,\nwe reduce the proof for the non-degraded sources\nto that for the degraded sources by using the\nenhancement technique introduced by\nWeingarten {\\em et.~al.} \\cite{weingarten:06}.\nIn the second step, we change the variable so that\nwe can use the entropy power inequality (EPI).\nIn the last step, we derive an upper bound\non $R_k - \\mu R_p$ by using the EPI,\nwhich turn out to be tight.\n\n\\noindent{\\em Step 1:}\nIn this step, in order to reduce the \nproof for the non-degraded sources to\nthat for the degraded sources, \nwe introduce the covariance matrix\n$\\Sigma_{\\tilde{W}_y}$ satisfying\n\\begin{eqnarray}\n\\lefteqn{ (1+\\mu) (\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1} } \\nonumber \\\\\n\\label{eq:enhancement}\n &=& (1+\\mu) (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1} + M.\n\\end{eqnarray}\nThen, we have (see Appendix \\ref{appendix:enhanced-1-2} for a proof)\n\\begin{eqnarray}\n\\label{eq:enhanced-1}\n0 \\prec \\Sigma_{\\tilde{W}_y} \\preceq \\Sigma_{W_y},&&\\\\\n\\label{eq:enhanced-2}\n\\Sigma_{\\tilde{W}_y} \\preceq \\Sigma_{W_z}. &&\n\\end{eqnarray}\nLet $\\tilde{W}_y$ be the Gaussian random vector whose \ncovariance matrix is $\\Sigma_{\\tilde{W}_y}$, and let \n\\begin{eqnarray}\n\\tilde{Y} = X + \\tilde{W}_y.\n\\label{eq:enhanced-y}\n\\end{eqnarray}\nFrom Eq.~(\\ref{eq:enhanced-2}), we can find that\nthe sources $(X,\\tilde{Y},Z)$\nsatisfy $X \\leftrightarrow \\tilde{Y} \\leftrightarrow Z$.\nFurthermore, from Eq.~(\\ref{eq:enhanced-1}), we can also \nfind that $X \\leftrightarrow \\tilde{Y} \\leftrightarrow Y$, which implies\n\\begin{eqnarray*}\n{\\cal R}(X,Y,Z) \\subset {\\cal R}(X,\\tilde{Y},Z).\n\\end{eqnarray*}\nThus, it suffice to show that \nEq.~(\\ref{eq:goal}) holds for any\n$(R_p,R_k) \\in {\\cal R}(X,\\tilde{Y},Z)$.\nIn steps 2 and 3, we will show that\n\\begin{eqnarray}\n\\label{eq:intermidiate}\nR_k - \\mu R_p \\le \n\\tilde{I}_k(\\Sigma_{x|u}^*) \n - \\mu \\tilde{I}_p(\\Sigma_{x|u}^*)\n\\end{eqnarray}\nfor any $(R_p,R_k) \\in {\\cal R}(X,\\tilde{Y},Z)$,\nwhere \n\\begin{eqnarray*}\n\\tilde{I}_p(\\Sigma_{x|u})\n&:=& \\frac{1}{2}\\log \\left|\\frac{\\Sigma_x}{\\Sigma_{x|u}}\\right|\n - \\frac{1}{2}\\log \\left|\\frac{\\Sigma_x + \\Sigma_{\\tilde{W}_y}}{\n \\Sigma_{x|u} + \\Sigma_{\\tilde{W}_y}}\\right|, \\\\\n\\tilde{I}_k(\\Sigma_{x|u})\n&:=& \\frac{1}{2}\\log \\left|\\frac{\\Sigma_x + \\Sigma_{\\tilde{W}_y}}{\n \\Sigma_{x|u} + \\Sigma_{\\tilde{W}_y}}\\right|\n- \\frac{1}{2}\\log \\left|\\frac{\\Sigma_x + \\Sigma_{W_z}}{\n \\Sigma_{x|u} + \\Sigma_{W_z}}\\right|.\n\\end{eqnarray*}\nThen, by using the relation\n(see Appendix \\ref{appendix:preservation} for a proof)\n\\begin{eqnarray}\n\\lefteqn{ (\\Sigma_x + \\Sigma_{\\tilde{W}_y})\n (\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1} } \\nonumber \\\\\n\\label{eq:reduction}\n&=& (\\Sigma_x + \\Sigma_{W_y})\n (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1},\n\\end{eqnarray}\nwe have $I_k(\\Sigma_{x|u}^*) = \\tilde{I}_k(\\Sigma_{x|u}^*)$\nand $I_p(\\Sigma_{x|u}^*) = I_p(\\Sigma_{x|u}^*)$. Thus, \nEq.~(\\ref{eq:intermidiate}) implies \nthat Eq.~(\\ref{eq:goal}) holds for any\n$(R_p,R_k) \\in {\\cal R}(X,\\tilde{Y},Z)$.\n\n\\noindent{\\em Step 2:}\nFirst, we show Eq.~(\\ref{eq:intermidiate})\nfor $\\mu = 0$. In this case, from\nEqs.~(\\ref{eq:kkt-1}) \nand (\\ref{eq:enhancement}), we have\n$\\Sigma_{\\tilde{W}_y} = \\Sigma_{W_z}$.\nThus, from Corollary \\ref{coro:cn-bound}, we have\n\\begin{eqnarray*}\nR_k - 0 \\cdot R_p \\le\n I(U;Y) - I(U;Z) = 0\n = \\tilde{I}_k(\\Sigma_{x|u}^*).\n\\end{eqnarray*}\nThus, we have the assertion.\n\nIn order to prove Eq.~(\\ref{eq:intermidiate})\nfor $\\mu > 0$,\nwe change the variable as follows.\nSince $(X,\\tilde{Y},Z)$ is jointly Gaussian, we can write\n\\begin{eqnarray*}\nX &=& K_{xz} Z + N_1,\\\\\n\\tilde{Y} &=& K_{\\tilde{y}x} X + \n K_{\\tilde{y}z} Z + N_2 \n\\end{eqnarray*}\nfor Gaussian random vectors $N_1, N_2$ with covariance\nmatrices\n\\begin{eqnarray*}\n\\Sigma_{N_1} &=& \\Sigma_{x|z} \n := \\Sigma_x - K_{xz} \\Sigma_{zx}, \\\\\n\\Sigma_{N_2} &=& \\Sigma_{\\tilde{y}|xz}\n := \\Sigma_{\\tilde{y}} - K_{\\tilde{y}x} \\Sigma_{x\\tilde{y}}\n - K_{\\tilde{y}z} \\Sigma_{z\\tilde{y}},\n\\end{eqnarray*}\nwhere the coefficients are given by\n\\begin{eqnarray*}\nK_{xz} = \\Sigma_{xz} \\Sigma_z^{-1}\n\\end{eqnarray*}\nand\n\\begin{eqnarray}\n\\left[ \\begin{array}{cc}\nK_{\\tilde{y}z} & K_{\\tilde{y}x}\n\\end{array} \\right]\n = \\left[ \\begin{array}{cc}\n\\Sigma_{\\tilde{y}z} & \\Sigma_{\\tilde{y}x}\n\\end{array} \\right]\n\\left[ \\begin{array}{cc}\n\\Sigma_z & \\Sigma_{zx} \\\\\n\\Sigma_{xz} & \\Sigma_x \n\\end{array} \\right]^{-1}.\n\\label{eq:coefficients}\n\\end{eqnarray}\nBy noting the relations\n\\begin{eqnarray*}\nI(U;X|\\tilde{Y}) &=& I(U;X) - I(U;\\tilde{Y}),\\\\\nI(U;\\tilde{Y}|Z) &=& I(U;\\tilde{Y}) - I(U;Z),\\\\\nI(U;X|Z) &=& I(U;X) - I(U;Z), \\\\\nI(U;X|\\tilde{Y}) &=& I(U;X|Z) - I(U;\\tilde{Y}|Z)\n\\end{eqnarray*}\nfor random variables satisfying\n$U \\leftrightarrow X \\leftrightarrow \\tilde{Y} \\leftrightarrow Z$, we have\n\\begin{eqnarray}\n\\lefteqn{ \\tilde{I}_k(\\Sigma_{x|u}) - \n \\mu \\tilde{I}_p(\\Sigma_{x|u}) } \\nonumber \\\\\n&=& I(U; \\tilde{Y}|Z) - \\mu I(U;X|\\tilde{Y}) \\nonumber \\\\\n&=& (1+\\mu) I(U;\\tilde{Y}|Z) - \\mu I(U;X|Z) \\nonumber \\\\\n&=& [(1+\\mu) h(\\tilde{Y}|Z) - \\mu h(X|Z)] \\nonumber \\\\\n&& + [\\mu h(X|U,Z) - (1+\\mu) h(\\tilde{Y}|U,Z)] \\nonumber \\\\\n&=& [(1+\\mu) h(\\tilde{Y}|Z) - \\mu h(X|Z)]\n - \\frac{1+\\mu}{2}\\log|K_{\\tilde{y}x} K_{\\tilde{y}x}^T| \\nonumber \\\\\n&& + \\mu[ h(X|U,Z) - \\gamma h(X + K_{\\tilde{y}x}^{-1} N_2|U,Z)] \\nonumber\\\\\n&=& [(1+\\mu) h(\\tilde{Y}|Z) - \\mu h(X|Z)]\n - \\frac{1+\\mu}{2}\\log|K_{\\tilde{y}x} K_{\\tilde{y}x}^T| \\nonumber \\\\\n&& + \\mu[ h(X|U,Z) - \\gamma h(X + N_3|U,Z)] \\nonumber \\\\\n&=& [(1+\\mu) h(\\tilde{Y}|Z) - \\mu h(X|Z)]\n - \\frac{1+\\mu}{2}\\log|K_{\\tilde{y}x} K_{\\tilde{y}x}^T| \\nonumber \\\\\n&+& \\hspace{-3mm} \\mu \\left[\n \\frac{1}{2} \\log (2 \\pi e)^m |\\Sigma_{x|uz}|\n - \\frac{\\gamma}{2} \n \\log (2 \\pi e)^m |\\Sigma_{x|uz} + \\Sigma_{N_3}| \\right],\\nonumber \\\\\n\\label{eq:variable-changed}\n\\end{eqnarray}\nwhere we set $\\gamma := \\frac{1+\\mu}{\\mu} > 1$ and\n$N_3 := K_{\\tilde{y}x}^{-1} N_2$.\nIt should be noted that\n\\begin{eqnarray}\n\\label{eq:invertibility}\n|K_{\\tilde{y}x}| \\neq 0\n\\end{eqnarray}\nfor $\\mu >0$,\nwhich will be proved in \nAppendix \\ref{appendix:invertibility}.\n\nFor the change of variable \n\\begin{eqnarray*}\n\\phi: \\Sigma_{x|u} \\mapsto \\Sigma_{x|uz}\n = (\\Sigma_{x|u}^{-1} + \\Sigma_{W_z}^{-1})^{-1}, \n\\end{eqnarray*}\nlet $\\Sigma_{x|uz}^* := \\phi(\\Sigma_{x|u}^*)$.\nFrom Eqs.~(\\ref{eq:kkt-1}) and (\\ref{eq:enhancement})\nand the relation \n\\begin{eqnarray*}\n\\lefteqn{ \\tilde{I}_k(\\Sigma_{x|u}) - \\mu \\tilde{I}_p(\\Sigma_{x|u}) } \\nonumber \\\\\n&=& \\frac{\\mu}{2}\\log (2 \\pi e)^m |\\Sigma_{x|u}| \n + \\frac{1}{2} \\log (2 \\pi e)^m |\\Sigma_{x|u} + \\Sigma_{W_z}| \\nonumber \\\\\n &&~~~~~~ - \\frac{(1+\\mu)}{2} \\log (2 \\pi e)^m |\\Sigma_{x|u} + \\Sigma_{\\tilde{W}_y}| \n\\nonumber\\\\\n&& + [(1+\\mu)h(\\tilde{Y}) - h(Z) - \\mu h(X)],\n\\end{eqnarray*}\nwe have\n\\begin{eqnarray*}\n\\nabla_{\\Sigma_{x|u}}\\left[\n\\tilde{I}_k(\\Sigma_{x|u}^*) - \\mu \\tilde{I}_p(\\Sigma_{x|u}^*) \\right] = 0.\n\\end{eqnarray*}\nBy the chain rule for the derivative, we have\n\\begin{eqnarray*}\n\\lefteqn{\n \\nabla_{\\Sigma_{x|uz}} \\left[\n \\tilde{I}_k(\\phi^{-1}(\\Sigma_{x|uz}^*)) - \\mu\n \\tilde{I}_p(\\phi^{-1}(\\Sigma_{x|uz}^*)) \\right] } \\\\\n&=& \\nabla_{\\Sigma_{x|uz}} \\phi^{-1}(\\Sigma_{x|uz}^*) \\cdot\n \\nabla_{\\Sigma_{x|u}} \\left[\n\\tilde{I}_k(\\Sigma_{x|u}^*) - \n\\mu \\tilde{I}_p(\\Sigma_{x|u}^*) \\right] \\\\\n&=& 0.\n\\end{eqnarray*}\nThus, from Eq.~(\\ref{eq:variable-changed}), we have\n\\begin{eqnarray}\n\\label{eq:kkt-4}\n(\\Sigma_{x|uz}^*)^{-1} = \\gamma\n (\\Sigma_{x|uz}^* + \\Sigma_{N_3})^{-1}.\n\\end{eqnarray}\n\n\\noindent{\\em Step 3:}\nBy noting that $(X,\\tilde{Y},Z)$ is degraded,\nfrom Corollary \\ref{coro:cn-bound}, \nfor any $(R_p,R_k) \\in {\\cal R}(X,\\tilde{Y},Z)$ we have\n\\begin{eqnarray}\n\\lefteqn{\n R_k - \\mu R_p } \\nonumber \\\\\n&\\le& I(U;\\tilde{Y}|Z) - \\mu I(U;X|\\tilde{Y}) \\nonumber \\\\\n&=& [(1+\\mu) h(\\tilde{Y}|Z) - \\mu h(X|Z)]\n - \\frac{1+\\mu}{2}\\log|K_{\\tilde{y}x} K_{\\tilde{y}x}^T| \\nonumber \\\\\n&& + \\mu \\left[ h(X|U,Z) - \\gamma h(X + N_3|U,Z) \\right],\n\\label{eq:upper-1}\n\\end{eqnarray}\nwhere $U$ is not necessarily Gaussian.\nBy using the conditional version of EPI \\cite{bergmans:74},\nwe have\n\\begin{eqnarray}\n\\lefteqn{\n h(X|U,Z) - \\gamma h(X+ N_3|U,Z) } \n\\label{eq:standard-extremal} \\\\\n&\\le& h(X|U,Z) \\nonumber \\\\\n&& -\\frac{\\gamma m}{2} \\log\\left(\n \\exp\\left[ \\frac{2}{m} h(X|U,Z) \\right]\n + \\exp\\left[ \\frac{2}{m} h(N_3) \\right] \\right) \\nonumber \\\\\n&\\le& f\\left( h(N_3) - \\frac{m}{2} \\log(\\gamma -1); h(N_3) \\right),\n\\label{eq:epi-bound}\n\\end{eqnarray}\nwhere we set\n\\begin{eqnarray*}\nf(t;a) := t - \\frac{\\gamma m}{2} \\log \\left(\n \\exp\\left[ \\frac{2}{m} t \\right] + \\exp\\left[ \\frac{2}{m} a \\right] \\right).\n\\end{eqnarray*}\nNote that the function $f(t;a)$ is concave function of $t$ and\ntakes the maximum at $t = a - \\frac{m}{2}\\log(\\gamma - 1)$ \\cite{liu:07}.\nFrom Eq.~(\\ref{eq:kkt-4}), we have\n\\begin{eqnarray*}\n(\\gamma - 1)^{-1} \\Sigma_{N_3}\n = \\Sigma_{x|uz}^*,\n\\end{eqnarray*} \nwhich implies\n\\begin{eqnarray*}\nh(N_3) - \\frac{m}{2}\\log(\\gamma-1)\n&=& \\frac{1}{2}\\log (2 \\pi e)^m (\\gamma -1)^{-m}|\\Sigma_{N_3}| \\\\\n&=& \\frac{1}{2} \\log(2 \\pi e)^m |\\Sigma_{x|uz}^*|.\n\\end{eqnarray*}\nFurthermore, since $\\Sigma_{x|uz}^*$ and \n$\\Sigma_{N_3}$ are proportional to each other,\nwe have\n\\begin{eqnarray*}\n|\\Sigma_{x|uz}^*|^{1\/m} + |\\Sigma_{N_3}|^{1\/m}\n = |\\Sigma_{x|uz}^* + \\Sigma_{N_3}|^{1\/m}.\n\\end{eqnarray*}\nThus, from Eqs.~(\\ref{eq:upper-1}) and (\\ref{eq:epi-bound}), we have\n\\begin{eqnarray*}\n\\lefteqn{\n R_k - \\mu R_p } \\\\\n&\\le& [(1+\\mu) h(\\tilde{Y}|Z) - \\mu h(X|Z)]\n - \\frac{1+\\mu}{2}\\log|K_{\\tilde{y}x} K_{\\tilde{y}x}^T| \\\\\n&& +\\mu \\left[\n \\frac{1}{2} \\log(2\\pi e)^m |\\Sigma_{x|uz}^*|\n - \\frac{\\gamma}{2} \\log (2 \\pi e)^m |\\Sigma_{x|uz}^* + \\Sigma_{N_3} | \\right] \\\\\n&=& \\tilde{I}_k(\\Sigma_{x|u}^*) - \\mu \\tilde{I}_p(\\Sigma_{x|u}^*).\n\\end{eqnarray*}\n\\hfill$\\square$\n\n\\begin{remark}\n\\label{remark:extremal}\nOne of the difficulties in the above proof\nis that, after Step 1,\n we have to show the extremal\ninequality of the form\n\\begin{eqnarray}\n&& \\hspace{-10mm} \\mu h(X|U) + h(X + W_z|U)\n- (1+\\mu) h(X + \\tilde{W}_y|U) \\nonumber \\\\\n&\\le& \\frac{\\mu}{2} \\log |\\Sigma_{x|u}^*|\n + \\frac{1}{2} \\log |\\Sigma_{x|u}^* + \\Sigma_{W_z} | \\nonumber \\\\\n&& ~~~~~- \\frac{(1+\\mu)}{2} \\log |\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y} |.\n\\label{eq:key-extremal-inequality}\n\\end{eqnarray}\nThis type of extremal inequality has appeared in\n\\cite[Corollary 2]{liu:10b} (scalar version has appeared in \\cite[Lemma 1]{chen:09}).\nIn \\cite{liu:10}, the extremal inequality was proved by using\n a vector generalization of Costa's entropy power \ninequality \\cite{costa:85}. On the otherhand,\nwe showed Eq.~(\\ref{eq:key-extremal-inequality}) by using\nthe change of variable in Step 2 and by reducing\nto more tractable\nform (Eq.~(\\ref{eq:standard-extremal})), \nwhich has appeared in the literature \\cite{liu:07}.\nBy this reduction, we only need the standard EPI in our proof\ninstead of Costa's type EPI, and our proof seems more\nelementary.\n\\end{remark} \n\n\\section{Proof of Theorem \\ref{theorem:general-case}}\n\\label{sec:proof-general}\n\nIn this section, we show Theorem \\ref{theorem:general-case}\nby using Theorem \\ref{theorem:aligned-case}.\nWe follow a similar approach as in \\cite[Section 4]{ly:09}.\nSince the direct part can be proved by taking\na Gaussian auxiliary random variable $U$ in \nProposition \\ref{proposition:direct-part} \n(see Section \\ref{subsec:direct-aligned}),\nwe concentrate on the converse part.\nWithout loss of generality, we can assume that\nthe matrices $B$ and $E$ are square (but not necessarily invertible).\nIf that is not the case, we can apply singular value decomposition (SVD)\nto show equivalent sources $(X^\\prime,Y^\\prime, Z^\\prime)$ on\n$\\mathbb{R}^{m_x} \\times \\mathbb{R}^{m_x} \\times \\mathbb{R}^{m_x}$ such\nthat ${\\cal R}(X^\\prime,Y^\\prime,Z^\\prime) = {\\cal R}(X,Y,Z)$\nin a similar manner as \\cite[Section 5-B]{weingarten:06}.\n\nBy using SVD, we can write the matrices as \n\\begin{eqnarray*}\nB &=& U_y \\Lambda_y V_y, \\\\\nE &=& U_z \\Lambda_z V_z,\n\\end{eqnarray*}\nwhere $U_y,V_y, U_z$ and $V_z$ are $m_x \\times m_x$ orthogonal matrices,\nand $\\Lambda_y$ and $\\Lambda_z$ are diagonal matrices. Let \n\\begin{eqnarray*}\n\\bar{B} &=& U_y (\\Lambda_y + \\alpha I) V_y, \\\\\n\\bar{E} &=& U_y (\\Lambda_z + \\alpha I) V_z\n\\end{eqnarray*}\nfor some $\\alpha > 0$. Then, let\n\\begin{eqnarray*}\n\\bar{Y} &=& \\bar{B} X + W_y, \\\\\n\\bar{Z} &=& \\bar{E} X + W_z.\n\\end{eqnarray*}\nSince $\\bar{B}$ and $\\bar{E}$ are invertible, \nTheorem \\ref{theorem:aligned-case} implies\n\\begin{eqnarray}\n\\label{eq:pertarbated-equality}\n{\\cal R}(X, \\bar{Y}, \\bar{Z}) = {\\cal R}_G(X,\\bar{Y},\\bar{Z}).\n\\end{eqnarray}\nIn the following, we will show the following lemma.\n\n\\begin{lemma}\n\\label{lemma:pertarbated}\nWe have\n\\begin{eqnarray*}\n{\\cal R}(X,Y,Z) \\subset {\\cal R}(X,\\bar{Y},\\bar{Z}) \n + {\\cal O}(X, \\bar{Y}, \\bar{Z}),\n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n{\\cal O}(X,\\bar{Y},\\bar{Z})\n &=& \\left\\{ (0, R_k) :\n 0 \\le R_k \\le \\frac{1}{2} \\log |\\bar{E} \\Sigma_x \\bar{E}^T + I| \\right. \n \\\\\n && \\left. ~~- \\frac{1}{2} \\log |E \\Sigma_x E^T + I| \\right\\}.\n\\end{eqnarray*}\n\\end{lemma}\n\nBy letting $\\alpha \\to 0$, ${\\cal R}_G(X, \\bar{X}, \\bar{Z})$\nconverges to ${\\cal R}_G(X,Y,Z)$ and ${\\cal O}(X, \\bar{Y},\\bar{Z})$\nconverges to $\\{(0,0) \\}$.\nThus, Eq.~(\\ref{eq:pertarbated-equality}) and Lemma \\ref{lemma:pertarbated} \nimply ${\\cal R}(X,Y,Z) \\subset {\\cal R}_G(X,Y,Z)$.\n\n\n\\noindent{\\em Proof of Lemma \\ref{lemma:pertarbated}}\n\nLet \n\\begin{eqnarray*}\nC_y &=& U_y \\Lambda_y (\\Lambda_y + \\alpha I)^{-1} V_y, \\\\\nC_z &=& U_z \\Lambda_z (\\Lambda_z + \\alpha I)^{-1} V_z.\n\\end{eqnarray*}\nThen, we have $C_y C_y^T \\prec I$ and $C_z C_z^T \\prec I$.\nThus, we can write \n\\begin{eqnarray*}\nY &=& C_y \\bar{Y} + W_y^\\prime, \\\\\nZ &=& C_z \\bar{Z} + W_z^\\prime\n\\end{eqnarray*}\nfor $W_y^\\prime \\sim {\\cal N}(0, I - C_y C_y^T)$\nand $W_z^\\prime \\sim {\\cal N}(0, I- C_z C_z^T)$, i.e., we have\n\\begin{eqnarray}\n\\label{eq:markov-ralations-1}\n&& X \\leftrightarrow \\bar{Y} \\leftrightarrow Y, \\\\\n\\label{eq:markov-ralations-2}\n&& X \\leftrightarrow \\bar{Z} \\leftrightarrow Z. \n\\end{eqnarray}\n\nFrom Proposition \\ref{prop:cn-bound}, for any $(R_p, R_k) \\in {\\cal R}(X,Y,Z)$,\nthere exist $(U,V)$ satisfying \n\\begin{eqnarray*}\nR_p &\\ge& I(U;X) - I(U; Y), \\\\\nR_k &\\le& I(U;Y|V) - I(U; Z|V),\n\\end{eqnarray*}\nand $(U,V) \\leftrightarrow X \\leftrightarrow (Y,Z)$.\nLet \n\\begin{eqnarray*}\n\\bar{R}_p &=& I(U; X) - I(U; \\bar{Y}), \\\\\n\\bar{R}_k &=& I(U; \\bar{Y}|V) - I(U; \\bar{Z}|V).\n\\end{eqnarray*}\nThen, we have\n\\begin{eqnarray*}\nR_p - \\bar{R}_p &\\ge&\n I(U; X) - I(U;Y) - [ I(U; X) - I(U; \\bar{Y}) ] \\\\\n &=& I(U; \\bar{Y}) - I(U; Y) \\\\\n &\\ge & 0,\n\\end{eqnarray*}\nwhere the second inequality follows from\nEq.~(\\ref{eq:markov-ralations-1}).\nOn the other hand, we have\n\\begin{eqnarray*}\nR_k - \\bar{R}_k &\\le&\n I(U; Y|V) - I(U;Z|V) \\\\\n && - [ I(U; \\bar{Y}|V) - I(U; \\bar{Z}|V) ] \\\\\n&=& I(U; \\bar{Z}|V) - I(U;Z|V) \\\\\n && - [ I(U;\\bar{Y}|V) - I(U;Y|V) ] \\\\\n&\\le& I(U; \\bar{Z}|V) - I(U; Z|V) \\\\\n&=& I(U,V; \\bar{Z}) - I(U, V; Z) \\\\\n && - [ I(V; \\bar{Z}) - I(V; Z) ] \\\\\n&\\le& I(U,V; \\bar{Z}) - I(U,V; Z) \\\\\n&=& I(X; \\bar{Z}) - I(X; Z) \\\\\n && - [ I(X; \\bar{Z}|U,V) - I(X; Z|U,V) ] \\\\\n&\\le& I(X; \\bar{Z}) - I(X; Z) \\\\\n&=& \\frac{1}{2} \\log |\\bar{E} \\Sigma_x \\bar{E}^T + I|\n - \\frac{1}{2} \\log |E \\Sigma_x E^T + I|,\n\\end{eqnarray*}\nwhere the second, third, and forth inequalities follow\nfrom the Markov relations in\nEqs.~(\\ref{eq:markov-ralations-1}) and (\\ref{eq:markov-ralations-2}).\n\\hfill$\\square$\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we investigated the secret key\nagreement from vector Gaussian sources by rate\nlimited public communication. We characterized\nthe optimal trade-off between the key rate\nand the public communication rate as a\n(covariance) matrix optimization problem.\nInvestigating an efficient method to solve\nthe optimization problem is a future research agenda.\n\n\\appendices\n\\section{Derivation of the KKT condition}\n\\label{appendix:kkt}\n\nWe first rewrite the optimization problem\nin Eq.~(\\ref{eq:opt1}) as a standard form\n\\begin{eqnarray}\n\\mbox{minimize} && - I_k(\\Sigma_{x|u}) \\nonumber \\\\\n\\mbox{subject to} && I_p(\\Sigma_{x|u}) - R_p^o \\le 0 \n \\label{eq:opt2} \\\\\n&& 0 \\prec \\Sigma_{x|u} \\preceq \\Sigma_x.\n\\nonumber\n\\end{eqnarray}\nLet $\\Sigma_{x|u}^*$ be an optimal solution\nfor this problem, which is also an optimal solution\nof Eq.~(\\ref{eq:opt1}).\nThen, we have $\\Sigma_{x|u}^* \\succ 0$ because\nof the constraint $I_p(\\Sigma_{x|u}^*) - R_p^o \\le 0$.\nThus, there exists a positive definite\nmatrix $L$ satisfying $L \\prec \\Sigma_{x|u}^*$.\n\nLet us consider another optimization problem\n\\begin{eqnarray}\n\\mbox{minimize} && - I_k(\\Sigma_{x|u}) \\nonumber \\\\\n\\mbox{subject to} && I_p(\\Sigma_{x|u}) - R_p^o \\le 0 \n \\label{eq:opt3} \\\\\n&& L \\preceq \\Sigma_{x|u} \\preceq \\Sigma_x.\n\\nonumber\n\\end{eqnarray}\nObviously, $\\Sigma_{x|u}^*$ is also an optimal solution\nfor the problem in Eq.~(\\ref{eq:opt3}), and\nthe optimal values for Eqs.~(\\ref{eq:opt2})\nand (\\ref{eq:opt3}) are the same.\nAlthough the optimization problem in\nEq.~(\\ref{eq:opt3}) is not convex, there exist\nLagrange multipliers $M_1 \\succeq 0$,\n$M_2 \\succeq 0$, and $\\mu \\ge 0$ satisfying\n\\begin{eqnarray}\n&& \\hspace{-40mm} - \\left(\n- \\nabla_{\\Sigma_{x|u}} I_k(\\Sigma_{x|u}^*)\n + \\mu \\nabla_{\\Sigma_{x|u}}(I_p(\\Sigma_{x|u}^*) - R_p^o)\n\\right) \\nonumber \\\\\n&=& M_2 - M_1, \\\\\nM_1(\\Sigma_{x|u}^* - L) &=& 0, \\label{kkt:app-2} \\\\\nM_2(\\Sigma_x - \\Sigma_{x|u}^*)) &=& 0, \\\\\n\\mu (R_p - I_p(\\Sigma_{x|u}^*)) &=& 0\n\\end{eqnarray}\nif the set of constraint qualifications\n(CQs) shown below are satisfied\n(see \\cite[Appendix 4]{weingarten:06} for\nthe detail).\nSince $\\Sigma_{x|u}^* \\succ L$, \nEq.~(\\ref{kkt:app-2}) implies $M_1 = 0$.\nThus, by noting the relation\n\\begin{eqnarray}\n\\lefteqn{ I_k(\\Sigma_{x|u}) - \\mu I_p(\\Sigma_{x|u}) } \\nonumber \\\\\n&=& \\frac{\\mu}{2}\\log (2 \\pi e)^m |\\Sigma_{x|u}| \n + \\frac{1}{2} \\log (2 \\pi e)^m |\\Sigma_{x|u} + \\Sigma_{W_z}| \\nonumber \\\\\n &&~~~~~~ - \\frac{(1+\\mu)}{2} \\log (2 \\pi e)^m |\\Sigma_{x|u} + \\Sigma_{W_y}| \n\\nonumber\\\\\n&& + [(1+\\mu)h(Y) - h(Z) - \\mu h(X)],\n\\label{eq:relation}\n\\end{eqnarray}\nand by setting $M = 2M_2$, we have\nthe KKT conditions in Eqs.~(\\ref{eq:kkt-1})--(\\ref{eq:kkt-3}).\n\nThe CQs shown in \\cite[Appendix 4]{weingarten:06},\nwhich is an interpretation of \n\\cite[CQ5a of Section 5.4]{bertsekas:03} are\nthe following:\nThere exists a matrix $A$ satisfying\n\\begin{enumerate}\n\\item \\label{cq-1}\nFor any $\\bol{u} \\neq 0$ in the null space\nof $\\Sigma_{x|u}^* - L$, we have $\\bol{u}^T A \\bol{u} > 0$.\n\n\\item \\label{cq-2}\nFor any $\\bol{v} \\neq 0$ in the null space \nof $\\Sigma_x - \\Sigma_{x|u}^*$, we have\n$\\bol{v}^T A \\bol{v} < 0$.\n\n\\item \\label{cq-3}\n\\begin{eqnarray*}\n\\rom{Tr}\\left[\n\\nabla_{\\Sigma_{x|u}} (I_p(\\Sigma_{x|u}^*) - R_p^o)\n A^T\n\\right] > 0.\n\\end{eqnarray*}\n\\end{enumerate}\n\nTo check whether the above CQs are satisfied,\nwe suggest $A$ given by\n\\begin{eqnarray*}\nA = \\alpha (L - \\Sigma_{x|u}^*) + (\\Sigma_x - \\Sigma_{x|u}^*)\n\\end{eqnarray*}\nfor $\\alpha > 0$.\nFirst we check (\\ref{cq-1}). For any\n$\\bol{u} \\neq 0$ in the null space \nof $\\Sigma_{x|u}^* - L$, we have\n\\begin{eqnarray*}\n\\bol{u}^T A \\bol{u} = \\bol{u}^T (\\Sigma_x - \\Sigma_{x|u}^*) \\bol{u}.\n\\end{eqnarray*}\nSuppose that $\\bol{u}^T (\\Sigma_x - \\Sigma_{x|u}^*) \\bol{u} = 0$.\nThen we have\n\\begin{eqnarray*}\n0 &=& \\bol{u}^T \\left(\n(\\Sigma_{x|u}^* - L) + (\\Sigma_x - \\Sigma_{x|u}^*)\n\\right) \\bol{u} \\\\\n&=& \\bol{u}^T (\\Sigma_x - L) \\bol{u},\n\\end{eqnarray*}\nwhich is a contradiction because $\\Sigma_x \\succ L$.\nThus the condition (\\ref{cq-1}) is satisfied.\n\nNext, we check (\\ref{cq-2}). For any\n$\\bol{v} \\neq 0$ in the null space of\n$\\Sigma_x - \\Sigma_{x|u}^*$, we have\n\\begin{eqnarray*}\n\\bol{v}^T A \\bol{v} \n= \\bol{v}^T (L-\\Sigma_{x|u}^*) \\bol{v} < 0\n\\end{eqnarray*}\nbecause $L \\prec \\Sigma_{x|u}^*$.\n\nFinally, we check (\\ref{cq-3}).\nBy noting\n\\begin{eqnarray*}\n\\nabla_{\\Sigma_{x|u}} I_p(\\Sigma_{x|u})\n = \\frac{1}{2} (\\Sigma_{x|u} + \\Sigma_{W_y})^{-1}\n - \\frac{1}{2} \\Sigma_{x|u}^{-1} \\prec 0\n\\end{eqnarray*}\nfor any $\\Sigma_{x|u} \\succ 0$, we have\n\\begin{eqnarray*}\n\\lefteqn{\n\\rom{Tr}\\left[\n\\nabla_{\\Sigma_{x|u}}(I_p(\\Sigma_{x|u}^*) - R_p^o)\nA \\right]\n} \\\\\n&=& \\frac{\\alpha}{2} \\rom{Tr} \\left[\n\\left\\{\n(\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1}\n - (\\Sigma_{x|u}^*)^{-1}\n\\right\\} (L - \\Sigma_{x|u}^*)\n\\right] \\\\\n&& \\hspace{-3mm} + \n\\frac{1}{2} \\rom{Tr} \\left[\n\\left\\{\n(\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1}\n - (\\Sigma_{x|u}^*)^{-1}\n\\right\\} (\\Sigma_x - \\Sigma_{x|u}^*)\n\\right].\n\\end{eqnarray*}\nSince $L - \\Sigma_{x|u}^* \\prec 0$, by taking\n$\\alpha > 0$ to be sufficiently large, \nthe condition (\\ref{cq-3}) is satisfied.\n\n\\begin{remark}\nWe need to introduce the optimization problem\nin Eq.~(\\ref{eq:opt3}) because the arguments\nin \\cite[Appendix 4]{weingarten:06} is guaranteed\nonly under the condition such that\nthe range of the variable $\\Sigma_{x|u}$\nis a closed set.\n\\end{remark}\n\n\\section{Proof of Eqs.~(\\ref{eq:enhanced-1}) and (\\ref{eq:enhanced-2})}\n\\label{appendix:enhanced-1-2}\n\nBy noting $M \\succeq 0$, we have\n\\begin{eqnarray*}\n(\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1}\n &=& (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1} + M \\\\\n&\\succeq& (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1}.\n\\end{eqnarray*}\nThus we have\n\\begin{eqnarray*}\n\\Sigma_{\\tilde{W}_y} \\preceq \\Sigma_{W_y}.\n\\end{eqnarray*}\n\nSince $\\Sigma_{W_z} \\succ 0$, by substituting\nEq.~(\\ref{eq:enhancement}) into Eq.~(\\ref{eq:kkt-1}),\nwe have\n\\begin{eqnarray}\n\\lefteqn{\n(\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1}\n} \\nonumber \\\\\n&=& \\frac{\\mu}{1+\\mu} (\\Sigma_{x|u}^*)^{-1}\n + \\frac{1}{1+\\mu}(\\Sigma_{x|u}^* + \\Sigma_{W_z})^{-1} \n\\label{eq:proof-enhanced-1} \\\\\n&\\prec& (\\Sigma_{x|u}^*)^{-1}\n\\nonumber\n\\end{eqnarray}\nwhen $\\mu > 0$.\nThus, we have\n\\begin{eqnarray*}\n\\Sigma_{\\tilde{W}_y} \\succ 0.\n\\end{eqnarray*}\nNote that $\\Sigma_{\\tilde{W}_y} = \\Sigma_{W_z} \\prec 0$\nwhen $\\mu = 0$.\n\nFrom Eq.~(\\ref{eq:proof-enhanced-1}), we have\n\\begin{eqnarray*}\n(\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1}\n \\succeq (\\Sigma_{x|u}^* + \\Sigma_{W_z})^{-1},\n\\end{eqnarray*}\nwhere the strict inequality holds for\n$\\mu > 0$. Thus we have\n\\begin{eqnarray*}\n\\Sigma_{\\tilde{W}_y} \\preceq \\Sigma_{W_z}\n\\end{eqnarray*}\nand especially\n\\begin{eqnarray}\n\\label{eq:strict-enhanced}\n\\Sigma_{\\tilde{W}_y} \\prec \\Sigma_{W_z}\n\\end{eqnarray}\nfor $\\mu > 0$.\n\\hfill$\\square$\n\n\n\n\\section{Proofs of Eq.~(\\ref{eq:reduction})}\n\\label{appendix:preservation}\n\nEq.~(\\ref{eq:reduction}) can be derived by\nthe following sequence of equalities:\n\\begin{eqnarray}\n\\lefteqn{\n(\\Sigma_x + \\Sigma_{\\tilde{W}_y})\n (\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1} } \\nonumber \\\\\n&=& \\left[ (\\Sigma_x - \\Sigma_{x|u}^*) +\n (\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y}) \\right]\n (\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1} \\nonumber \\\\\n&=& (\\Sigma_x - \\Sigma_{x|u}^*)\n (\\Sigma_{x|u}^* + \\Sigma_{\\tilde{W}_y})^{-1}\n + I \\nonumber \\\\\n&=& (\\Sigma_x - \\Sigma_{x|u}^*)\n \\left[ (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1} + M \\right]\n + I \\label{eq:proof-preservation-1} \\\\\n&=& (\\Sigma_x - \\Sigma_{x|u}^*)\n (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1} + I \n \\label{eq:proof-preservation-2} \\\\\n&=& \\left[ (\\Sigma_x - \\Sigma_{x|u}^*)\n + (\\Sigma_{x|u}^* + \\Sigma_{W_y}) \\right]\n (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1} \\nonumber \\\\\n&=& (\\Sigma_x + \\Sigma_{W_y}) \n (\\Sigma_{x|u}^* + \\Sigma_{W_y})^{-1},\n\\end{eqnarray}\nwhere Eq.~(\\ref{eq:proof-preservation-1})\nfollows from Eq.~(\\ref{eq:enhancement}) and\nEq.~(\\ref{eq:proof-preservation-2}) follows\nfrom Eq.~(\\ref{eq:kkt-2}).\n\\section{Proof of Eq.~(\\ref{eq:invertibility})}\n\\label{appendix:invertibility}\n\nFrom Eqs.~(\\ref{eq:enhanced-y}),\n(\\ref{eq:aligned-z}) and (\\ref{eq:strict-enhanced}),\nwe can write\n\\begin{eqnarray}\nZ = X + \\tilde{W}_y + W^\\prime,\n\\end{eqnarray}\nwhere $W^\\prime \\sim {\\cal N}(0,\\Sigma_{\\tilde{W}_y} - \\Sigma_{W_z})$.\nThus, we have\n\\begin{eqnarray*}\n\\Sigma_{\\tilde{y}z} &=& \\Sigma_{\\tilde{y}}, \\\\\n\\Sigma_{zx} &=& \\Sigma_{xz}\n = \\Sigma_x.\n\\end{eqnarray*}\nFurthermore, we have\n\\begin{eqnarray}\n\\label{eq:proof-invertiblity-0}\n\\Sigma_{\\tilde{y}} \\prec \\Sigma_z.\n\\end{eqnarray}\n\nFrom the block inversion of the matrix\n(e.g.~see \\cite[Appendix 5.5]{boyd-book:04}) and\nEq.~(\\ref{eq:coefficients}), we have\n\\begin{eqnarray}\nK_{\\tilde{y}x}\n &=& \n \\left[ \\begin{array}{cc}\n\\Sigma_{\\tilde{y}} & \\Sigma_x \n \\end{array}\\right]\n\\left[\n\\begin{array}{c}\n- \\Sigma_z^{-1} \\Sigma_x S^{-1} \\\\\n S^{-1}\n\\end{array}\\right] \n \\nonumber \\\\\n &=& (I - \\Sigma_{\\tilde{y}} \\Sigma_z^{-1}) \\Sigma_x S^{-1},\n\\label{eq:proof-invertibility-1}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\nS = \\Sigma_x - \\Sigma_x \\Sigma_z^{-1} \\Sigma_x\n\\end{eqnarray*}\nis the Schur complement.\n\nFrom Eq.~(\\ref{eq:proof-invertiblity-0}),\nwe have\n\\begin{eqnarray*}\nI - \\Sigma_{\\tilde{y}}^{\\frac{1}{2}}\n \\Sigma_z^{-1} \\Sigma_{\\tilde{y}}^{\\frac{1}{2}}\n \\succ I - \\Sigma_{\\tilde{y}}^{\\frac{1}{2}}\n \\Sigma_{\\tilde{y}}^{-1} \\Sigma_{\\tilde{y}}^{\\frac{1}{2}}\n = 0.\n\\end{eqnarray*}\nThus we have\n\\begin{eqnarray}\n\\label{eq:proof-invertibility-2}\n| I - \\Sigma_{\\tilde{y}} \\Sigma_z^{-1}|\n = | I - \\Sigma_{\\tilde{y}}^{\\frac{1}{2}}\n \\Sigma_z^{-1} \\Sigma_{\\tilde{y}}^{\\frac{1}{2}}|\n\\neq 0.\n\\end{eqnarray}\nBy combining Eqs.~(\\ref{eq:proof-invertibility-1}) \nand (\\ref{eq:proof-invertibility-2}),\nwe have Eq.~(\\ref{eq:invertibility}).\n\\hfill$\\square$\n\n\n\n\n\n\\section*{Acknowledgment}\n\nThe first author would like to thank \nProf.~Ryutaroh Matsumoto\nfor valuable discussions and comments.\nThe authors also would like to thank\nProf.~Jun Chen for informing the literatures\n\\cite{liu:10b,chen:09}.\nThis research is partly supported by \nGrant-in-Aid for Young Scientists (Start-up):\nKAKENHI 21860064.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nWhile low mass stars dominate the mass of galaxies, massive stars regulate\ntheir energy budget. Understanding how and when the mass distribution of stars\nis determined is therefore essential in establishing a comprehensive picture\nof galactic evolution, and star formation, throughout the Universe.\n\n\\begin{figure*}[!th!]\n\\hspace{0cm}\n\\includegraphics[width=16.5cm,angle=0]{fig1.eps}\n\\caption{Filamentary infrared dark clouds. ({\\it a}) Spitzer 3 colour image of\n a region containing IRDCs (Blue: 3.6$\\mu$m ; Green: 8$\\mu$m ; Red:\n 24$\\mu$m ). The IRDCs are the long filamentary structures seen in\n extinction. On this figure the blue stars are foreground stars, the\n red\/yellow stars are young stars currently forming in the IRDC. ({\\it b})\n H$_2$ column density map constructed from the 8$\\mu$m extinction seen in\n ({\\it a}). While the outer contour delimits the boundary of each of the\n three IRDCs, the fragments are the substructures seen within each IRDCs. The\n 28 fragments identified in these clouds are marked by blue dashed ellipses.}\n\\end{figure*}\n\nSince stars form in molecular clouds the comparison of the internal structure\nof the clouds and the initial mass function (IMF) of stars can provide\ninsights on the processes responsible for the formation of stars. The mass\ndistribution of molecular clouds, and cores within them have been extensively\nstudied in the past twenty years. Until recently, it was believed that the\nmass distribution of CO clumps was described by $\\Delta N_{\\rm {CO}}\/\\Delta\n\\log M=M^{-\\alpha}$ with $\\alpha=0.7\\pm 0.2$ for the Milky Way (Kramer et\nal. 1998, Rosolowski 2004). The mass distribution of prestellar cores, the\ndirect progenitors of stars and stellar systems, as observed in dust continuum\nis much steeper, resembling the Salpeter IMF with a power law index of\n$\\alpha=1.35$ (Motte et al. 1998; Enoch et al. 2008). However, recent papers\nquestioned the impact of the source extraction scheme used to segment the data\non the final mass distribution shape (Pineda et al. 2009). Buckle et\nal. (2010) found a steeper mass distribution for small scale CO clumps. Also,\nin most cases, different tracers are required to trace different structures\nsuch as dense cores and molecular clumps, raising the question of detection\nbiases. Statistics is often a problem too, binning small number of objects\nintroduce artifacts (Reid \\& Wilson 2006). Therefore some confusion exists on\nwhat is the real mass structure of molecular clouds.\n\nAnother important physical aspect of molecular cloud structure is the\nprobability density function (PDF) of the gas volume density. This quantity\nhas received only little attention (e.g. Dring et al. 1996 for HI; Smith\n\\& Scalo 2009 for CO) but potentially contains crucial\ninformation on the processes at the origin of the density fluctuations.\n For instance, turbulence-driven fragmentation models develop\ninitial lognormal density fluctuations (e.g. Padoan et al. 1997), which could\nbe the main driver of the lognormal part of the IMF (Chabrier 2003). Studying\nthe density distribution of fragments within molecular clouds could set\nimportant constraints on such models.\n\nTo perform such studies, we decided to focus on a specific type of molecular clouds, i.e. \ninfrared dark clouds (IRDCs). IRDCs are dense molecular clouds seen in silhouette\nagainst the bright emission of the galactic plane (e.g. Perault et al. 1996;\nTeyssier et al. 2002; Rathborne et al. 2006; Simon et al. 2006). They are\ncold and only slightly processed by star formation activity, still containing\nthe initial conditions of star formation. Peretto \\& Fuller (2009; hereafter\nPF09) recently constructed the column density maps of more than 11,000 of such\nIRDCs, the largest database of such structures to date. This catalogue\nprovides the opportunity to probe molecular clouds in the Galaxy over a wide\nrange of size scales and column density at high angular resolution using the\n8$\\mu$m dust absorption and a new source extraction scheme.\n\n\nIn section 2 of the present paper we discuss the dataset we used. In section 3\nwe describe and re-analyze previous results on the distance distribution of\nIRDCs, while in Section 4 we estimate completeness limits. Section 5 displays\nour main results on the size, density and mass distributions of IRDCs and\nfragments. Discussion is in Section 6. And finally we summarize the\nmain findings of this paper in Section 7.\n\n\\section{Data set}\n\n\nThe analysed IRDCs come from a new catalogue of clouds identified in the\nSpitzer GLIMPSE data (PF09). IRDCs were defined as connected structures with\ncolumn density peaks above $N_{\\rm H_2}=2\\times10^{22}$ cm$^{-2}$ and\nboundaries defined by the contour at $N_{\\rm H_2}=1\\times10^{22}$ cm$^{-2}$.\nSingle peaked structures lying within the IRDCs were identified as\nfragments. The boundary of a fragment being defined by the contour of the\nlocal minimum between a fragment and its closest neighbour, the same criterion\nused to define the {\\it leaves} of the dendogram analysis of Rosolowsky et\nal. (2008). As column density peaks, these fragments are particularly\nimportant in the context of star formation since they are the likely birth\nplace of the future generation of stars.\nThe catalogue includes opacity maps at 4\\hbox{$^{\\prime\\prime}$}\\ resolution and physical\nproperties for over 11,000 IRDCs. Extracting the densest structures, a\ntotal of $\\sim$50,000 fragments have been catalogued within the full sample of\nclouds (PF09). Figure 1a shows a Spitzer three colour image of a region\ncontaining three filamentary IRDCs from the PF09 catalogue. Figure 1b shows\nthe column density map of these IRDCs and identifies the fragments within the\nclouds.\n\n\n\\begin{figure}[!t!]\n\\hspace{-0.5cm}\n\\includegraphics[width=6cm,angle=270]{histo_sdc_pk.eps\n\\caption{Histogram of the IRDC peak column density. We can see a steady decrease down to \n$N_{\\rm H_2} \\simeq 1\\times10^{23}$~cm$^{-2}$. Above this limit saturation does not allow us to probe the true peak column density. \n \\label{NH2_pk_av}}\n\\end{figure}\n\n\\subsection{IRDC saturation}\n\nAs discussed in PF09, some of the absorption towards IRDCs is saturated,\nmeaning the infrared background is not strong enough in order to fully probe\nthe internal structure of an IRDC. Based on photometric noise limitation and\nbackground strength PF09 estimated the fraction of saturated IRDCs to be 3\\%,\ncorresponding to roughly 340 IRDCs over the entire sample. \n\nAn alternative estimate of the number of saturated clouds can be \nmade from an inspection of the distribution of peak column density of IRDCs\nshown in Fig.~\\ref{NH2_pk_av}.\nThere appears to be a break in the distribution of peak $N_{\\rm H_2}$\\ at $\\sim\n1\\times10^{23}$~cm$^{-2}$, which likely reflects the effect of saturation.\nThe fraction of IRDCs lying above this limit is 4\\%, very similar to the value\nestimated in PF09. However, even for these saturated clouds only a small\nfraction of their area is above the saturation limit, only marginally affecting\nthe averaged IRDC column density (and therefore any estimate of the cloud\nmass). But, the saturation has a much stronger effect on some fragments. For\nthis reason, in the analysis presented here IRDCs containing saturated pixels\nare considered, but fragments with saturated pixels are excluded.\n\nOur estimated saturation limit is roughly twice as large as the one found by\nVasyunina et al. (2009) from millimetre emission in their study of\nparticularly high column density IRDCs. \nThe discrepancy between the low absorption column densities Vasyunina et\nal. determined by assuming the minimum possible foreground emission, that due\nto the zodical light, and the high values they determined from millimeter dust\ncontinuum led Vasyunina et al. to derive a relatively low saturation\nlimit. However, the majority of their clouds do not in fact appear saturated\nas considerable substructure can be seen in the 8$\\mu$m extinction maps.\n\n\n\n\n\n\n\\subsection{Column density and angular size distributions}\n\nThis study aims to statistically analyze the density and mass distributions of\nIRDCs and their fragments. To derive such quantities we first need to know the\nangular size and column density distributions as measured on the column\ndensity maps constructed by PF09. \n\nFigure \\ref{NH2_distrib} shows the distribution of the angular size\nand column density for the $\\sim11,000$ IRDCs and the $\\sim 50,000$\nfragments identified within them. Fragments with saturated pixels have\nbeen excluded (see Sec.~2.1). In addition IRDCs which are not\nfragmented\n($\\sim40\\%$ of the IRDC sample) \nhave also been removed to maintain a clear definition of\na fragment as a substructure within a cloud. However, in practice\nkeeping these single peak clouds has little effect on the results.\n\n\nIt is important to note that the column densities we plot here are the {\\it\n background substracted} column densities, equivalent to the one obtained in\nthe {\\it clipping} option of the dendogram analysis of Rosolowsky et\nal. (2008). In the context of centrally concentrated structures, these column\ndensities are the relevant ones when interested in the physical properties of\nthe gas enclosed in a given radius. Figure~\\ref{NH2_distrib} clearly shows that\nthe distributions are dominated by small structures of low column density. We\ncan also clearly see the effect of incompleteness on the distributions with\nthe decrease in the number of sources at low radius\/column density,\nresponsible for the formation of artificial peaks. The incompleteness in the\nsample and these distributions are discussed in Section~\\ref{sec:complete}.\n \n\n\\begin{figure}[!t!]\n\\hspace{-0.5cm}\n\\includegraphics[width=4cm,angle=270]{req_nh2_sdc.eps}\n\n\\includegraphics[width=4cm,angle=270]{req_nh2_sdcfrag.eps}\n\\caption{Distributions of angular radius R$_{eq}$ and average column densities\n over each structure for IRDC (top) and fragments (bottom). The estimates for\n the radius completeness limit are discussed in Section~\\ref{sec:complete}.\n \\label{NH2_distrib}}\n\\end{figure}\n\n\n\n\n\\begin{figure*}[!t!]\n\\hspace{-0.0cm}\n\\includegraphics[width=7cm,angle=270]{dist_irdc.eps}\n\\includegraphics[width=7cm,angle=270]{dist_gl.eps}\n\\caption{{\\it (left):} Distributions of distances of IRDCs (Simon et al. 2006, \n\tJackson et al. 2008, Marshall et al. 2009). {\\it (right):} Plot of the galactic longitude\n dependance of average IRDC distance determined from kinematics (top panel)\n and extinction (bottom panel). The horizontal dashed lines show the average\n distances over the indicated longitude range (Table~\\ref{tab:dist}).\n \\label{dist}}\n\\end{figure*}\n\n\n\\section{IRDCs distance distribution}\n\nTo calculate the density and mass of the clouds the distance of each IRDC is\nrequired, however this is not yet known for most of the 11,000 IRDCs. For this\nanalysis we have therefore adopted a statistical approach based on previous\nmeasurements of the distances to samples of IRDCs.\n\nSeveral studies have measured the distance distribution of subsamples of IRDCs\nin both the 1st and 4th quadrant of the Galactic plane\n(Simon et al. 2006; Jackson et al. 2008; Marshall et al. 2009). Both kinematic and dust\nextinction techniques have been used to infer these distances, and although\nthey lead to similar results, there are some differences\n(Fig.~\\ref{dist}). The properties of these distance distributions are\nsummarised in Table~\\ref{tab:dist}. Using dust extinction, Marshall et\nal. (2009) found a centrally peaked, Gaussian-like distance distribution very\nsimilar for both the 1st and 4th quadrant of the Galaxy (Fig.~\\ref{dist} -\nbottom panel). In a similar way, kinematic distances\\footnote{These kinematic\n distances have been recalculated by taking the CS(2-1) and $^{13}$CO(1-0)\n velocities published in Jackson et al. (2008) and Simon et al. (2006),\n respectively, and using the Reid et al. (2009) revised galactic rotation\n model.} shown in Fig.~\\ref{dist} (top panel) show a good agreement between\nthe 1st and 4th quadrant . Although in the 1st quadrant a tail at 5~kpc\nclearly emerges. However most significant difference between the extinction\nand kinematic distances is the position of the peak, being located at\n$\\sim3$~kpc in one case and at $\\sim5$~kpc in the other. Both techniques have\ntheir own biases and advantages, it is therefore difficult to favor one\ndistance distribution over another. However a Gaussian distribution is a\nrather good approximation to the distance distribution in both quadrants.\n\n\\begin{table*}\n \\begin{tabular}{lccc}\nSample & Mean distance & FWHM & Dispersion, $\\sigma$\\\\\n & (kpc) & (kpc) & (kpc) \\\\ \\hline\nMarshall, 1st \\& 4th quad. & 4.7 & 1.5 & 1.2 \\\\\nJackson, 4th quad. & 3.0 & 0.8 & 1.2\\\\\nSimon,1st quad. & 3.6 & 2.5 & 1.3\\\\\n\\hline\\hline\n \\end{tabular}\n \\caption{Properties of the distance distributions of IRDCs shown in\n Fig.~\\ref{dist}. If the distributions were truly Gaussian the FWHM would\n equal 2.35$\\sigma$.}\\label{tab:dist}\n\\end{table*}\n\n\nFig.~\\ref{dist} (right) shows the average distance to the IRDCs as a function\nof galactic longitude for the sources with distances measured by these two\ntechniques. Any systematic trend with longitude could introduce a bias in\nanalysis adopting a statistical distribution for the distance of IRDCs. It is\nclear that there is very little variation of the IRDC distances with respect\nto the galactic longitude. The only region where there may be such an effect\nis towards galactic centre seems, an area which is not covered in our IRDC\nSpitzer catalogue which only extends into $l=\\pm10\\hbox{$^\\circ$}$. It is worth noting\nthat distance variations around the mean as a function of longitude are very\nsimilar for both methods, emphasising that is it predominantly only the\naverage distance which differs between the two methods.\n\n\\begin{figure}[!th!]\n\\hspace{-0.5cm}\n\\includegraphics[width=6cm,angle=270]{cs_dist.eps}\n\\caption{Heliocentric distance distribution of IRDCs observed in CS J=1-0\n (Peretto et al. in prep). The 3 histograms correspond to 3 different range\n of IRDC size. There is no evidence of any statistical difference in IRDC distance as a function of their size. \n \\label{cs_dist}}\n\\end{figure}\n\nAnother possible bias is with respect to the size of the IRDCs. The studies\nshown in Fig.~\\ref{dist} do not contain IRDCs as small as those in the Spitzer\nbased sample and so it is possible that the small and large clouds have\ndifferent distance distributions. \nHowever, recent observations with ATNF Mopra in CS J=1-0 (Peretto et al. in\nprep) of a square degree of the galactic plane ($29.8\\hbox{$^\\circ$} < l < 31.8\\hbox{$^\\circ$}$,\n$-0.27\\hbox{$^\\circ$}< b < 0.27\\hbox{$^\\circ$}$) can be used to investigate this possibility.\nThis area covers 196 Spitzer IRDCs in total\nand more than 80$\\%$ of the clouds can be associated\nwith CS emission.\n\nThe distances of these clouds can be calculated using the Reid et al. (2009)\ngalactic rotation model. Figure \\ref{cs_dist} shows the distance distribution\nof the Spitzer IRDCs detected in CS. In this figure, the IRDCs have been\ndivided int to three size ranges. The 80 smallest IRDCs, those with\n$R_{eq}$ (PF09) less than 15\\hbox{$^{\\prime\\prime}$}~have a mean distance and standard deviation of\n$5.2$~kpc and $0.8$~kpc. This is indistinguishable from the values of $5.3$~kpc\nand $0.7$~kpc and $5.3$~kpc and $0.8$~kpc for the IRDCs in the next two size\nranges, $15''30''$, which contain $52$ and $39$ objects\nrespectively. These distributions therefore show no\nindication that large and small IRDCs have different distributions of\ndistance.\n\n\\begin{figure*}[!t!]\n\\hspace{0.5cm}\n\\includegraphics[width=5cm,angle=270]{pk_sdc.eps}\n\\includegraphics[width=5cm,angle=270]{pk_frag.eps}\n\\caption{Peak column density distributions for different IRDC (left) and\n fagment(right) sizes. For both structures we show examples for which we are\n sensitivity limited and for which we are not. The location of the peaks of\n these distributions as a function of size of the clouds and fragments are\n used to construct Fig.~\\ref{rad_comp}. } \\label{pk_r}\n\\end{figure*}\n\n\n\\begin{figure*}[!th!]\n\\hspace{0.0cm}\n\\includegraphics[width=5cm,angle=270]{req_pk_sdc.eps}\n\\includegraphics[width=5cm,angle=270]{req_pk_frag.eps}\n\\caption{Maximum peak column density for all IRDCs (left) and fragments (right) falling in a given bin of radius. We can see\nthat this maximum differs from the sensitivity limit $N_{\\rm H_2}^{amp}$ when reaching a certain radius. This radius is taken as being the completeness radius.\n\\label{rad_comp}}\n\\end{figure*}\n\n\n\n\\section{Completeness limits}\n\\label{sec:complete}\n\n\\subsection{Column Density and Mass}\n\nThe mass completeness limit for the IRDCs and fragments can be written as \n\\begin{equation}\nM_{\\rm c}= \\pi(R_{\\rm c} d_{\\rm c})^2 \\times _{\\rm c} \n\\label{eqn:mass}\n\\end{equation}\nwhere $R_{\\rm c}$ is the smallest radius above which the sample is complete,\n$d_{\\rm c}$ is the distance within which the majority of the sources occur,\nand $_{\\rm c}$ is the typical average column density of the\nstructures with a radius $R_{\\rm c}$.\nFigure~\\ref{dist}(left) shows that about 95$\\%$ of the IRDCs in that plot have\ndistances below 6~kpc and so we conservatively adopt $d_{\\rm c}=6$~kpc.\n\nEstimating $R_{\\rm c}$ is less straighforward. The completeness limits of our\nsurvey are related to two parameters of the source extraction: $N_{\\rm\n H_2}^{\\rm amp}$, the minimum column density amplitude of a source (which is\nrelated to sensitivity) from the boundary of a cloud to its peak; and the\nangular resolution, 4\\hbox{$^{\\prime\\prime}$}\\ both for IRDCs and fragments. In order to\ninvestigate how these contribute to the completeness limits we look at the\ndistribution of average column density of IRDCs for objects of a given range of\nsizes as plotted in Fig.~\\ref{pk_r}. We then plot the column density at the\npeak of these distributions as a function of cloud size. This is also done\nfor the fragments. Figure \\ref{rad_comp} shows these plots.\n\n\nThe plots show a similar structure for both the IRDCs and fragments. Up to some\nsize, $55\\hbox{$^{\\prime\\prime}$}$ for the IRDCs and $9\\hbox{$^{\\prime\\prime}$}$ for the fragments, the peak of\nthe column density distributions is constant. Above these values it increases\nwith increasing size. This constant column density for small sizescales\nsuggests that the sample is not fully probing the populations of objects at\nthese sizescales. There are objects in these size ranges which have lower\ncolumn densities and are not sampled by the objects in the catalogue. These\nplots therefore show the size limit completeness of the catalogue, for both\nIRDCs, $R_c=55\\hbox{$^{\\prime\\prime}$}$, and fragments, $R_c=9\\hbox{$^{\\prime\\prime}$}$.\n\nAdopting these sizes, the average column density of\nclouds\/fragments below these sizes gives average column density\ncompleteness limits of $ _{\\rm c}^{\\rm frag}=\n2.5\\times10^{21}$~cm$^{-2}$ and $ _{\\rm c}^{\\rm IRDC}=5\\times10^{21}$~cm$^{-2}$. Using Eq.~\\ref{eqn:mass}, these\nvalues give the mass completeness of the catalogue as $M_{\\rm c}^{\\rm\n IRDC}= 800$~M$_{\\odot}$ and $M_{\\rm c}^{\\rm frag}=9$~M$_{\\odot}$.\n \n\n\n\n\n\n\n\n\n\\subsection{Density}\n\n\n \nThe density distribution of IRDCs is difficult to interpret since the\nclouds are defined based on a column density threshold. Therefore we\nconfine our discussion of the density distributions to the fragments.\nBoth sensitivity and angular resolution are important limiting factors\nin the context of density completeness of the sample: both compact\nlow-mass fragments and large, diffuse, high-mass fragments could remain\nundetected. For any fragment mass, $M_{\\rm lim}$ there is a minimum\nradius for which the peak column density of the fragment becomes\nhigher than the threshold for identifying fragments\n($3\\times10^{21}$cm$^{-2}$). If this minimum radius is larger than the\nangular resolution then such a fragment is detected. Therefore, we can\ndefine a density completeness limit for all fragments more massive\nthan $M_{\\rm lim}$. The minimum radius, $R_{\\rm min}$, and the\ncorresponding maximum density, $\\rho^{\\rm low}_{\\rm c}$, are given by\n\\begin{eqnarray}\n R_{\\rm min} &=& \\sqrt{\\frac{M_{\\rm lim}}{\\pi \\mu }}\\\\\n \\rho_{\\rm c}^{low} &=& \\frac{3}{4}\\sqrt{\\frac{\\pi}{M_{\\rm lim}}} \\mu^{3\/2}^{3\/2}\n\\end{eqnarray}\nwhere $\\mu$ is the mean mass per molecule and we adopt a column\ndensity $ $ which is the average column density\nof fragments of mass $M_{\\rm lim}$ (which by definition have peak\ncolumn densities greater than the threshold to be identified as\nfragments).\nAt a distance of 6~kpc, for fragment mass of 2\/8\/32~M$_{\\odot}$ and $\n =1.5\\times10^{21}$~cm$^{-2}$ we get R$_{\\rm min}=5\/10\/20$\\hbox{$^{\\prime\\prime}$}\\\nand $\\rho_{\\rm c}^{\\rm low} =2.4\/1.2\/0.6\\times10^3$cm$^{-3}$, respectively.\n\nFor the same given mass there is also an upper density limit which corresponds to the point where\nthe size of the fragment becomes smaller than the resolution of the observations.\nThis provides an upper limit on the density, $\\rho_{\\rm\n c}^{\\rm up}=0.75\\times M_{lim}\/(\\pi R_{res}^3)$. For the 3 masses discussed\nbefore we get $\\rho_{\\rm c}^{up} =0.4\/1.7\/6.8\\times10^5$~cm$^{-3}$.\n\n\n\n\\section{Size, mass and density structure of IRDCs}\n\\subsection{Physical size distribution}\n\\label{sec:size}\n\nTo calculate the density and mass of the IRDCs and fragments requires a\ndistance for each object. Two different approaches to statistically attribute\na particular distance to a particular cloud have been adopted. The first is\nto simply assign a unique distance to all clouds. Doing this, the physical\nsize distribution of IRDCs and fragments will be exactly the same as the\nangular size distribution. Given the well peaked distribution of distances\nfor clouds with measured distances (Fig.~\\ref{dist}), this should be a\nreasonable first approximation. However, a more sophisticated approach is to\nmake use of the distribution of distances (rather than just its peak\nposition). To do this we adopt a distance distribution for the IRDCs and then\nrandomly assign a distance drawn from this distribution to each cloud. Doing\nthis for the whole sample of clouds repeatedly provides a statistical sampling\nof the distance distribution. The final physical size distribution is the\nconvolution of the true physical size distribution by the chosen distance\ndistribution. However this does not have a crucial impact on the\ninterpretation of the physical size distribution if the dispersion of the\ndistance distribution is much smaller than the angular size distribution. This\nis clearly the case since the angular sizes, both for IRDCs and fragments,\nextends over 2 order of magnitudes while IRDC distances span only over a\nfactor of 3 at most. In other words, the dispersion in distance has relatively\nlittle effect on the final physical size distribution (see Appendix A).\nTo assign distances to the clouds using this sampling technique we adopt a\nGaussian distribution of distances with a peak at 4 kpc and a\ndispersion of 1 kpc, consistent with observed distance distributions (Fig.~\\ref{dist}).\n\n\n\\subsection{Mass distributions}\n\n\\begin{figure*}[!t!]\n\\hspace{0.5cm}\n\\vspace{-.cm}\n\\includegraphics[width=8cm,angle=270]{mass_dist.eps}\n\\caption{ Mass distribution of infrared dark clouds (left) and\n fragments (right) for two different distance distributions: The\n filled square symbols correspond to the adoption of an unique\n distance of 4~kpc for each single cloud, while the open triangles\n and associated shaded area corresponds to adopting and sampling a\n Gaussian distance distribution (see Section 5.1). The vertical\n dashed lines show the incompleteness limits. The best-fit is linear\n for the IRDCs (red solid line) with $\\Delta N_{\\rm IRDC}\/\\Delta\n \\log(M) = M^{-\\alpha}$ with $\\alpha=0.85\\pm0.07$, while the best-fit\n for fragments (blue solid line) is a lognormal function. For\n comparison the slope of the mass distribution of CO molecular clouds\n and clumps and the Salpeter part of the IMF are also shown.}\n \\label{mass}\n\\end{figure*}\n\n\nMass distributions of molecular cloud structures have been extensively studied\nin the past, therefore they represent a good point of comparison for this\ncurrent study. We defined mass as:\n\\begin{equation}\nM=\\pi R_{\\rm eq}^2\n\\end{equation}\nwhere $$ is the average column density across the IRDC or\nfragment and $R_{\\rm eq}$ its equivalent radius (PF09).\n\nFigure~\\ref{mass} shows the mass distributions for IRDCs and fragments\ncalculated adopting a single distance of 4kpc (filled square symbols) and for\nrandomly attributed distances as described in Section~\\ref{sec:size}. The\nshaded band on the figures shows the range (3 times the dispersion) spanned by\nthe 100 different distance realizations and the open triangles the mean for\nthe different realizations. The completeness limits are shown by the dashed\nlines. For comparison the power-law slopes of the CO clump mass function\n(slope $= -0.7$) and the Salpeter mass function (slope $= -1.35$) are also\nshown. Using the MPFITS IDL package (Markwardt 2009) we have fitted the two\ndistributions above their respective completeness limits. For the IRDCs we\nfind a linear function (in a log-log plot) provides a good fit with\n$dN_{\\rm{IRDC}}\/d\\log M=M^{-\\alpha}$ with $\\alpha=0.85 \\pm 0.07$. The mass\ndistribution of fragments is better fitted by a lognormal function defined\nas \\begin{equation} \\frac{dN_{\\rm{IRDC}}}{d\\log M}= A \\exp( -\n [\\log(M)-\\log(M_{\\rm{peak}})]^2\/2\\sigma^2)\n\\end{equation}\nwhere A is a normalization constant, $M_{\\rm{peak}}$ is the peak mass of the\ndistribution, and $\\sigma$ is the dispersion. However, since we do not map the\npeak, the precise parameters of the lognormal function fit are not well\nconstrained, several provide adequate fits to the data points. The function\nshowed in Fig.~\\ref{mass} has A=4610, M$_{peak} =1.55$~M$_{\\odot}$, and\n$\\sigma=0.78$. As argued above, the figure confirms that as a consequence of\nthe nature of the distance distribution, there is relatively little\ndifference in the derived mass distributions whether a single distance is\nadopted for the clouds or statistical approach is adopted. Also, the results of this\nanalysis are also not strongly dependent on exact parameters of the assumed\ndistance distribution as demonstrated in Appendix~A.\n\n\nA number of previous studies have attempted to construct, with samples at\nleast an order of magnitude smaller, the mass distributions of IRDCs (Simon et\nal. 2006; Marshall et al. 2009) and fragments within them (Rathborne et\nal. 2006; Ragan et al. 2009). Except for the Ragan et al. study, the mass\ndistributions in these studies agree: the IRDC mass distribution is similar to\nthat of CO clumps, while the distribution for the sub-structures are steeper,\nmore like the Salpeter IMF.\n\n\nIn their analysis of 11 IRDCs, Ragan et al. (2009) found that the mass\ndistributions of what they called {\\it clumps}, which correspond to\nfragments here, is quite flat, similar to the CO clump mass distribution, in\ncontrast with the present study.\nHowever it is difficult to understand the Ragan et al. result as the radii and\nmasses they quote for their clumps imply 8\\micron\\ opacities over 10 times\nlarger than the 8\\micron\\ opacities they quote.\n\n\n\n\\subsection{Density distribution}\n\n\\begin{figure*}[!th!]\n\\hspace{0.5cm}\n\\includegraphics[width=6.5cm,angle=270]{dens_plot_vir.eps}\n\\caption{Distribution of the number density of fragments normalised to 100\n cm$^{-3}$. {\\it(left):} Fragments with a mass $2 32$~M$_{\\odot}$. The dashed lines mark the density\n completeness limits, lower and upper for the left hand side panel and only\n lower for the 2 others (since the upper limits are off the plots). The\n square and triangle symbols and shaded area have the same meaning\n as for Fig.~\\ref{mass}.\n \\label{density}}\n\\end{figure*}\n\nThe density distribution of fragments may provide important insights on the\nphysical process generate these structures. We define the number density of a\nfragment as\n\\begin{equation}\nn = \/d_t\n\\end{equation}\nwhere d$_t$ is the line of sight size of the fragments which is assumed to be\ntwice the projected radius. Figure \\ref{density} shows the fragment density\ndistributions for the following mass ranges: $232$M$_{\\odot}$. For each range, the density\ncompleteness limit, the dashed line, is calculated for $M_{\\rm lim} =\n2\/8\/32$~M$_{\\odot}$, respectively. The density distribution over the entire\nmass range is very similar to that for the lower mass range (left panel). The\nfigures show that going from low mass to high mass fragments, the\ndistributions become flatter. Compared to low density fragments, there is a\nhigher probability of finding high density fragments for high mass\nfragments. One of the main issue in interpreting such a plot in terms of the\nformation of the fragments is that the density of gravitationally bound\nfragments is increasing over the time as they evolve (i.e. contract), and\ntherefore might {\\it pollute} the initial density PDF of the parental\nIRDCs. In the next section we will discuss the impact of such effects on the\ndensity distributions.\n\n\n\n\n\\section{Discussion: Turbulent vs gravitational dominated structures}\n\n\nThe mass distributions of IRDCs and fragments plotted in Fig.~\\ref{mass}\nclearly shows a steepening, from large structures to smaller fragments. While\nthe mass distribution of IRDCs is similar to that of CO clumps, the fragment\nmass distribution has a slope at high masses which is reminiscent of the slope\nof the Salpeter IMF, although it is best fitted with a lognormal function. \nHowever, two biases could affect the shape of the high-mass end of the\ndistribution and the interpretation that its slope is related to the Salpeter\nIMF. The detailed structure of this part of the distribution may be\nparticularly sensitive to the adopted Gaussian distance distribution. Also\nhigh-mass fragments might evolve more rapidly than their low-mas analogues and\ntherefore be under represented in extinction observations at 8$\\mu$m \n(c.f. Hatchell \\& Fuller 2008).\n\n To the first order, the mass distributions are\nin agreement with the theoretical work of Hennebelle \\& Chabrier (2008) who interpreted\nthe transition from a flat mass distribution to a steeper one \n as the transition from turbulence-generated structures to \n gravity dominated structures. In this context it is interesting to measure the gravitational binding of IRDCs and fragments. To investigate this, we have used Larson's relation (Larson 1981) to compute the kinetic\nsupport, and calculate the virial mass. Following Hennebelle \\& Chabrier (2008) we assume the effective\nvelocity dispersion is given by\n\\begin{equation}\nc_{\\rm eff}= (c_s^2 +0.33V_0^2d^{2\\eta})^{1\/2}\n\\end{equation}\nwhere $c_s$ is the sound speed $V_0 $ is the normalization velocity of\nLarson's relation, $\\eta$ is the power law index of Larson's relation, and $d$ is the size of the structure. We can then use this to compute the corresponding virial mass M$_{\\rm vir}$\n for every IRDC\/fragment. Figure \\ref{boundfrac} shows the fraction\nof IRDCs and fragments having a ratio $M_{\\rm H_2}\/M_{\\rm vir} > 0.5$,\nassuming $c_s = 0.2$~km\/s (T=10~K), $V_0 = 1$~km\/s, and $\\eta=0.4$. Above the\ncompleteness limit all the IRDCs appear gravitationally bound, as do the\nmajority of the fragments. Of course large uncertainties exist on the use of\nLarson's relation and its normalization. However, different normalizations\nstill give similar conclusions about the fraction of IRDCs and fragments which\nare bound.\n\n\\begin{figure}[!t!]\n\\hspace{0.cm}\n\\vspace{-.cm}\n\\includegraphics[width=5.5cm,angle=270]{boundfrac_vir.eps}\n\\caption{Fraction of bound IRDCs and fragments as a function of their mass. The\n typical error due to distance uncertainty is 0.1. The dashed lines show the\n incompleteness limits of both fragments and IRDCs.\n \\label{boundfrac}}%\n\\end{figure}\n\nA consequence of the IRDCs being gravitationally bound is that the observed change of slope of the mass distributions shown in Fig.~\\ref{mass}\ndoes not represent the transition from turbulence dominated to gravity dominated structures: \nmost of the IRDCs down to the completion limit are bound.\nHowever, as shown in Fig.~\\ref{boundfrac} a significant fraction of the fragments lying\nabove the completeness limit are unbound. \n In other words, even if globally gravitationally bound, IRDCs may contain turbulence-generated over-densities which will probably disperse and not form stars. The physical properties of these unbound fragments are likely to be \nrepresentative of the initial conditions of star formation within IRDCs.\n\n\n\n\n\n \n\n\nUsing the previously defined ratio to separate bound fragments to\nunbound ones we constructed the density distributions for both type of\nfragments as shown in Fig.~\\ref{densbound}. The mass ranges are the\nsame as in Fig.~\\ref{density}. The unbound fragments have all very\nsimilar distributions, independent of their mass range. In particular,\nthe high mass end of the distribution is well fit by the following\nrelation $\\Delta N\/ \\Delta \\log(n) \\propto n^{-4.0\\pm0.5}$, {\\bf the error bar arising from \nthe uncertainties on Larson's relation.} The location of\nthe peak seems to move with the completeness limit and is therefore\nquestionable. From these plots we cannot exclude a possible lognormal\ndistribution for unbound fragments,\n but the peak of such distribution would have to be below\nn$\\sim10^3$~cm$^{-3}$.\n\nIn contrast, the density distribution of the gravitationally bound\nfragments show a well defined peak between n=$10^3$~cm$^{-3}$ and\n$10^4$~cm$^{-3}$ in each mass range and a shape which broadens to\nlower densities as the mass range increase. \nThis could result from the higher mass fragments of a given density\nevolving to being bound more rapidly that lower mass fragments.\n\n\n\n\\begin{figure*}[!th!]\n\\hspace{0.cm}\n\\vspace{-.cm}\n\\includegraphics[width=6.5cm,angle=270]{unboundbound_vir.eps}\n\\caption{Same as Fig.~\\ref{density}, the shaded areas are the same as\n in Fig.~\\ref{density}. The red solid line represent the density\n distributions of gravitationally unbound fragments, while the blue\n dashed line represent the gravitationally bound\n fragments.} \\label{densbound}\n\\end{figure*}\n\n\\section{Summary}\n\nWe used the largest sample of IRDC column density maps to date in\norder to better characterize the size, mass, and density structure of\ndense molecular clouds. The large number of objects, 11,000 IRDCs and\n50,000 fragments, allows a detailed analysis of the completeness of\nthe sample. Using a statistically attribute distances to each IRDC, we\nhave demonstrated that above the completeness limit the mass\ndistribution of the IRDCs are consistent with a power-law $\\Delta\nN_{\\rm IRDC}\/\\Delta \\log(M) = M^{-0.8}$, where $N_{\\rm IRDC}$ is the\nnumber of clouds. For the fragments the high mass end of the mass\ndistribution shows a steeper slope, consistent with the slope of the\nSalpeter IMF, with the overall distribution well fitted by a lognormal\nfunction.\n\nUsing Larson's law to estimate the linewidth of each IRDC and\nfragment, we have shown that above our completeness limit all the\nIRDCs and the majority of fragments are likely to be bound. This\nimplies that the transition in the shape of the mass distribution does\nnot reflect a transition from unbound to graviationally bound\nstructures. Looking at the distribution of fragment density shows that\nbound fragments dominate the high density ($n\\gtrsim10^4$~cm$^{-3}$)\nend of the distribution for all mass ranges, and dominate the whole\ndistribution for the highest range of fragment masses. There is also\na distinct broadening of the distribution with increasing fragment\nmass. This could be a result of the higher mass fragments evolving to\nbeing bound more rapidly that lower mass fragments. The number of\nunbound fragments as a function of number density has the form $\\Delta\nN_{\\rm f}\/ \\Delta \\log(n) \\propto n^{-4.0\\pm0.5}$ (where $N_{\\rm f}$ is the\nnumber of fragments) down to a density of $\\sim10^3$~cm$^{-3}$ where\nthe completeness limit is reached.\n\nThe absence of bright infrared sources embedded in IRDCs indicates that the mass distributions and density distributions as a function of mass\nand degree of gravitational binding derived here are representative of the initial conditions of star formation within dense molecular clouds. These results should serve as\nconstraints on theoretical and numerical models in order to identify and\ncharacterize the physical processes responsible for the formation and\nearly fragmentation of molecular clouds.\n\n\\begin{acknowledgements}\n{\\it Acknowledgements. We thank the anonymous referee and John Scalo for their thorough reports which helped significantly improve the initial version of the paper. We would also like to thank Patrick Hennebelle for some useful discussions.}\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\nThe standard ``Lambda Cold Dark Matter'' (hereafter {\\it plain}\n$\\Lambda$CDM, or $\\Lambda$CDM with {\\it POwer LAw Inflation}\\footnote{\nHere ``power law\" indicates that primordial power spectrum is of a ``power-law\" form. \nDo not confuse with power law inflation model in which the inflaton potential is given by an exponential function.\n}) model of\ncosmology is founded on three key assumptions: firstly, that the\nUniverse contains, in addition to baryonic matter, a collisionless,\n``cold'' dark matter component that accounts for $\\sim 5\/6$ of the\nmatter density; secondly, that the expansion of the Universe is\ndescribed by the Friedmann equations with a cosmological constant\nterm, $\\Lambda$, and, thirdly, the implicit assumption that structure\nformation has its origin in the primordial perturbations seeded by\ninflation.\n\nThe plain $\\Lambda$CDM model has successfully predicted structure\nformation over many epochs and orders of magnitude, from the\nstructures observed in the Cosmic Microwave Background (CMB) at\n$z=1100$ to the sizes and distributions of clusters and galaxies at\nmuch later time. However, tensions between the CMB and measurements on\nsmall scales have also been reported. These have either been\nattributed to poorly understood systematic effects in various data\nsets, or interpreted as an indication of new physics.\n\nOn small scales, predictions for structure formation concern the\nabundance and internal structure of low mass dark matter halos in the\nLocal Universe. Here, observations of the Milky Way and Andromeda\nsatellite populations in particular, appear to be in disagreement with\nN-body simulations based on the plain $\\Lambda$CDM model.\n\nThis tension is twofold. The ``missing satellites'' problem\n\\cite{Klypin:1999uc, Moore:1999nt} refers to the apparent paucity of\nluminous satellite galaxies compared to the large number of dark\nmatter substructures predicted in plain $\\Lambda$CDM. While processes\nincluding supernova feedback (e.g. \\cite{Larson:1974}), and cosmic\nreionisation (e.g. \\cite{Efstathiou:1992}) are expected to reduce the\nnumber of observed satellite galaxies, the apparent excess of\nsubstructures in the plain $\\Lambda$CDM is not limited to the lowest\nmass scales: simulations also predict the presence of subhalos so\nmassive that they should not be affected by reionisation (and hence\ndeemed ``too big to fail'', \\cite{BoylanKolchin:2011de}), but whose\ninternal structure seems incompatible with that of the brightest\nobserved satellites.\n\nRecent hydrodynamic simulations have shown that, when baryonic effects\nare included, the observed Local Group satellite populations can be\nreproduced in $\\Lambda$CDM (e.g. \\cite{Sawala:2015cdf}), but only if\nthe Local Group mass is $\\sim 2-3\\times 10^{12}\\mathrm{M}_\\odot$, at the lower end\nof the observational limits \\cite{Gonzalez:2013pqa}. It has also been\nshown that, if the CDM assumption of the plain $\\Lambda$CDM model is\nrelaxed, simulations of warm dark matter \\cite{Lovell:2016fec,\n Bozek:2018ekc} and photon-coupled dark matter\n\\cite{Schewtschenko:2015rno}, which both reduce the small-scale power\nafter decoupling, give a better agreement to satellite kinematics than\nthe equivalent plain $\\Lambda$CDM models. Conversely, observed\nstructures in the Lyman-$\\alpha$ forest \\cite{Viel:2013apy,\n Irsic:2017ixq}, and the abundance of Milky Way satellites\n\\cite{Kennedy:2013uta} also provide upper limits for any reduction of\nsmall scale power relative to plain $\\Lambda$CDM.\n\nWarm dark matter models prevent the hierarchical formation of\nstructures below the free-streaming scale (e.g. \\cite{Bode:2000gq}),\nbut above that, the abundance of structures relative to CDM differs\nsignificantly over only a narrow range. Consequently, within current\nlimits, WDM effects are testable only at or below the scale of\nultra-faint dwarf galaxies \\cite{Lovell:2016fec}. This gives WDM\nmodels limited predictive power. Furthermore, the motivation for the\nrequired WDM particle appears somewhat ad-hoc. Another, perhaps more\nattractive possibility, is to modify the power spectrum of primordial\nperturbations. The modification should be such that it persists over\n$\\sim 4$ orders of magnitude; hence e.g. local features in the\ninflation potential are unlikely to do the job.\n\nInstead, here we propose a unified description of structure formation\nthat relies on two periods of inflation and accounts both for the\npower on CMB scales and for the apparent suppression of power on\nsubgalactic scales.\n\nThere exist models for two periods of inflation making use e.g. of a\nsuitably arranged inflaton potential with an intervening phase\ntransition \\cite{Kamionkowski:1999vp,Yokoyama:2000tz}\\footnote{\nIt has been argued that the CMB spectral $\\mu$ distortion would be\nuseful to differentiate models with a small-scale suppression of the\nmatter power spectrum due late-time effects (such as different dark\nmatter properties) from those caused by a modification of the\nprimordial power spectrum \\cite{Nakama:2017ohe}.\n}.\n\nIn this paper we consider a model where the inflaton field, giving\nrise to a slow roll inflation, is complemented by another scalar\nfield, which is dynamically irrelevant during the first period of\ninflation. Such a scalar is called a spectator field. We know that\nspectator fields exist: the Higgs field is one example (see, e.g.,\n\\cite{Enqvist:2014bua,Enqvist:2014tta,Enqvist:2015sua} for\nfluctuations of the mean Higgs field during inflation), barring the\npossibility that the Higgs field itself is the inflaton. Another, much\nstudied spectator field is the curvaton\n\\cite{Enqvist:2001zp,Lyth:2001nq,Moroi:2001ct} which imprints the\nperturbations it receives during inflation on radiation and matter\nafter inflation\\footnote{Another example would be the modulated\n reheating model \\cite{Dvali:2003em,Kofman:2003nx}}. Rather than the\ninflaton, the curvaton could then be the origin of the whole of the\nobserved spectrum of perturbations.\n\nHere we do not assume that the spectator field contributes to the\nprimordial perturbation in a significant manner. Instead, we assume\nthat it gives rise to second period of inflation, which happens under\nthe conditions to be discussed below. Such a second period of\ninflation affects the power spectrum both on the CMB scales as well as\non subgalactic scales. In this paper we shall discuss a model with\ntwo periods of inflation that yields the desired suppression at small\nscales while remaining in agreement with the observed properties of\nthe CMB power spectrum, making it possible to address the apparent\nshortfalls of the plain $\\Lambda$CDM model at small scales in\ncompletely different, and astrophysically decoupled regimes. Such\nagreement, however, makes certain demands on the inflaton model; not\nall inflationary potentials are consistent with two inflationary\nperiods. We will also demonstrate that for our working example, the\nfact that some subgalactic structure has actually been observed gives\nrise to a lower limit on the number of $e$-folds of the first\ninflationary period.\n\nThis paper is organized as follows. In Section~\\ref{sec:spectator}, we\nbriefly introduce the general concept of a spectator field, and in\nSections~\\ref{sec:background} and \\ref{sec:perturbation} we describe\nthe background evolution and perturbation, respectively. In\nSection~\\ref{sec:model}, we introduce a concrete inflation model. In\nSection~\\ref{sec:small} we discuss the predictions for small scale\npower. We conclude with a summary in Section~\\ref{sec:summary}.\n\n\n\n\n\\section{Spectator fields}\\label{sec:spectator}\n\nFields whose energy densities during inflation are irrelevant for the\nexpansion rate of the Universe are called spectators. If they are\nlight, both their mean field values and the field perturbations will\nbe subject to inflaton-driven inflationary expansion. At the onset of\ninflation the spectator energy density is subdominant. The spectator\nfield $\\sigma$ is assumed to begin to oscillate during the\nradiation-dominated period after the inflaton field $\\phi$ has decayed\n(or during the period dominated by the oscillations of the\ninflaton). Its energy density depends on the initial spectator field\nvalue $\\sigma_\\ast$, as well as on the exact form of the spectator\npotential. During inflation, the mean spectator field is subject to\nfluctuations and in a given inflationary patch is one realization of\nthe probability distribution, which is determined by the Fokker-Planck\nequation\\footnote{Assuming slow-roll inflation, as will be done here.}\nas was first pointed out by Starobinsky \\cite{Starobinsky:1986fx} (for\na discussion in the context of spectators, see e.g\n\\cite{Enqvist:2012xn,Hardwick:2017fjo}).\n\nAt the end of inflation, the spectator square-mean-field has a value\n$\\sigma_\\ast$, which serves as the initial spectator field value for\nits subsequent evolution. As a consequence of its stochastic evolution\nduring inflation, it may have a value greater than the Planck mass\n$M_{\\rm pl}$. If inflation lasts long enough, the probability\ndistribution for the initial curvaton field equilibrates; otherwise\nthere will be a dependence on the curvaton field value before the\nonset of inflation (however, equilibration is not automatic; for a\nrecent discussion on the fluctuations of spectator fields, see\n\\cite{Hardwick:2017fjo}). In any case, if $\\sigma_\\ast > M_{\\rm pl}$,\nthe spectator may end up dominating the Universe even before it starts\nto oscillate. This happens, provided that the spectator is still\nslowly rolling down its potential, whence a period of secondary\ninflation can\nensue\\cite{Langlois:2004nn,Moroi:2005kz,Ichikawa:2008iq,Dimopoulos:2011gb}. Thus,\nvery crudely, first there is a period of inflation driven by the\n``usual\" inflaton field; then the inflaton decays; after a while, the\nenergy density of the slowly rolling spectator becomes dominant and\ngenerates a second period of inflation, which ends when the spectator\ndecays. This scenario, which we investigate in the present paper, has\nsome very interesting consequences, in particular for the spectrum of\nperturbations at small scales. As we will discuss, these consequences\nwill also depend on the details of the slow roll inflation model.\n\nDepending on the duration of the secondary inflation driven by the\nspectator, the current observable scales (such as CMB) may have exited\nthe horizon either during the primary slow roll inflationary phase or\nduring the secondary, spectator-driven phase. In the standard case\nwith no inflating spectator, the required number of $e$-folds is\nusually taken to be $N \\sim 50 - 60$ $e$-folds. However, depending on\nthe duration of the secondary inflationary phase, we may tolerate\nprimary $e$-folds as low as $N \\sim 10$. In such a case, the\npredictions for the primordial power spectrum can be drastically\nmodified.\n\nThe spectator modifies the spectrum of perturbations already at the\nCMB scales. Because the first phase of inflation ends early, the modes\ncorresponding to the CMB scales which exited the horizon closer to the\nend of the inflaton period, at which the inflaton starts to move\nfaster, even if it is still slowly rolling. As a result, there will be\na large running of the spectral index, which may also significantly modify\nthe predictions for astrophysical scales.\n\n\n\n\\subsection{Background evolution}\\label{sec:background}\n\nBefore describing primordial density fluctuations, let us first look\nat the background evolution when a primary inflationary period is\nfollowed by a secondary period driven by a spectator.\n\n\nIf the spectator is to drive a secondary phase of inflation, it has to\ndominate the Universe before it begins to oscillate. This condition\nis given by\n\\begin{equation}\nU(\\sigma_\\ast) \\ge \\rho_r (t_{\\rm osc}),\n\\end{equation}\nwhere $\\sigma_\\ast$ is the spectator field value set during the first\ninflationary phase, $\\rho_r (t_{\\rm osc})$ is the radiation energy\ndensity at the beginning of the spectator oscillation $t=t_{\\rm osc}$\nand $U(\\sigma)$ is the potential of $\\sigma$. The spectator\noscillation starts when $H \\sim m_\\sigma$, where $m_\\sigma$ is the\nmass of the spectator. Throughout this paper we will assume that the\nspectator potential reads simply\n\\begin{equation}\n\\label{curvpotential}\nU(\\sigma) = \\frac 12 m_\\sigma^2 \\sigma^2~,\n\\end{equation}\nwhence the above condition can be\nrewritten as\n\\begin{equation}\n\\label{condforinf}\n\\frac12 m_\\sigma^2 \\sigma_\\ast^2 \\ge 3 M_{\\rm pl}^2 m_\\sigma^2,\n\\end{equation}\nwhere $ M_{\\rm pl}$ is the reduced Planck mass. From\n(\\ref{condforinf}) we obtain the condition for the spectator-driven\nsecondary inflation as\n\\begin{equation}\n\\sigma_\\ast \\gtrsim \\sqrt{6} M_{\\rm pl}.\n\\end{equation}\n\n\n\nThe mass and the initial field value of the spectator are generally\nassumed to be model parameters. If one has a long period of initial\ninflaton-driven inflation so that the curvaton reaches the\nFokker-Planck equilibrium distribution, a typical value of the\namplitude of the spectator field is given by\n\\cite{Enqvist:2012xn,Hardwick:2017fjo}\n\\begin{equation}\n\\label{eq:equil_value}\n\\sigma_{\\ast} \\simeq \\frac{H_\\ast^2}{m_\\sigma},\n\\end{equation}\nwith $H_\\ast$ being the Hubble rate during inflation, in which\nde-Sitter (a constant $H_\\ast$) background is assumed. Although\nwhether this distribution can be reached or not depends on inflation\nmodels, when an inflation model with a potential of the plateau type\nis adopted, the spectator field can obtain the de-Sitter equilibrium\n\\cite{Hardwick:2017fjo}. On the other hand, for the case of\n large-field inflation models, the equilibrium solution would not be\n reached although the spectator can still acquire a super-Planckian\n amplitude $\\sigma_\\ast > M_{\\rm pl}$ in the regime of an eternal\n inflation \\cite{Hardwick:2017fjo}. \n\nOnce the equilibrium value is reached, by using the fact that the\ninflationary Hubble scale is directly related to the slow-roll\nparameter $\\epsilon$ as $H_\\ast^2 \/ M_{\\rm pl}^2 \\simeq 1.6 \\times\n10^{-7} \\epsilon$\\footnote{\nHere we assume that primordial perturbations are generated only from\nan inflaton.\n},\nthe above expression can be recast as\n\\begin{equation}\n\\label{eq:sigma_ast}\n\\sigma_\\ast \\sim 8 M_{\\rm pl} \\left( \\frac{10^{6}~{\\rm GeV}}{m_\\sigma} \\right) \\left( \\frac{\\epsilon}{2 \\times 10^{-5}} \\right) \\,,\n\\end{equation}\nfrom which one can see that the amplitude of a spectator field can be\nas large as $\\sigma_\\ast > {\\cal O}(M_{\\rm pl})$ for some set of parameters. As\nnoted before, when $\\sigma_\\ast > M_{\\rm pl}$, the spectator can give\nrise to the secondary inflation. From Eq.~\\eqref{eq:sigma_ast} we can\nsee that such a scenario may in fact be generic, given the stochastic\nbehaviour of $\\sigma$ during the primary inflation and the relation\n\\eqref{eq:equil_value}.\nEven if the equilibrium value is not reached, the initial value of the spectator can be regarded as \na model parameter, which does not prohibit $\\sigma_\\ast$ of having a value as large as $\\sigma_\\ast > {\\cal O}(M_{\\rm pl})$. \nTherefore a scenario with a second period of inflation driven by a spectator could be realized in a broad class of model. \n\n\n\nIn Fig.~\\ref{fig:rho_evolution}, we show a typical thermal history in\nthe scenario with two periods of inflation, the second driven by a\nspectator. We display the evolution of energy densities of the\ninflaton, the spectator and radiation, denoted as $\\rho_\\phi,\n\\rho_\\sigma$ and $\\rho_{\\rm rad}$, respectively. In the following, we\nfocus on the case where the spectator gives a second inflationary\nperiod after the CMB scales have already exited the horizon close to\nthe end of the first period of inflation so that the small scale\nfluctuations are suppressed.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\vspace{10mm}\n\\resizebox{120mm}{!}{\n \\includegraphics{rho_evolution.eps}\n}\n\\end{center}\n\\caption{Evolution of energy densities of the inflaton field,\n spectator field and radiation in models where the secondary\n inflation is driven by a spectator. The number of $e$-folds $N$ is\n normalized to 0 at the end of the first inflation in this figure.\n Depending on the decay rate of the inflaton, the Universe may have\n been dominated by oscillating $\\phi$ from the end of the first\n inflation to the beginning of the second inflation. In this case,\n there is no intermediate radiation-dominated phase.}\n\\label{fig:rho_evolution}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Density perturbation}\\label{sec:perturbation}\n\nIn a scenario where a spectator field is present, both the inflaton\nand the spectator may contribute to the primordial fluctuations (in\nwhich case the spectator is a curvaton) for the mode which exited the\nhorizon during the first inflationary epoch driven by the inflaton.\nFor the modes which exited the horizon during the second inflationary\nepoch driven by a spectator, primordial fluctuations are sourced from\nthe spectator alone. Although we consider a case where fluctuations\nfrom a spectator field can be neglected in the next section, here we\nalso discuss the curvature perturbation generated by a spectator in\norder to clarify in what case spectator fluctuations can be\nneglected. Here we consider the curvaton type model.\n\nAs mentioned above, for modes which exited the horizon during the\nfirst inflationary period generated by the inflaton field, the\ncurvature perturbation $\\zeta$ is in general given by the sum of two\ncontributions\\footnote{\nFor a general discussion on a scenario where the inflaton and the\nspectator can both contribute to the curvature perturbation, we refer\nthe readers to\n\\cite{Langlois:2004nn,Lazarides:2004we,Moroi:2005kz,Moroi:2005np,Ichikawa:2008iq,Fonseca:2012cj,Enqvist:2013paa,Vennin:2015vfa,Fujita:2014iaa,Haba:2017fbi}\nfor the curvaton model and \\cite{Ichikawa:2008ne} for modulated\nreheating scenario.\n}: \n\\begin{equation}\n\\zeta_I = \\zeta^{(\\phi)}_I + \\zeta^{(\\sigma)}_I,\n\\end{equation}\nwhere $\\zeta^{(\\phi)}_I$ and $\\zeta^{(\\sigma)}_I$ are respectively the\ncurvature perturbations generated from the inflaton and the spectator\n(which in this case should be called the curvaton). A subscript $I$\nindicates that the perturbations correspond to modes which exited the\nhorizon during the first inflation. The inflaton part can be written\nas\n\\begin{equation}\n\\label{eq:zeta_phi1}\n\\zeta^{(\\phi)}_I = \\frac{1}{M_{\\rm pl}^2} \\frac{V (\\phi)}{V_\\phi (\\phi) } \\delta \\phi_\\ast\n= \\frac{1}{\\sqrt{ 2 \\epsilon M_{\\rm pl}^2}} \\delta \\phi_\\ast,\n\\end{equation}\nin which $V(\\phi)$ and $V_\\phi (\\phi)$ are the potential for the\ninflaton and its derivative with respect to the inflaton field $\\phi$\nand the fluctuation of the inflaton is given by $\\delta \\phi_\\ast =\nH_\\ast \/(2\\pi)$. Therefore the primordial power spectrum sourced by\ninflaton fluctuations is given by\n\\begin{equation}\n\\label{eq:P_prim_inflaton}\n{\\cal P}_{\\zeta}^{(\\phi)} (k) = \\frac{1}{12\\pi^2 M_{\\rm pl}^6} \\frac{V(\\phi)^3}{V_\\phi(\\phi)^2} \\,.\n\\end{equation}\n\nFor the curvaton part, here we give the expression for the case where\nthe curvaton drives the secondary inflation since we mainly consider\nthis kind of scenario in this paper. By adopting the $\\delta N$\nformalism \\cite{Starobinsky:1986fxa,Sasaki:1995aw,Sasaki:1998ug}, the\ncurvature perturbation generated from the spectator is given by\n$\\zeta^{(\\sigma)} = (\\partial N \/\\partial \\sigma_\\ast) \\delta\n\\sigma_\\ast$ where $N$ is the number of $e$-folds from the epoch when\nthe mode exited the horizon to that at the decay of the curvaton.\nHowever, in the case where the second inflation is driven by a\nspectator, the $\\sigma_\\ast$-dependence of $N$ comes from the period\nduring the second inflation. The number of $e$-folds during such a\nsecondary inflation is given by \\cite{Ichikawa:2008iq}\n\\begin{equation}\nN_2 = - \\frac{1}{M_{\\rm pl}^2} \\int_{\\sigma_\\ast}^{\\sigma_{\\rm end}} \\frac{U(\\sigma)}{U_\\sigma (\\sigma)} d\\sigma,\n\\end{equation}\nwhere $U(\\sigma)$ and $U_\\sigma (\\sigma)$ are the potential of the\ncurvaton and the derivative with respect to $\\sigma$, respectively.\nHere $\\sigma_{\\rm end}$ is the value of $\\sigma$ at the end of the\nsecond inflation. Here we adopt the potential \\eqref{curvpotential}\nand hence $N_2$ is given by\n\\begin{equation}\nN_2 = \\frac{1}{4 M_{\\rm pl}^2} (\\sigma_\\ast^2 - \\sigma^2_{\\rm end} ),\n\\end{equation}\nfrom which we can obtain \\cite{Ichikawa:2008iq}\n\\begin{equation}\n\\label{eq:zeta_inf_cur}\n\\zeta^{(\\sigma)}_I = \\frac{\\sigma_\\ast}{2M_{\\rm pl}^2} \\delta \\sigma_\\ast,\n\\end{equation}\nwith $\\delta \\sigma_\\ast = H_\\ast \/(2\\pi)$. The condition where\nfluctuations from the curvaton are negligible can be written as\n\\begin{equation}\n\\frac{\\zeta^{(\\sigma)}_I}{\\zeta^{(\\phi)}_I} = \\sqrt{2 \\epsilon} \\frac{\\sigma_\\ast}{M_{\\rm pl}} \\ll 1, \n\\end{equation}\nfrom which one can see that, even when $\\sigma_\\ast > M_{\\rm pl}$,\nfluctuations from a spectator can be negligible if small-field\ninflation models with very small $\\epsilon$ are assumed.\n\nOn small scales where modes exited the horizon during the second\ninflation driven by a spectator, the curvature perturbation is given\nby the same expression as for the ones generated from the inflaton\nduring the first inflationary period. Hence one can write\n\\begin{equation}\n\\zeta^{(\\sigma)}_{II} = \\frac{1}{M_{\\rm pl}^2} \\frac{U}{U_\\sigma} \\delta \\sigma_{\\ast II}\n\\simeq 3 \\frac{m_\\sigma}{M_{\\rm pl}} \\left( \\frac{\\sigma_\\ast}{10 M_{\\rm pl}} \\right)^2,\n\\end{equation}\nwith $\\delta \\sigma_{\\ast II} = H_{\\ast II} \/ 2 \\pi$ being determined\nby the Hubble parameter during the second inflation~$H_{\\ast II}$.\nThe mass for the spectator should be much smaller than $ H_\\ast (< M_{\\rm pl}) $\nsince otherwise the spectator field plays the role of the inflaton,\nand hence $\\zeta^{(\\sigma)}_{II} \\ll 10^{-5}$. Therefore primordial\nfluctuations on small scales would be much smaller than those\ngenerated on large scales from the inflaton fluctuations. This gives\nan effective cutoff in the power spectrum at some scale, which may\nhave interesting implications for small scale structure while CMB\nscale is not affected. In the next section, we assume the above kind\nof scenario in an inflaton model with very small $\\epsilon$.\n\n\n\n\\section{A concrete model and power spectrum}\\label{sec:model}\n\n\nTo compute the power spectrum in this kind of scenario explicitly, we\nalso need to specify the inflaton model. Although many inflationary\npotentials would be admissible to have a model with a consistent $n_s$\nand $r$ with current observations, here let us consider the following\ninflaton model, which is a hybrid inflation with a fractional power\nwhose potential is given by\n\\begin{equation}\n\\label{eq:V_VHI}\nV(\\phi) = V_0 \\left( 1+ \\left( \\frac{\\phi}{\\mu} \\right)^p \\right),\n\\end{equation}\nwhere $V_0$ represents the scale of inflation, which is determined by\nthe normalization condition. Here $\\mu$ and $p$ are model parameters\nwhich will be chosen to give the spectral index $n_s$ consistent with\ncurrent observations. This type of inflation model is also called\n``valley hybrid inflation (VHI)\" in \\cite{Martin:2013tda}.\n\nThe slow-roll parameters in this model are given by \n\\begin{equation}\n\\label{eq:slow-roll}\n\\epsilon = \\frac{p^2}{2} \\left( \\frac{M_{\\rm pl}}{\\mu} \\right)^2\n\\frac{ x^{2p-2}}{ (1+x^p)^2}, \\qquad \\eta = p (p-1) \\left(\n\\frac{M_{\\rm pl}}{\\mu} \\right)^2 \\frac{ x^{p-2}}{ 1+x^p},\n\\end{equation}\nwhere we have defined \n\\begin{equation}\nx \\equiv \\frac{\\phi}{\\mu}.\n\\end{equation}\nThe number of $e$-folds counted from the end of the first inflationary\nperiod is given by\n\\begin{equation}\n N = \\frac1p \\left( \\frac{\\mu}{M_{\\rm pl}} \\right)^2\n \\left[ \\frac{1}{2-p} (x_\\ast^{2-p} - x_{\\rm end}^{2-p} ) + \\frac12 (x_\\ast^2 - x_{\\rm end}^2 ) \\right].\n\\end{equation}\n\nNow we consider the case of $p <1$, which corresponds to $n_s < 1$,\nhowever when we assume $N = 50 - 60$, $n_s$ is very close to unity.\nAs described below, if the secondary inflation is driven by a\nspectator field, $N$ can be much reduced, which can lead to a value\nconsistent with current observations ($n_s \\sim 0.96$). To consider\nthe situation where the spectator does not contribute to primordial\nfluctuations but only affects the background dynamics, we require a\nvery small value for $\\epsilon$ as shown in the previous section. To\nrealize this, here we assume a large value for $\\mu$ i.e. $\\mu \\gg\nM_{\\rm pl}$ (for concreteness, we assume $\\mu = 10^3 M_{\\rm pl}$ in\nthe following). We also need a very small $x = \\phi\/\\mu \\ll 1$. In\nthis case, the approximate number of $e$-folds is\n\\begin{equation}\n\\label{eq:N_hybrid_fractional}\nN = \\frac{1}{p(2-p)} \\left( \\frac{\\mu}{M_{\\rm pl}} \\right)^2 x_\\ast^{2-p}.\n\\end{equation}\nWhen $x_\\ast \\ll 1$ and $p<1$, we have $\\epsilon \\ll |\\eta|$\\footnote{\nWith the value of the model parameters assumed here, we have very\nsmall $\\epsilon$ as $\\epsilon = 10^{-6} - 10^{-5}$.\n}, which, with the help of Eq.~\\eqref{eq:slow-roll} gives\nthe spectral index $n_s$ as \n\\begin{equation}\nn_s \\simeq 1 + 2 \\eta \\simeq 1 - \\frac{2(1-p)}{(2-p)N}.\n\\end{equation}\nIf we take $p=0.8$, the spectral index is $n_s \\simeq 0.993$ for\n$N=50$, which is outside the region allowed by Planck. However, when the curvaton generates a secondary inflation to\nmake $N$ much smaller, say $N=10$, the spectral index becomes $n_s\n\\simeq 0.966$, which gives a good fit to the current Planck data.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n \\resizebox{160mm}{!}{\n \\includegraphics{P_primordial.eps} \\hspace{5mm}\n \\includegraphics{matter_power.eps}\n }\n\\end{center}\n\\caption{Primordial power spectrum (left) and linear matter power\n spectrum (right) for a spectator model with VHI model whose\n potential is given by Eq.~\\eqref{eq:V_VHI}. Here we assume $p =\n 0.84$. For the primordial power spectrum, there are two\n characteristic damping scales: $k_2$ and $k_{\\rm end}$, which\n correspond to the modes that crossed the horizon at the time of the\n start of the second inflation and the end of the first inflation,\n respectively. In the matter power spectrum, the additional effect of\n oscillatory damping of the transfer function is visible. Depending\n on model parameters, the matter power spectrum can be either\n enhanced or suppressed at intermediate scales.}\n\\label{fig:power_spectrum}\n\\end{figure}\n\n\nIn Fig.~\\ref{fig:power_spectrum}, we show the primordial power\nspectrum (left) and matter power spectrum (right) for the case of\n$p=0.84$ in the VHI inflation model, in which the CMB scale\ncorresponds to the modes exited the horizon at $N\\sim 8$ during the\nfirst inflationary period. For comparison, the power spectrum is also\nshown for the case where the power-law form $P_{\\zeta} \\propto k^{n_s\n - 1}$ is assumed. As can be understood from\nEq.~\\eqref{eq:zeta_phi1} in the previous section, the curvature\nperturbations are generally suppressed on small scales where the modes\nexit the horizon close to the end of inflation, since $V_\\phi$ is\ngetting larger. As discussed in the previous section, in our\nscenario, small scale fluctuations of the modes which exited the\nhorizon during the second inflationary period are much smaller that\nthose on larger scales generated during the first inflation. Therefore\nwe can see the cutoff of the primordial power spectrum at around\n$k_{\\rm end} \\sim 10^2 ~{\\rm Mpc}^{-1}$ which corresponds to the mode\nwhich exited the horizon at the end of the first inflationary\nperiod. This can be clearly seen in the left panel of\nFig.~\\ref{fig:power_spectrum}, which gives interesting implications to\nthe tension between CMB and subgalactic scales.\n\n\nTo discuss the implication of our scenario for the small scale\nstructure, we calculate the matter power spectrum in the present\nUniverse. In the right panel of Fig~\\ref{fig:power_spectrum}, we plot\nthe matter power spectrum at $z=0$ for the same model. To\ncalculate the matter power spectrum, we need to incorporate the\neffects of the evolution of fluctuations after the modes crossed the\nhorizon. In particular, there are two periods of inflation in our\nmodel, which is quite similar to the case of thermal inflation where a\nmini-inflation occurs after the first inflation driven by the\ninflaton. The transfer function in thermal inflation model has been\ninvestigated in \\cite{Hong:2015oqa}, where an analytic formula is\nprovided as\n\\begin{align}\n\\label{eq:transfer}\nT (k) =& \n\\cos \\left[ \\left( \\frac{k}{k_2} \\right) \\int_0^{\\infty} \\frac{d\\alpha}{\\sqrt{\\alpha ( 2 + \\alpha^3)}} \\right] \\notag \\\\\n&\n\\qquad\\qquad\n+ 6 \\left( \\frac{k}{k_2} \\right) \\int_0^{\\infty} \\int^\\gamma_0 d\\beta \\left( \\frac{\\beta}{2 + \\beta^3} \\right)^{3\/2} \n\\sin \\left[ \\left( \\frac{k}{k_2} \\right) \\int_\\gamma^{\\infty} \\frac{d\\alpha}{\\sqrt{\\alpha ( 2 + \\alpha^3)}} \\right] \\,.\n\\end{align}\nHere $k_2$ is the wave number which corresponds to the mode which\n``touched\" the horizon at the start of the second inflation ($k_2$ is\ndenoted as $k_b$ in \\cite{Hong:2015oqa}). \nThe above transfer function\nexhibits oscillatory damping at scales smaller than $k_2$ which can be\nrelated to $k_{\\rm end}$, the mode which exited the horizon at the end\nof the first inflation once we fix the background evolution. Since $k\n= aH$ holds at the time of the horizon crossing, one has\n\\begin{equation}\n\\frac{k_2}{k_{\\rm end}} = \\frac{a_2}{a_{\\rm end}} \\frac{H_2}{H_{\\rm end}},\n\\end{equation}\nwhere a subscript ``2\" denotes that the quantity is evaluated at the\nbeginning of the second inflation. \nIn Fig.~\\ref{fig:scale}, a schematic figure of the horizon crossings for two characteristic scales $k_2$ and $k_{\\rm end}$ is shown.\nAssuming that the Universe is\n$\\phi$ oscillation-dominated\\footnote{\nDepending on the decay rate of the inflaton, the Universe may have\nbecome radiation-dominated before the second inflation\nstarts. However, we assume that $\\phi$ oscillation-domination in the\nfollowing.\n}, in which $H \\propto a^{-3\/2}$, we obtain \n\\begin{equation}\nk_2 = k_{\\rm end} \\left( \\frac{H_2}{H_{\\rm end}} \\right)^{1\/3}.\n\\end{equation}\n\nAssuming that the Hubble parameter during inflation does not\n change much and hence we can approximate the Hubble parameter at the\n end of the first inflation as $H_{\\rm end}^2 \\simeq H_\\ast^2 \\simeq\n 1.6 \\times 10^{-7} \\epsilon M_{\\rm pl}^2$ as mentioned just above\n Eq.~\\eqref{eq:sigma_ast}. The Hubble rate at the beginning of the\n second inflation can be written as $H_2^2 \\simeq m_\\sigma^2\n \\sigma_\\ast^2 \/(3 M_{\\rm pl}^2)$, from which one has\n\\begin{equation}\nk_2 \\sim 0.01 ~k_{\\rm end} \\left( \\frac{m_\\sigma}{10^6~{\\rm GeV}} \\right)^{1\/3} \\left( \\frac{\\sigma_\\ast}{10 M_{\\rm pl}} \\right)^{1\/3} \\left( \\frac{10^{-5}}{\\epsilon} \\right)^{1\/6}.\n\\end{equation}\nFor the case of $p=0.84$ depicted in Fig.~\\ref{fig:power_spectrum},\nthe damping scale corresponding to the end of the first inflation is\ngiven by $k_{\\rm end} \\sim 10^{2}~{\\rm Mpc}^{-1}$, and hence the\ndamping scale in the transfer function is estimated as $k_2 \\sim {\\cal\n O}(1)~{\\rm Mpc}^{-1}$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\vspace{10mm}\n \\resizebox{100mm}{!}{\n \\includegraphics{scales.eps}\n }\n \n\\end{center}\n\\caption{Schematic figure of corresponding scales during the two\n periods of inflation, the final radiation dominated era, and the\n (possible) intermediate radiation (or matter-like $\\phi$\n oscillation) dominated phase.}\n\\label{fig:scale}\n\\end{figure}\n\n\nIn the large and small scale limits ($k \\ll k_2$ and $k \\gg k_2$,\nrespectively), Eq.~\\eqref{eq:transfer} gives the transfer function\n\\cite{Hong:2015oqa}\n\\begin{equation}\nT(k) \\rightarrow\n\\begin{cases}\n1 & (k \\ll k_2), \\\\ \\\\\n-\\displaystyle\\frac15 \\cos \\left( \\nu_1 \\frac{k}{k_2} \\right) & (k \\gg k_2),\n\\end{cases}\n\\end{equation}\nwhere $ \\nu_1 \\simeq 2.2258 $. Cosmological N-body simulations with\nthe above transfer function have been studied by \\cite{Leo:2018kxp}\nfor the thermal inflation model. It should be noted that, in our\nmodel, the primordial power spectrum also gives the suppression on\nsmall scales due to the mechanism discussed in the previous section,\nand hence the power spectrum after the second inflation is given by\n\\begin{equation}\n\\left. {\\cal P}(k) \\right|_{\\rm after~2nd~inf.} = T^2(k) {\\cal P}_\\zeta (k), \n\\end{equation}\nwhere $T(k)$ is given by Eq.~\\eqref{eq:transfer} and ${\\cal P}_\\zeta\n(k)$ can be calculated by Eq.~\\eqref{eq:P_prim_inflaton} when only the\ninflaton contributes to the primordial curvature perturbation.\n\nWe input $ \\left. {\\cal P}(k) \\right|_{\\rm after~2nd~inf.} $ to the\npublic code {\\tt CAMB} \\cite{Lewis:1999bs} to obtain the matter power\nspectrum at late time, which is shown in the right panel of\nFig.~\\ref{fig:power_spectrum}. As discussed above, there are two\ndamping scales in the model, $k_2$ and $k_{\\rm end}$, corresponding to\nthe modes which exited the horizon at the end of the first inflation,\nand those which ``touched\" the horizon at the start of the second\ninflation. This may give interesting consequences for structure\nformation on subgalactic scales.\n\n\n\n\\section{The power spectrum at small scales}\\label{sec:small}\n\n\\begin{figure}[htbp]\n\\begin{center}\n \\resizebox{140mm}{!}{\n \\includegraphics{Pk_ratio.eps}\n }\n \n\\end{center}\n\\caption{Ratio of the linear matter power spectra at $z=0$ between\n either spectator VHI models (solid lines) or warm dark matter models\n (dashed lines) relative to to the plain $\\Lambda$CDM model. The top\n $x$-axis shows the equivalent mass, $M$ of a sharp $k$-space filter,\n indicative of the halo mass corresponding to a given length scale\n $k$.}\n\\label{fig:wdm_comparison}\n\\end{figure}\n\nAlternatives to, or extensions of, the plain $\\Lambda$CDM model are\nmost easily distinguished at small scales. Whereas plain $\\Lambda$CDM\npredicts bottom-up hierarchical structure formation after decoupling\non all observable scales, alternatives deviate in several\ncharacteristic ways. For example, the free-streaming motions of\nrelativistic ``warm'' dark matter particles, such as sterile\nneutrinos, dampen and erase initial perturbations, greatly reducing\nthe number of structures below a certain ``free streaming mass''\n\\cite{Bode:2000gq}. They also require that lower-mass halos and the\ngalaxies within them only form ``top down'' from the fragmentation of\nmore massive objects. Models where dark matter is coupled to photons\nin the early Universe erase structures similar to warm dark matter,\nbut with some resonances at particular scales\n(e.g. \\cite{Boehm:2001hm, Schewtschenko:2015rno}). Models where dark\nmatter is self-interacting can change the inner structure of dark\nmatter halos, as well as lower the abundance of satellites\n\\cite{Vogelsberger:2012ku, Vogelsberger:2018bok}.\n\nThe most sensitive observations of small scale structures are the\nabundance of present-day Local Group dwarf galaxies in halos of $\\sim\n10^{8}-10^{9.5}\\mathrm{M}_\\odot$, and structures observed in the Lyman-$\\alpha$\nforest at redshifts $z= 3-4$. Based on Lyman-$\\alpha$ forest\nmeasurements, warm dark matter models with thermal relics of less than\n$3.5~{\\rm keV}$ can be excluded at 99\\% confidence level\n\\cite{Viel:2013apy}, but in the presence of Lepton asymmetry, the\ncorrespondence between particle mass and structure damping differs\n\\cite{Shi:1998km}. However, the effects of WDM on structure formation\nare similar in all models, and it is convenient to parameterise them\nby the equivalent thermal relic mass.\n\nFor Local Group dwarf galaxies, several caveats exist: while a simple\ncomparison between the number of observed satellite galaxies, and the\nnumber of halos appears to strongly disfavour plain $\\Lambda$CDM,\nastrophysical processes are understood to prevent the formation of\ndwarf galaxies in low-mass halos, and also reduced the abundance of\nlow mass halos compared to collisionless simulations\n(e.g. \\cite{Sawala:2012cn}). Furthermore, the uncertain mass of the\nMilky Way and of the Local Group has a significant effect on the\nexpected number of satellite halos. Indeed, for a massive Milky Way,\nplain $\\Lambda$CDM is strongly disfavoured, while for a low mass Milky\nWay, many WDM models can be excluded \\cite{Kennedy:2013uta}. In\nsimulations with Local Group analogues in the allowed mass range, it\nappears that moderate WDM models are slightly favoured over CDM, but\nobservations of dwarf galaxies are not discriminant enough to\ndistinguish them \\cite{Lovell:2016fec, Newton:2018izu}. A clearer\ndistinction may be possible based on the discovery or non-detection of\neven lower mass dark matter halos, e.g. through their perturbations of\nstellar streams, Milky Way halo stars, or via gravitational\nlensing. While CDM predicts thousands of subhalos in the\n$10^6-10^8\\mathrm{M}_\\odot$ range surrounding the MW, most WDM models predict very\nfew \\cite{Sawala:2016tlo, Bose:2016irl}.\n\nA modification of the initial power spectrum could act similar to warm\ndark matter at the dwarf galaxy scale in the present Universe. In\nFig.~\\ref{fig:wdm_comparison}, the ratio of the linear matter power\nspectra at $z=0$ between the spectator VHI model and the plain\n$\\Lambda$CDM model, $P(k) \/ P(k)_{\\rm CDM}$ is shown. In both models,\nthe spectral index at the reference scale of $k=0.05~{\\rm Mpc}^{-1}$\nis taken as $n_s = 0.9645$ and other cosmological parameters are\nassumed as $\\Omega_b h^2 = 0.02225, \\Omega_c h^2 = 0.1198, h=0.6727$\nand $A_s =2.2 \\times 10^{-9}$ \\cite{Ade:2015xua}, where $\\Omega_b$ and $\\Omega_c$ are\ndensity parameters for baryon and CDM, $h$ is the Hubble parameter and\n$A_s$ is the amplitude of primordial power spectrum at the reference\nscale. For the spectator VHI model, we have chosen the model\nparameters, $\\mu$ and $V_0$, for a fixed value of $p$ in such a way\nthat we obtain the above values of $n_s$ and $A_s$ at the reference\nscale. By fixing the model parameters in this way, the number of\n$e$-folds corresponding to the mode $k=0.05~{\\rm Mpc}^{-1}$ are $N =\n10.2, 9.4, 7.6$ and $7.8$ for $p=0.78, 0.8, 0.82$ and $0.84$,\nrespectively. For guidance, as dashed lines, we also include several\nWDM models, assuming thermal relics of masses in the range\n$1-10$~keV. It can be seen that, compared to WDM models, the power\nspectrum in the spectator model begins to deviate at significantly\nlarger scales.\n\nDepending on the model parameters, the reduction of the primordial\npower, and the modification of the transfer function, discussed in\nSection~\\ref{sec:model}, can result in either a reduction or boost of\nthe linear matter power spectrum on galactic and supergalactic\nscales. Below the cut-off scale $k_{\\rm end}$, there is a sharp\nsuppression, more rapid than in the WDM case. Defining the ``filtering\nmass'' as the scale where the abundance of halos is suppressed by half\nrelative to the standard model \\cite{Bode:2000gq}, we see that the\nspectator-VHI model with $p \\sim 0.82$ has a similar filtering scale\nto a WDM model with $m_{WDM} \\sim 2$ keV, near the lower limits\nallowed by Lyman-$\\alpha$ observations \\cite{Viel:2013apy}.\n\nThe scale-dependence of the WDM or VHI models relative to the standard\nmodel, however, is quite different: We see that the WDM models change\nthe power by less than $1\\%$ only half an order of magnitude above the\nfiltering scale, while in the spectator model, there is an even weaker\nsuppression on large scales, followed by a boost of $\\sim 1\/3$ on\nintermediate scales. Models based on observed stellar kinematics,\nabundance matching, and direct hydrodynamic cosmological simulations,\nplace the lowest-mass dwarf galaxies in dark matter halos around\n$10^{8}-10^9\\mathrm{M}_\\odot$. While WDM solutions to the postulated plain\n$\\Lambda$CDM failures in this regime would affect only a very narrow\nmass range, the signature of the spectator model would lead to a weak\nchange in galaxy abundance over a range of masses and affect different\nscales in the Lyman-$\\alpha$ forest. At the very low mass end, just\nbelow the filtering scale, the spectator model predicts a very sharp\ndecline in power. For a filtering mass of $\\sim 10^9\\mathrm{M}_\\odot$\n(corresponding to the peak mass of typical Milky Way dwarf\nspheroidals), the abundance of substructures below $10^7\\mathrm{M}_\\odot$ is\ngreatly reduced in the WDM model, but for the spectator model, such\nsubstructures are non-existent. In this regime, where structures can\nonly be detected indirectly, even a very small number of detections\ncould therefore place significant constraints on spectator models.\n\nIn combination, this makes the spectator model clearly\nfalsifiable. Quantitative predictions will require full, cosmological\nand hydrodynamic simulations, and will be the subject of future work.\n\n\\section{Conclusion and Discussion} \\label{sec:summary}\n\nWe have argued that, in models with a spectator field in the framework\nof a small-field inflation, after the inflaton-driven\nexpansion there generically arises a second inflationary epoch which\nis driven by the spectator field. Depending on model parameters, the\nsecond inflation can generate the large number of $e$-folds, possibly\neven $N \\sim 40-50$. In this case, CMB scale fluctuations correspond\nto the modes which exited the horizon at $N \\sim 10$ when counted from\nthe end of the first inflation. Let us recall that this is a very\nsmall number compared to the standard scenario where $N \\sim 50 -60$.\nGalactic scales correspond to the perturbations exiting around the\nvery end of the first inflation.\n\nAs we discussed in Section~\\ref{sec:spectator}, in the case of\nsmall-field inflation, the spectator fluctuations tend to give a\nnegligible contribution. As a result, primordial power spectrum on\nsmall scales can be much suppressed compared to the standard plain\n$\\Lambda$CDM model while on large scales, the prediction can be\nconsistent with observations of CMB. This is demonstrated in\nFig.~\\ref{fig:wdm_comparison}. Hence a model of the type with two\nperiods of inflation would give interesting implications for the\ntension between small and large scale structure such as is manifest in\n``missing satellites\" and ``too big to fail\" problems. As baryonic\nphysics strongly regulate galaxy formation on these scales, a\nquantitative investigation of this issue will require cosmological\nhydrodynamic simulations, which are left for future work.\n\nRecent weak lensing results from KiDS \\cite{Joudaki:2017zdt} continue\nto yield $\\sigma_8$ smaller than the value implied by the Planck data\nby $2$ to $3 \\sigma$. There are extensions of the plain $\\Lambda$CDM\nmodel devised to alleviate the apparent tension; these include\nallowing for the curvature of the Universe, adopting dark energy\nmodels with a time-varying equation of state, modifying general\nrelativity, assuming decaying dark matter (see e.g.,\n\\cite{Joudaki:2017zdt} for a recent discussion).\n\nWhile the linear matter power spectrum shown in\nFig.~\\ref{fig:wdm_comparison} also suggests a large-scale effect of\nthe modified transfer function, at low redshifts, this is likely to be\nwashed out due to mode coupling in the full, non-linear\nevolution. More promising are observations of the 21cm forest around\nthe time of reionisation, which could be able to either detect or rule\nout the characteristic bump in the power spectrum resulting from our\nmodel. With SKA promising to make these measurements during the next\ndecade \\cite{Koopmans:2015sua}, more detailed numerical studies of the\nnon-linear evolution in different inflation models seem to be\nparticularly timely.\n\n\\section*{Acknowledgments}\nThe authors also thank Baojiu Li, Matteo Leo, Mark Lovell and Kasper\nSiilin for helpful comments and suggestions. T.T. would like to thank\nthe Helsinki Institute of Physics for the hospitality during the\nvisit, where this work was done. T.S. is an Academy of Finland\nResearch Fellow and supported by grant number 314238. T.T is supported\nby JSPS KAKENHI Grant Number 15K05084, 17H01131 and MEXT KAKENHI Grant\nNumber 15H05888.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nA {\\it weighted oriented graph\\\/} $D$ is a triplet $(G,\\mathcal{O},w)$, where $G$ is a simple graph whose vertex set is $V(G)$; $\\mathcal{O}$ is an edge orientation of $G$ (an assignment of a direction to each edge of $G$); and $w$ is a function $w:V(G) \\to\\mathbb{N}$. In this case, $G$ is called the {\\it underlying graph\\\/} of $D$. The vertex set of $D$ is $V(D):=V(G)$, the edge set of $D$, denoted by $E(D)$, is the set of oriented edges of the oriented graph $(G,\\mathcal{O})$. The \\emph{weight} of $x\\in V(D)$ is $w(x)$ and we denote the set $\\{x\\in V(D)\\mid w(x)>1\\}$ by $V^{+}$. If $R=K[x_{1},\\ldots, x_{n}]$ is a polynomial ring over a field $K$, then the edge ideal of $D$ is $I(D)=(x_{i}x_{j}^{w(x_{j})}\\mid (x_i,x_j)\\in E(D))$ where $V(D)=\\{ x_1,\\ldots ,x_n\\}$. These ideals (introduced in \\cite{WOG}) generalize the usual edge ideals of graphs (see \\cite{Villa}), since if $w(x)=1$ for each $x\\in V(D)$, then $I(D)$ is the edge ideal of $G$, i.e. $I(D)=I(G)$. An interest in $I(D)$ comes from coding theory in some studies of Reed--Muller typed codes, (see \\cite{Carvalho,pit-villa}). Furthermore, some algebraic and combinatorial invariants and proper\\-ties of $I(D)$ have been studied in some papers \\cite{vivares,reyes-villa,WOG,V-R-P,Zhu}. In particular, in \\cite{WOG} is given a characterization of the irredundant irreducible decomposition of $I(D)$. This characterization permits studying when $I(D)$ is unmixed, using the strong vertex covers (Definition \\ref{strong-cover-defn} and Theorem \\ref{theorem42}). The unmixed property of $I(D)$ have been studied when $G$ is one of the following graphs: cycles in \\cite{WOG}; graphs with whiskers in \\cite{vivares,WOG}; bipartite graphs in \\cite{reyes-villa,WOG}; graphs without $3$- $5$- and $7$-cycles and K\\\"onig graphs in \\cite{V-R-P}.\n\n\\noindent\nIn this paper, we study the unmixed property of $I(D)$ for some families of weighted oriented graphs. In Section 2, we give the known results and definitions that we will use in the following sections. In Section 3 (in Theorem \\ref{theorem-oct29}), we characterize when a subset of $V(D)$ is contained in a strong vertex cover. Using this result, we characterize the unmixed property of $I(D)$, when $G$ is a perfect graph (see Theorem \\ref{Perf-Unm}). In Section 4 (in Theorem \\ref{SCQ-char}), we characterize the unmixed property of $I(D)$ when $G$ is an $SCQ$ graph (see Definition \\ref{CondSCQ}). These graphs generalize the graph defined in \\cite{Rand} and in the context of this paper, they are important because if $G$ is well--covered such that $G$ is simplicial, $G$ is chordal or $G$ has no some small cycles, then $G$ is an $SCQ$ graph (see Remark \\ref{rem-nov13} and Theorem \\ref{wellcovered-characterization2}). In \\cite{SCQ-ivan}, using the $SCQ$ graphs the authors characterize the vertex decomposable property of $G$ when each $5$-cycle of $G$ has at least $4$ chords. Also, in Section 4, we characterize the unmixed property of $I(D)$ when $G$ is K\\\"oning, $G$ is simplicial or $G$ is chordal (see Corollaries \\ref{Koning-Unm} and \\ref{Simp-Chor-Unm}). In Section 5, we characterize the unmixed property of $I(D)$, when $G$ has no $3$- or $5$-cycles; or $G$ has no $4$- or $5$-cycles; or $G$ has girth greater than 4 (see Theorems \\ref{theorem-oct31}, \\ref{No4,5Cyc-Unmix} and \\ref{No3,4,5Cyc-Unmix}). Finally, in Section 6, we give some examples. Our results generalize the results about the unmixed property of $I(D)$ given in \\cite{vivares,reyes-villa,WOG,V-R-P}, since if $G$ is well-covered and $G$ is one of the following graphs: cycles, graphs with whiskers, bipartite graphs, K\\\"oning graphs, or graphs without $3$-, $5$- and $7$-cycles, then $G$ is an $SCQ$ graph.\n\n\\section{Preliminaries}\nIn this Section, we give some definitions and well-known results that we will use in the following sections. Let $D=(G,\\mathcal{O}, w)$ be a weighted oriented graph, recall that $V^{+}=\\{ x\\in V(D) \\mid w(x)>1\\}$ and $I(D)=\\big( x_ix_{j}^{w(x_j)} \\mid (x_i,x_j)\\in E(D)\\big)$.\n\n\\begin{definition}\\rm\nLet $x$ be a vertex of $D$, the sets \n$$N_{D}^{+}(x):=\\{y\\mid(x,y)\\in E(D)\\} \\quad {\\rm and} \\quad N_{D}^{-}(x):=\\{y\\mid(y,x)\\in E(D)\\}$$ are called the {\\it out-neighbourhood\\\/} and the {\\it in-neighbourhood\\\/} of $x$, respectively. The {\\it neighbourhood\\\/} of $x$ is the set $N_{D}(x):=N_{D}^{+}(x)\\cup N_{D}^{-}(x)$. Furthermore, $N_D[x]:=N_D(x)\\cup \\{ x\\}$. Also, if $A\\subseteq V(D)$ then $N_D(A):=\\{ b\\in V(D)\\mid b\\in N_D(a) \\ {\\rm for \\ some} \\ a\\in A\\}$.\n\\end{definition}\n\n\\begin{definition}\\rm\\label{Sink-source}\nLet $x$ be a vertex of $D$. If $N_{D}^{+}(x)=\\emptyset$, then $x$ is called a {\\it sink\\\/}. On the other hand, $x$ is a {\\it source\\\/} if $N_{D}^{-}(x)=\\emptyset$.\n\\end{definition}\n\n\\begin{remark}\\rm\\label{rem-V-R-P}\nConsider the weighted oriented graph $\\tilde{D}=(G,\\mathcal{O},\\tilde{w})$ with $\\tilde{w}(x)=1$ if $x$ is a source and $\\tilde{w}(x)=w(x)$ if $x$ is not a source. Hence, $I(\\tilde{D})=I(D)$. Therefore, in this paper, we assume that if $x$ is a source, then $w(x)=1$. \n\\end{remark}\n\n\\begin{definition}\\rm\nThe {\\it degree of $x\\in V(D)$\\\/} is $deg_G(x):=|N_D(x)|$ and $N_G(x):=N_D(x)$. \t \n\\end{definition}\n\n\\begin{definition}\\rm\nA {\\it vertex cover\\\/} $\\mathcal{C}$ of $D$ (resp. of $G$) is a subset of $V(D)$ (resp. of $V(G)$), such that if $(x,y)\\in E(D)$ (resp. $\\{ x,y\\} \\in E(G)$), then $x\\in \\mathcal{C}$ or $y\\in \\mathcal{C}$. A vertex cover $\\mathcal{C}$ of $D$ is {\\it minimal\\\/} if each proper subset of $\\mathcal{C}$ is not a vertex cover of $D$.\n\\end{definition}\n\n\\begin{remark}\\rm\\label{Compl-einC}\nLet $\\mathcal{C}$ be a vertex cover of $D$ and $e\\in E(G)$. Then, $\\mathcal{C} \\cap e\\neq \\emptyset$. Furthermore, $e\\cap (\\mathcal{C}\\setminus a)\\neq \\emptyset$ if $a\\notin e$, $b\\in N_D(a)$ and $e=\\{ a,b\\}$. Hence, $(\\mathcal{C} \\setminus a)\\cup N_D(a)$ is a vertex cover of $D$.\n\\end{remark}\n\n\\begin{definition}\\rm\\label{L-sets}\nLet $\\mathcal{C}$ be a vertex cover of $D$, we define the following three sets: \n\\begin{itemize}[noitemsep]\n\\item $L_1(\\mathcal{C}):=\\{x\\in \\mathcal{C}\\mid N_{D}^{+}(x)\\cap \\mathcal{C}^{c}\\neq \\emptyset \\}$ where $\\mathcal{C}^{c}=V(D)\\setminus \\mathcal{C}$,\n\\item $L_2(\\mathcal{C}):=\\{x\\in \\mathcal{C}\\mid\\mbox{$x\\notin L_1(\\mathcal{C})$ and $N^{-}_{D}(x)\\cap \\mathcal{C}^c\\neq\\emptyset$}\\}$,\n\\item $L_3(\\mathcal{C}):=\\mathcal{C}\\setminus(L_1(\\mathcal{C})\\cup L_2(\\mathcal{C}))$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{remark}\\rm\\label{einC}\nIf $\\mathcal{C}$ is a vertex cover of $G$, $x\\in V(G)\\setminus \\mathcal{C}$ and $y\\in N_G(x)$, then $e:=\\{ x,y\\} \\in E(G)$ and $e\\cap \\mathcal{C} \\neq \\emptyset$. So, $y\\in \\mathcal{C}$, since $x\\notin \\mathcal{C}$. Hence, $N_G(x)\\subseteq \\mathcal{C}$.\n\\end{remark}\n\n\\begin{remark}\\rm\\label{VertexL3}\nLet $\\mathcal{C}$ be a vertex cover of $D$, then $x\\in L_3(\\mathcal{C})$ if and only if $N_{D}[x]\\subseteq \\mathcal{C}$. Hence, $L_3(\\mathcal{C})= \\emptyset$ if and only if $\\mathcal{C}$ is minimal.\n\\end{remark}\n\n\\begin{definition}\\rm\\label{strong-cover-defn}\nA vertex cover $\\mathcal{C}$ of $D$ is {\\it strong\\\/} if for each $x\\in L_3(\\mathcal{C})$ there is $(y,x)\\in E(D)$ such that $y\\in L_2(\\mathcal{C})\\cup L_{3}(\\mathcal{C})=\\mathcal{C} \\setminus L_1(\\mathcal{C})$ with $y\\in V^{+}$ (i.e. $w(y)>1$).\n\\end{definition}\n\n\\begin{definition}\\rm\nAn ideal $I$ of a ring $R$ is {\\it unmixed\\\/} if each one of its associated primes has the same height. \n\\end{definition}\n \n\\begin{theorem}\\label{theorem42}{\\rm \\cite[Theorem 31]{WOG}}\nThe following conditions are equivalent:\n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (1)}] $I(D)$ is unmixed.\n\\item[{\\rm (2)}] Each strong vertex cover of $D$ has the same cardinality. \n\\item[{\\rm (3)}] $I(G)$ is unmixed and $L_{3}(\\mathcal{C})=\\emptyset$ for each strong vertex cover $\\mathcal{C}$ of $D$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{definition}\\rm\nThe {\\it cover number\\\/} of $G$ is $\\tau(G):={\\rm min \\ }\\{ |\\mathcal{C}|\\ \\mid \\mathcal{C} \\ {\\rm \\ is\\ a\\ vertex\\ cover\\ of\\ } G\\}$. Furthermore, a {\\it $\\tau$-reduction\\\/} of $G$ is a collection of pairwise disjoint induced subgraphs $H_1,\\ldots ,H_s$ of $G$ such that $V(G)=\\cup_{i=1}^{s} V(H_i)$ and $\\tau(G)=\\sum_{i=1}^{s} \\tau(H_i)$.\n\\end{definition}\n\n\\begin{remark}\\rm\\label{MinimalStrongProp}\nWe have $\\tau (G)=|\\mathcal{C}_1|$, for some vertex cover $\\mathcal{C}_1$. So, $\\mathcal{C}_1$ is minimal. Thus, by Remark \\ref{VertexL3}, $L_3(\\mathcal{C}_1)=\\emptyset$. Hence, $\\mathcal{C}_1$ is strong. Now, if $I(D)$ is unmixed, then by {\\rm (2)} in Theorem \\ref{theorem42}, $|\\mathcal{C}|=|\\mathcal{C}_1|=\\tau (G)$ for each strong vertex cover $\\mathcal{C}$ of $D$.\n\\end{remark}\n\n\\begin{definition}\\rm\nA {\\it stable set\\\/} of $G$ is a subset of $V(G)$ containing no edge of $G$. The {\\it stable number of $G$\\\/}, denoted by $\\beta(G)$, is $\\beta(G):={\\rm max \\ } \\{ |S|\\ \\mid S {\\rm \\ is\\ a\\ stable\\ set\\ of\\ }G\\}$. Furthermore $G$ is {\\it well--covered\\\/} if $|S|=\\beta(G)$ for each maximal stable set $S$ of $G$.\n\\end{definition}\n\n\\begin{remark}\\rm\\label{tau-beta}\n$S$ is a stable set of $G$ if and only if $V(G)\\setminus S$ is a vertex cover. Hence, $\\tau (G)=|V(G)|-\\beta (G)$.\n\\end{remark}\n\n\\begin{remark}\\rm\\label{1star}{\\rm \\cite[Remark 2.12]{V-R-P}}\n$G$ is well-covered if and only if $I(G)$ is unmixed.\n\\end{remark}\n\n\\begin{definition}\\rm\nA collection of pairwise disjoint edges of $G$ is called a {\\it matching\\\/}. A {\\it perfect matching\\\/} is a matching whose union is $V(G)$. On the other hand, $G$ is a {\\it K\\\"{o}nig graph\\\/} if $\\tau(G) =\\nu(G)$ where $\\nu(G)$ is the maximum cardinality of a matching of $G$.\n\\end{definition}\n\n\\begin{definition}\\rm\\label{defi-VPR2.14}\nLet $e$ be an edge of $G$. If $\\{a,a^{\\prime}\\}\\in E(G)$ for each pair of edges, $\\{a,b\\}$, $\\{a^{\\prime},b^{\\prime}\\}\\in E(G)$ and $e=\\{b,b^{\\prime}\\}$, then we say that $e$ {\\it has the property \\bf{(P)}\\\/}. On the other hand, we say that a matching $P$ of $G$ {\\it has the property \\bf{(P)}\\\/} if each edge of $P$ has the property \\bf{(P)}.\n\\end{definition}\n\n\\begin{theorem}\\label{Koning-Char}{\\rm \\cite[Proposition 15]{Ivan-Reyes}}\nIf $G$ is a K\\\"oning graph without isolated vertices, then $G$ is well--covered if and only if $G$ has a perfect matching with the property {\\bf(P)}.\n\\end{theorem}\n\n\\begin{definition}\\rm \n$\\mathcal{P}=(x_1,\\ldots ,x_n)$ is a {\\it walk\\\/} (resp. an {\\it oriented walk\\\/}) if $\\{ x_i,x_{i+1}\\} \\in E(G)$ for $i=1,\\ldots ,n-1$. In this case, $\\mathcal{P}$ is a {\\it path\\\/} (resp. an {\\it oriented path\\\/}) if $x_1,\\ldots ,x_n$ are different. On the other hand, a walk (resp. an oriented walk), $C=(z_1,z_2,\\ldots ,z_n,z_1)$ is a {\\it $n$-cycle\\\/} (resp. an {\\it oriented $n$-cycle\\\/}) if $(z_1,\\ldots ,z_n)$ is a path (resp. is an oriented path).\n\\end{definition}\n\n\\begin{definition}\\rm\nLet $A$ be a subset of $V(G)$, then the {\\it graph induced by $A$\\\/}, denoted by $G[A]$, is the subgraph $G_1$ of $G$ with $V(G_1)=A$ and $E(G_1)=\\{ e\\in E(G)\\mid e\\subseteq A\\}$. On the other hand, a subgraph $H$ of $G$ is {\\it induced\\\/} if there is $B\\subseteq V(G)$ such that $H=G[B]$.\n\n\\noindent\nA cycle $C$ of $G$ is {\\it induced\\\/} if $C$ is an induced subgraph of $G$.\n\\end{definition}\n\n\\begin{definition}\\rm\nA weighted oriented graph $D^{\\prime}=(G^{\\prime},\\mathcal{O}^{\\prime},w^{\\prime})$ is a {\\it weighted oriented subgraph of $D=(G,\\mathcal{O},w)$\\\/}, if $(G^{\\prime},\\mathcal{O}^{\\prime})$ is an oriented subgraph of $(G,\\mathcal{O})$ and $w^{\\prime}(x)=w(x)$ for each $x\\in V(G^{\\prime})$. Furthermore, $D^{\\prime}$ is an {\\it induced weighted oriented subgraph of $D$\\\/} if $G^{\\prime}$ is an induced subgraph of $G$.\n\\end{definition}\n\n\\begin{definition}\\rm\nA vertex $v$ is called {\\it simplicial} if the induced subgraph $H=G[N_{G}[v]]$ is a complete graph with $k=|V(H)|-1$, in this case, $H$ is called $k$-{\\it simplex} (or {\\it simplex}). The set of simplexes of $G$ is denoted by $S_G$. $G$ is a {\\it simplicial graph\\\/} if every vertex of $G$ is a simplicial vertex of $G$ or is adjacent to a simplicial vertex of $G$. \n\\end{definition}\n\n\\begin{definition}\\rm\nThe minimum length of a cycle (contained) in a graph $G$, is called the {\\it girth of $G$\\\/}. On the other hand, $G$ is a {\\it chordal graph\\\/} if the induced cycles are $3$-cycles.\n\\end{definition}\n\n\\begin{theorem}{\\rm \\cite[Theorems 1 and 2]{Prisner}}\\label{T1-T2Prisner}\nIf $G$ is a chordal or simplicial graph, then $G$ is well-covered if and only if every vertex of $G$ belongs to exactly one simplex of $G$. \n\\end{theorem}\n\n\\begin{definition}\\rm\\label{CondSCQ}\nAn induced $5$-cycle $C$ of $G$ is called {\\it basic} if $C$ does not contain two adjacent vertices of degree three or more in $G$. $G$ is an $SCQ$ graph (or $G\\in SCQ$) if $G$ satisfies the following conditions:\n\\begin{enumerate}[noitemsep]\n\\item[$(i)$] There is $Q_G$ such that $Q_G=\\emptyset$ or $Q_G$ is a matching of $G$ with the property {\\bf{(P)}}.\n\\item[$(ii)$] $\\{V(H)\\mid H\\in S_G\\cup C_G\\cup Q_G\\}$ is a partition of $V(G)$, where $C_G$ is the set of basic $5$-cycles. \n\\end{enumerate}\n\\end{definition}\n\n\\noindent\nIn the following three results, we use the graphs of Figure \\ref{specialgraphs}.\n\n\\begin{figure}[h]\n\\centering\n\\begin{tikzpicture}[dot\/.style={draw,fill,circle,inner sep=1pt},scale=.82] \n\n\n \\foreach \\l [count=\\n] in {{},{},{},{},{},{},{}} {\n \\pgfmathsetmacro\\angle{90-360\/7*(\\n-1)}\n \\node[dot,label={\\angle:$\\l$}] (n\\n) at (\\angle:1) {};\n }\n \\draw (n6) -- (n7) -- (n1) -- (n2) -- (n3) -- (n4)--(n5)--(n6);\n \\node (0) at (0,-1.5){$\\mathbf{C_{7}}$};\n\n\n\\node[draw,fill,circle,inner sep=1pt] (2) at (-5.8,.9){};\n\\node (12) at (-5.6,1.1){{\\small $d_{\\tiny 1}$}};\n\\node[draw,fill,circle,inner sep=1pt] (5) at (-5.8,-.5) {};\n\\node (15) at (-5.9,-.8){{\\small $d_{\\tiny 2}$}};\n\\node[draw,fill,circle,inner sep=1pt](3) at (-4.4,.9) {};\n\\node (13) at (-4.1,.9){{\\small $b_{\\tiny 2}$}};\n\\node[draw,fill,circle,inner sep=1pt] (6) at (-4.4,-.5){};\n\\node (16) at (-4.1,-.5){{\\small $a_{\\tiny 2}$}};\n\\node[draw,fill,circle,inner sep=1pt] (1) at (-7.2,.9){};\n\\node (11) at (-7.5,.9){{\\small $a_{\\tiny 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(5,-3.5){\\textcolor{blue}{\\small $C_{1}$}};\n\\node (25) at (5,-2.75){{\\small $\\mathbf{\\tilde{e}_{\\tiny 1}}$}};\n\\node (26) at (4.5,-7.75){{\\small $\\mathbf{\\tilde{e}_{\\tiny 2}}$}};\n\\node (27) at (5.5,-7.75){{\\small $\\mathbf{\\tilde{e}_{\\tiny 3}}$}};\n\n\\draw[-] (1) -- (2)-- (5)--(9)--(11)--(12)--(3)--(4)--(6)--(8)--(7)--(10)--(13)--(5)--(4);\n\\draw[-] (1) -- (3)-- (7);\n\\draw[-](8)--(9);\n\\end{tikzpicture}\n{\\bf \\caption{}\\label{specialgraphs}}\n\\end{figure}\n\n\\begin{theorem}\\label{wellcovered-characterization1}{\\rm \\cite[Theorem 1.1]{Finbow2}}\nIf $G$ is connected without $4$- and $5$-cycles, then $G$ is well-covered if and only if $G\\in \\{C_7,T_{10}\\}$ or $\\{V(H) \\mid H\\in S_G\\}$ is a partition of $V(G)$.\n\\end{theorem}\n\n\\begin{remark}\\rm\\label{rem-nov13}\nSuppose $G$ is well-covered. If $G$ is simplicial, or $G$ is chordal or $G$ is a graph without $4$- and $5$-cycles and $G\\notin \\{ C_7,T_{10}\\}$. Then, by Theorems \\ref{T1-T2Prisner} and \\ref{wellcovered-characterization1}, $\\{ V(H)\\mid H \\in S_G\\}$ is a partition of $V(G)$. Therefore, $G$ is an $SCQ$ graph with $C_G=Q_G=\\emptyset$. \n\\end{remark}\n\n\\begin{theorem}\\label{wellcovered-characterization2}{\\rm \\cite[Theorem 2 and Theorem 3]{Finbow}}\nIf $G$ is a connected graph without $3$- and $4$-cycles, then $G$ is well-covered if and only if $G\\in\\{K_1,C_7,P_{10},P_{13},P_{14},Q_{13}\\}$ or $\\{V(H) \\mid H\\in S_G \\cup C_G\\}$ is a partition of $V(G)$.\n\\end{theorem} \n\n\\begin{definition}\\rm\nThe {\\it complement\\\/} of $G$, denoted by $\\overline{G}$, is the graph with $V(\\overline{G})=V(G)$ such that for each pair of different vertices $x$ and $y$ of $D$, we have that $\\{ x,y\\} \\in E(\\overline{G})$ if and only if $\\{ x,y\\} \\notin E(G)$.\n\\end{definition}\n\n\\begin{definition}\\rm\\label{PerfectGraph}\nA {\\it $k$-colouring of $G$\\\/} is a function $c:V(G)\\rightarrow \\{ 1,2,\\ldots ,k\\}$ such that $c(u)\\neq c(v)$ if $\\{ u,v\\} \\notin E(G)$. The smallest integer $k$ such that $G$ has a $k$-colouring is called the {\\it chromatic number of $G$\\\/} and it is denoted by $\\chi (G)$. On the other hand, the {\\it clique number\\\/}, denoted by $\\omega(G)$ is the size of the largest complete subgraph of $G$. Finally, $G$ is {\\it perfect\\\/} if $\\chi (H)=\\omega (H)$ for every induced subgraph $H$ of $G$.\n\\end{definition}\n\n\\begin{remark}\\rm\\label{stableset-char}\nLet $A$ be a subset of $V(G)$, then $A$ is a stable set of $G$ if and only if $\\overline{G}[A]$ is a complete subgraph of $\\overline{G}$. Hence, $\\beta(G)=\\omega(\\overline{G})$.\n\\end{remark}\n\n\\begin{theorem}\\label{prop-Perfect}{\\rm \\cite[Theorem 5.5.3]{Diestel}}\n$G$ is perfect if and only if $\\overline{G}$ is perfect.\n\\end{theorem}\n\n\\section{Strong vertex cover and $\\star$-semi-forest}\nLet $D=(G,\\mathcal{O},w)$ be a weighted oriented graph. In this Section, we introduce the unicycle oriented subgraphs (Definition \\ref{Unicycle}), the root oriented trees (Definition \\ref{ROT}), and the $\\star$-semi-forests of $D$ (Definition \\ref{semi-forest}). With this definitions, we characterize when a subset of $V(G)$ is contained in a strong vertex cover (see Theorem \\ref{theorem-oct29}). Using this result, we characterize when $I(D)$ is unmixed if $G$ is a perfect graph (see Definition \\ref{PerfectGraph} and Theorem \\ref{Perf-Unm}). \n\n\\begin{proposition}\\label{GeneratinAStrongVC}\nIf $\\mathcal{C}$ is a vertex cover of $D$ such that $N_{D}^{+}(A)\\subseteq \\mathcal{C}$ and $A\\subseteq V^{+}$, then there is a strong vertex cover $\\mathcal{C}^{\\prime}$ of $D$, such that $N_{D}^{+}(A)\\subseteq \\mathcal{C}^{\\prime}\\subseteq \\mathcal{C}$.\n\\end{proposition}\n\\begin{proof}\nFirst, we prove that there is a vertex cover $\\mathcal{C}^{\\prime}$ such that $L_3(\\mathcal{C}^{\\prime})\\subseteq N_{D}^{+}(A)\\subseteq \\mathcal{C}^{\\prime}\\subseteq \\mathcal{C}$. We take $L:=N_{D}^{+}(A)$. If $L_{3}(\\mathcal{C})\\subseteq L$, then we take $\\mathcal{C}^{\\prime}=\\mathcal{C}$. Now, we suppose there is $a_1\\in L_{3}(\\mathcal{C})\\setminus L$, then by Remark \\ref{VertexL3}, $N_{D}[a_1]\\subseteq \\mathcal{C}$. Thus, $\\mathcal{C}_{1}=\\mathcal{C}\\setminus\\{a_1\\}$ is a vertex cover and $L\\subseteq \\mathcal{C}_1$, since $L\\subseteq \\mathcal{C}$ and $a_1\\notin L$. Now, we suppose that there are vertex covers $\\mathcal{C}_0,\\ldots, \\mathcal{C}_k$, such that $L\\subseteq \\mathcal{C}_i=\\mathcal{C}_{i-1}\\setminus \\{a_i\\}$ and $a_i\\in L_{3}(\\mathcal{C}_{i-1})\\setminus L$ for $i=1,\\ldots, k$ where $\\mathcal{C}_0=\\mathcal{C}$ and we give the following recursively process: If $L_{3}(\\mathcal{C}_k)\\subseteq L$, then we take $\\mathcal{C}^{\\prime}=\\mathcal{C}_k$. Now, if there is $a_{k+1}\\in L_{3}(\\mathcal{C}_k)\\setminus L$, then by Remark \\ref{VertexL3}, $N_D[a_{k+1}]\\subseteq \\mathcal{C}_k$. Consequently, $\\mathcal{C}_{k+1}:=\\mathcal{C}_{k}\\setminus \\{a_{k+1}\\}$ is a vertex cover. Also, $L\\subseteq \\mathcal{C}_{k+1}$, since $L\\subseteq \\mathcal{C}_k$ and $a_{k+1}\\not\\in L$. This process is finite, since $|V(D)|$ is finite. Hence, there is $m$ such that $L_3(\\mathcal{C}_m)\\subseteq L\\subseteq \\mathcal{C}_m\\subseteq \\mathcal{C}$. Therefore, we take $\\mathcal{C}^{\\prime}=\\mathcal{C}_m$. \n\n\\noindent\nNow, we prove that $\\mathcal{C}^{\\prime}$ is strong. We take $x\\in L_{3}(\\mathcal{C}^{\\prime})$, then $x\\in L=N_{D}^{+}(A)$, since $L_3(\\mathcal{C}^{\\prime})\\subseteq L$. Thus, $(y,x)\\in E(D)$ for some $y\\in A\\subseteq V^{+}$. Hence, $y\\in \\mathcal{C}^{\\prime}$, since $x\\in L_{3}(\\mathcal{C}^{\\prime})$. Also, $y\\not\\in L_{1}(\\mathcal{C}^{\\prime})$, since $N_{D}^{+}(y)\\subseteq N_{D}^{+}(A)\\subseteq \\mathcal{C}^{\\prime}$. Hence, $y\\in \\big( \\mathcal{C}^{\\prime}\\setminus L_1(\\mathcal{C}^{\\prime})\\big) \\cap V^{+}$. Therefore, $\\mathcal{C}^{\\prime}$ is strong. \\qed\n\\end{proof}\n\n\\begin{definition}\\rm\\label{Unicycle}\nIf $B$ is a weighted oriented subgraph of $D$ with exactly one cycle $C$, then $B$ is called {\\it unicycle oriented graph\\\/} when $B$ satisfies the following conditions:\n\\begin{enumerate}[noitemsep]\n\\item[$(i)$] $C$ is an oriented cycle in $B$ and there is an oriented path from $C$ to $y$ in $B$, for each $y\\in V(B)\\setminus V(C)$.\n\\item[$(ii)$] If $x\\in V(B)$ with $w(x)=1$, then $deg_B(x)=1$. \n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\\rm\\label{ROT}\nA weighted oriented subgraph $T$ of $D$ without cycles, is a {\\it root oriented tree\\\/} ({\\it ROT\\\/}) with {\\it parent\\\/} $v\\in V(T)$ when $T$ satisfies the following properties:\n\\begin{enumerate}[noitemsep]\n\\item[$(i)$] If $x\\in V(T)\\setminus \\{ v \\}$, there is an oriented path $\\mathcal{P}$ in $T$ from $v$ to $x$.\n\\item[$(ii)$] If $x\\in V(T)$ with $w(x)=1$, then $deg_T(x)=1$ and $x\\neq v$ or $V(T)=\\{ v\\}$ and $x=v$. \n\\end{enumerate} \n\\end{definition} \n\n\\begin{definition}\\rm\\label{semi-forest} \nA weighted oriented subgraph $H$ of $D$ is a {\\it $\\star$-semi-forest\\\/} if there are root oriented trees $T_1, \\ldots , T_r$ whose parents are $v_1, \\ldots , v_r$ and unicycle oriented subgraphs $B_1, \\ldots , B_s$ such that $H=\\big( \\cup_{i=1}^{r} \\ T_i \\big) \\cup \\big( \\cup _{j=1}^{s} \\ B_j \\big)$ with the following conditions:\n\\begin{enumerate}[noitemsep]\n\\item[$(i)$] $V(T_1), \\ldots , V(T_r), V(B_1), \\ldots , V(B_s)$ is a partition of $V(H)$. \n\\item[$(ii)$] There is $W=\\{ w_1, \\ldots , w_r \\} \\subseteq V(D)\\setminus V(H)$ such that $w_i \\in N_{D}(v_i)$ for $i=1, \\ldots , r$ (it is possible that $w_i=w_j$ for some $1\\leqslant i2$, then $deg_H(y_2) \\geqslant deg_{\\mathcal{P}}(y_2)=2$. If $|V(\\mathcal{P})|=2$, then $y_2 \\in V(C)$ and $deg_H (y_2)>deg_C (y_2)=2$. Therefore, $y_2 \\in \\tilde{H}$ and $x\\in N_{D}^{+}(\\tilde{H})$. \\smallskip\n\n\\noindent\nCase 2) $x\\in V(T_i)$ for some $1\\leqslant i\\leqslant r$. First, assume $x=v_i$, then there is $w_i \\in W$ such that $x\\in N_D(w_i)$. Consequently, $x\\in N_D(W_1)$ if $w_i\\in W_1$ and, by $(iii)$ of Definition \\ref{semi-forest}, $x\\in N_{D}^{+}(w_i)\\subseteq N_{D}^{+}(W_2)$ if $w_i\\in W_2$. Now, we suppose $x\\neq v_i$, then there is an oriented path $\\mathcal{L}$, from $v_i$ to $x$. Consequently, there is $y_3\\in V(\\mathcal{L})$ such that $(y_3, x)\\in E(D)$. If $y_3\\neq v_i$, then $deg_H(y_3)\\geqslant deg_{\\mathcal{L}}(y_3)=2$. Thus, $y_3\\in \\tilde{H}$ and $x\\in N_{D}^{+}(\\tilde{H})$. Finally, if $y_3=v_i$, then $deg_H(y_3)\\geqslant 1$. Hence, by Remark \\ref{SubsetH}, $y_3\\in \\tilde{H}$ and $x\\in N_{D}^{+}(\\tilde{H})$. \\qed \n\\end{proof}\n\n\\begin{remark}\\rm\\label{Relat-W-H}\nSometimes to stress the relation between $W$ and $H$ in Definition \\ref{semi-forest}, $W$ is denoted by $W^{H}$. Similarly, $W_1^{H}$ and $W_2^{H}$. If $\\{ T_1,\\ldots ,T_r\\}=\\emptyset$, then $W^{H}=W_1^{H}=W_2^{H}=\\emptyset$.\n\\end{remark}\n\n\\begin{lemma}\\label{lemma-dic3}\nLet $K$ be a weighted oriented subgraph of $D$. If $H$ is a maximal ROT in $K$ with parent $v$, or $H$ is a maximal unicycle oriented subgraph in $K$ whose cycle is $C$, then there is no $(y,x)\\in E(K)$ with $x\\in V(K)\\setminus V(H)$ and $y\\in V^{+}\\cap V(H)$.\n\\end{lemma}\n\\begin{proof}\nBy contradiction suppose there is $(y,x)\\in E(K)$ with $x\\in V(K)\\setminus V(H)$ and $y\\in V^{+}\\cap V(H)$. Thus, $H\\subsetneq H_1:=H\\cup \\{ (y,x)\\} \\subseteq K$. If $H$ is a unicycle oriented subgraph with cycle $C$ (resp. $H$ is a ROT), then there is an oriented path $\\mathcal{P}$ from $C$ (resp. from $v$) to $y$. Consequently, $\\mathcal{P}\\cup \\{ (y,x)\\}$ is an oriented path from $C$ (resp. from $v$) to $x$ in $H_1$. Furthermore, $H_1$ has exactly one cycle (resp. has no cycles), since $deg_{H_1}(x)=1$ and $V(H_1)\\setminus V(H)=\\{ x\\}$. \n\n\\noindent\nNow, we take $z\\in V(H_1)$ with $w(z)=1$, then $z=x$ or $z\\in V(H)$. We prove $deg_{H_1}(z)=1$. If $z=x$, then $deg_{H_1}(x)=1$. Now, if $z\\in V(H)$, then $z\\neq y$, since $y\\in V^{+}$. So, $deg_{H_1}(z)=deg_H(z)$, since $N_{H_1}(x)=\\{ y\\}$. If $H$ is a ROT with $V(H)=\\{ v\\}$, then $y=z=v$. A contradiction, since $w(z)=1$ and $y\\in V^{+}$. Consequently, by $(ii)$ in Definitions \\ref{Unicycle} and \\ref{ROT}, $deg_{H_1}(z)=deg_H(z)=1$. Hence, $H_1$ is a unicycle oriented subgraph with cycle $C$ (resp. is a ROT with parent $v$) of $K$. This is a contradiction, since $H\\subsetneq H_1\\subseteq K$ and $H$ is maximal. \\qed\n\\end{proof}\n\n\\begin{definition}\\rm\nLet $K$ be a weighted oriented subgraph of $D$ and $H$ a $\\star$-semi-forest of $D$. We say $H$ is a {\\it generating $\\star$-semi-forest\\\/} of $K$ if $V(K)=V(H)$.\n\\end{definition}\n\n\\begin{theorem}\\label{theorem-oct29}\nLet $K$ be an induced weighted oriented subgraph of $D$. Hence, the fo\\-llo\\-wing conditions are equivalent:\n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (1)}] There is a strong vertex cover $\\mathcal{C}$ of $D$, such that $V(K) \\subseteq\\mathcal{C}$.\n\\item[{\\rm (2)}] There is a generating $\\star$-semi-forest $H$ of $K$. \n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n${\\rm (2)} \\Rightarrow {\\rm (1)}$\nLet $\\mathcal{C}_1$ be a minimal vertex cover of $D$. By {\\rm (2)}, $K$ has a generating $\\star$-semi-forest $H$. Now, using the notation of Definition \\ref{semi-forest}, we take $\\mathcal{C}_2=\\big( \\mathcal{C}_1\\setminus W_1\\big) \\cup N_{D}(W_1)\\cup N_{D}^{+}(W_2\\cup \\tilde{H})$. By Remark \\ref{Compl-einC}, $\\mathcal{C}_2$ is a vertex cover of $D$. Since $W_1$ is a stable set, $N_D(W_1)\\cap W_1=\\emptyset$. Then, $\\mathcal{C}_2 \\cap W_1=\\emptyset$, since $N_{D}^{+}(W_2\\cup \\tilde{H})\\cap W_1=\\emptyset$. By Remark \\ref{SubsetH} and $(iii)$ in Definition \\ref{semi-forest}, $\\tilde{H}\\cup W_2\\subseteq V^{+}$. So, by Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}$ of $D$ such that $N_{D}^{+}(W_2\\cup \\tilde{H})\\subseteq \\mathcal{C} \\subseteq \\mathcal{C}_2$. Consequently, $\\mathcal{C} \\cap W_1=\\emptyset$, since $\\mathcal{C}_2 \\cap W_1=\\emptyset$. Thus, $N_D(W_1)\\subseteq \\mathcal{C}$, since $\\mathcal{C}$ is a vertex cover. Then, by Lemma \\ref{lemma-oct16}, $V(H)\\subseteq N_D(W_1)\\cup N_{D}^{+}(W_2\\cup \\tilde{H})\\subseteq \\mathcal{C}$. Hence, $V(K)\\subseteq \\mathcal{C}$, since $H$ is a generating $\\star$-semi-forest of $K$. \\medskip\n\n\\noindent\n${\\rm (1)} \\Rightarrow {\\rm (2)}$\nWe have, $\\mathcal{C}$ is a strong vertex cover such that $V(K)\\subseteq \\mathcal{C}$. If $A:=L_1(\\mathcal{C})\\cap V(K)=\\{ v_1,\\ldots , v_s\\}$, then there is $w_i\\in V(D)\\setminus \\mathcal{C}\\subseteq V(D)\\setminus V(K)$ such that $(v_i, w_i)\\in E(D)$. We take the ROT's $M_1=\\{ v_1\\} ,\\ldots ,M_s=\\{ v_s\\}$ and sets $W_{1}^{i}=\\{ w_i\\}$ and $W_{2}^{i}=\\emptyset$ for $i=1, \\ldots , s$. \n\n\\noindent\nNow, we will give a recursive process to obtain a generating $\\star$-semi-forest of $K$. For this purpose, suppose we have connected $\\star$-semi-forests $M_{s+1},\\ldots ,M_l$ of $K\\setminus A$ with subsets $W_{1}^{s+1},\\ldots ,W_{1}^l, W_{2}^{s+1},\\ldots ,W_{2}^l\\subseteq V(D)\\setminus V(K)$ and $V^{s+1},\\ldots ,V^{l}\\subseteq V(K)$ such that for each $s|V(G)|-s=\\tau (G)$, since $s=\\beta(G)$. A contradiction, by Remark \\ref{MinimalStrongProp}, since $I(D)$ is unmixed. \\medskip\n\n\\noindent\n$\\Leftarrow )$ Let $\\mathcal{C}$ be a strong vertex cover, then $\\mathcal{C}\\cap V_i$ is a vertex cover of $H_i$. So, $|\\mathcal{C} \\cap V_i|\\geqslant \\tau (H_i)=|V_i|-1$ for $i=1, \\ldots ,s$. Furthermore, by Theorem \\ref{theorem-oct29}, $V_i\\not\\subseteq \\mathcal{C}$. Consequently, $|\\mathcal{C} \\cap V_i|=|V_i|-1$. Thus, $|\\mathcal{C}|=\\sum_{i=1}^{s} \\big( |V_i|-1\\big)$, since $V_1,\\ldots ,V_s$ is a partition of $V(G)$. Therefore, by {\\rm (2)} in Theorem \\ref{theorem42}, $I(D)$ is unmixed. \\qed\n\\end{proof}\n\n\\section{Unmixedness of weighted oriented $SCQ$ graphs}\nLet $D=(G,\\mathcal{O},w)$ be a weighted oriented graph. If $P$ is a perfect matching of $G$ with the property {\\bf (P)}, then in Proposition \\ref{prop-unmixed}, we characterize when $|\\mathcal{C}\\cap e|=1$, for each strong vertex cover $\\mathcal{C}$ of $D$ and each $e\\in P$. Using Proposition \\ref{prop-unmixed} in Corollary \\ref{Koning-Unm}, we characterize when $I(D)$ is unmixed if $G$ is K\\\"oning. In Proposition \\ref{Basic5Cycle-Equ}, we characterize the basic $5$-cycles, $C$ such that $|\\mathcal{C}\\cap V(C)|=3$ for each strong vertex cover $\\mathcal{C}$ of $D$. Furthermore, in Theorem \\ref{SCQ-char}, we characterize when $I(D)$ is unmixed if $G$ is an $SCQ$ graph (see Definition \\ref{CondSCQ}). Finally, using this result we characterize the unmixed property of $I(D)$, when $G$ is simplicial or $G$ is chordal (see Corollary \\ref{Simp-Chor-Unm}).\n\n\\begin{proposition}\\label{prop-unmixed} \nLet $e$ be an edge of $G$. Hence, the following conditions are equivalent: \n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (1)}] $|\\mathcal{C}\\cap e|=1$ for each strong vertex cover $\\mathcal{C}$ of $D$. \n\\item[{\\rm (2)}] $e$ has the property {\\bf (P)} and $N_D(b)\\subseteq N_{D}^{+}(a)$ if $(a,b^{\\prime})\\in E(D)$ with $a\\in V^{+}$ and $e=\\{ b,b^{\\prime} \\}$.\n \n\\end{enumerate}\n\\end{proposition}\n\\begin{proof} \n${\\rm (1)} \\Rightarrow {\\rm (2)}$\nFirst, we show $e$ has the property {\\bf (P)}. By contradiction, suppose there are $\\{ a,b\\}, \\{ a^{\\prime},b^{\\prime} \\} \\in E(G)$ such that $\\{ a,a^{\\prime}\\} \\notin E(G)$. This implies, there is a maximal stable set $S$ such that $\\{ a,a^{\\prime}\\} \\subseteq S$. So, $\\tilde{\\mathcal{C}}=V(G)\\setminus S$ is a minimal vertex cover. Consequently, $\\tilde{\\mathcal{C}}$ is strong. Furthermore, $a,a^{\\prime} \\notin \\tilde{\\mathcal{C}}$, then $b,b^{\\prime}\\in \\tilde{\\mathcal{C}}$, since $\\{ a,b\\},\\{ a^{\\prime},b^{\\prime}\\} \\in E(G)$. A contradiction by {\\rm (1)}. Now, assume $(a,b^{\\prime})\\in E(D)$ with $a\\in V^{+}$ and $e=\\{ b,b^{\\prime}\\}$, then we will prove that $N_D(b)\\subseteq N_{D}^{+}(a)$. By contradiction, suppose there is $c\\in N_D(b)\\setminus N_{D}^{+}(a)$. We take a maximal stable set $S$ such that $b\\in S$. Thus, $\\mathcal{C}_1=V(G)\\setminus S$ is a minimal vertex cover such that $b\\notin \\mathcal{C}_1$. By Remark \\ref{Compl-einC}, $\\mathcal{C}=\\big( \\mathcal{C}_1\\setminus \\{ c \\} \\big)\\cup N_{D}(c)\\cup N_{D}^{+}(a)$ is a vertex cover. Furthermore, $c\\notin \\mathcal{C}$, since $c\\notin N_{D}^{+}(a)$. By Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}^{\\prime}$ such that $N_{D}^{+}(a)\\subseteq \\mathcal{C}^{\\prime}\\subseteq \\mathcal{C}$, since $a\\in V^{+}$. Also, $b^{\\prime}\\in N_{D}^{+}(a)\\subseteq \\mathcal{C}^{\\prime}$ and $c\\notin \\mathcal{C}^{\\prime}$, since $(a,b^{\\prime})\\in E(D)$ and $c\\notin \\mathcal{C}$. Then, $b\\in N_{D}(c)\\subseteq \\mathcal{C}^{\\prime}$. Hence, $\\{ b,b^{\\prime} \\} \\subseteq \\mathcal{C}^{\\prime}$. This is a contradiction, by {\\rm (1)}. \\medskip\n\n\\noindent\n${\\rm (2)} \\Rightarrow {\\rm (1)}$\nBy contradiction, assume there is a strong vertex cover $\\mathcal{C}$ of $D$ such that $|\\mathcal{C}\\cap e|\\neq 1$. So, $|\\mathcal{C}\\cap e|=2$, since $\\mathcal{C}$ is a vertex cover. Hence, by Theorem \\ref{theorem-oct29}, there is a generating $\\star$-semi-forest $H$ of $e$. We set $e=\\{ z,z^{\\prime}\\}$. First, assume $H$ is not connected. Then, using the Definition \\ref{semi-forest}, we have $H=M_1\\cup M_2$ where $M_1=\\{ v_1\\}$, $M_2=\\{ v_2\\}$ and $w_1,w_2\\in W$ such that $w_i\\in N_D(v_i)$ for $i=1,2$. Thus, $\\{ z,z^{\\prime}\\} =\\{ v_1,v_2\\}$ and $\\{ w_1,w_2\\} \\in E(G)$, since $e$ satisfies the property {\\bf(P)}. This implies $|W_1\\cap \\{ w_1,w_2\\}|\\leqslant 1$, since $W_1$ is a stable set. Hence, we can suppose $w_2\\in W_2$, then $w_2\\in V^{+}$ and $(w_2,z^{\\prime})\\in E(D)$. Consequently, by {\\rm (2)}, $w_1\\in N_D(z)\\subseteq N_{D}^{+}(w_2)$, then $(w_2,w_1)\\in E(D)$. Furthermore, by $(iii)$ in Definition \\ref{semi-forest}, $N_{D}^{+}(W_2)\\cap W_1=\\emptyset$, then $w_1\\in W_2$. So, $w_1\\in V^{+}$ and $(w_1,z)\\in E(D)$. By {\\rm (1)} with $a=w_1$, we have $(w_1,w_2)\\in E(D)$. A contradiction, then $H$ is connected. Thus, $H$ is a ROT with $V(H)=\\{ z,z^{\\prime}\\}$. We can suppose $v_1=z$ and $W^{H}=\\{ w_1\\}$, then $(z,z^{\\prime})\\in E(D)$, $w_1\\in N_D(z)$ and $z=v_1\\in \\tilde{H}$, since $deg_H(v_1)=1$. If $w_1\\in N_{D}^{+}(z)$, then $w_1\\in W_1$, since $z=v_1$. A contradiction, since $N_{D}^{+}(\\tilde{H})\\cap W_1=\\emptyset$. Then, $w_1\\notin N_{D}^{+}(z)$. By Remark \\ref{SubsetH}, $z=v_1\\in \\tilde{H}\\subseteq V^{+}$. Therefore, by {\\rm (1)} (taking $a=b=z$ and $b^{\\prime}=z^{\\prime}$), we have $N_D(z)\\subseteq N_{D}^{+}(z)$, since $e=\\{ z,z^{\\prime}\\}$ and $z^{\\prime}\\in N_{D}^{+}(z)$. A contradiction, since $w_1\\in N_D(z)\\setminus N_{D}^{+}(z)$. \\qed \n\\end{proof}\n\n\\begin{corollary}\\label{Koning-Unm}{\\rm \\cite[Theorem 3.4]{V-R-P}}\nLet $D=(G,\\mathcal{O},w)$ be a weighted oriented graph, where $G$ is K\\\"oning without isolated vertices. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following two conditions:\n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (a)}] G has a perfect matching $P$ with the property {\\bf (P)}.\n\\item[{\\rm (b)}] $N_D(b)\\subseteq N_{D}^{+}(a)$, when $a\\in V^{+}$, $\\{ b,b^{\\prime}\\} \\in P$ and $b^{\\prime}\\in N_{D}^{+}(a)$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof}\n$\\Rightarrow )$\nBy Theorem \\ref{theorem42}, $I(G)$ is unmixed. Thus, by Remark \\ref{1star} and Theorem \\ref{Koning-Char}, $G$ has a perfect matching $P$ with the property {\\bf (P)}. Consequently, $\\nu(G)=|P|$. Also, $\\tau(G)=\\nu(G)$, since $G$ is K\\\"oning. So, $\\tau(G)=|P|$. Now, we take a strong vertex cover $\\mathcal{C}$ of $D$ and $e\\in P$. Then, $|\\mathcal{C}\\cap e|\\geqslant 1$. Furthermore, by Remark \\ref{MinimalStrongProp}, $|\\mathcal{C}|=\\tau(G)=|P|$. Hence, $|\\mathcal{C}\\cap e|=1$, since $\\mathcal{C}=\\cup_{\\tilde{e}\\in P} \\ \\mathcal{C}\\cap \\tilde{e}$. Therefore, by Proposition \\ref{prop-unmixed}, $D$ satisfies {\\rm (b)}. \\medskip\n\n\\noindent\n$\\Leftarrow )$\nWe take a strong vertex cover $\\mathcal{C}$ of $D$. By Proposition \\ref{prop-unmixed}, $|\\mathcal{C}\\cap e|=1$ for each $e\\in P$, since $D$ satisfies {\\rm (a)} and {\\rm (b)}. This implies $|\\mathcal{C}|=|P|$, since $P$ is a perfect matching. Therefore, by $(2)$ in Theorem \\ref{theorem42}, $I(D)$ is unmixed. \\qed\n\\end{proof}\n\n\\begin{lemma}\\label{lemma-sep11}\nIf there is a basic $5$-cycle $C=(z_1, z_2, z_3, z_4, z_5, z_1)$ with $(z_{1},z_{2})$, $(z_{2},z_{3})\\in E(D)$, $z_2 \\in V^{+}$ and $C$ satisfies one of the following conditions:\n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (a)}] $(z_{3},z_{4})\\in E(D)$ with $z_{3} \\in V^{+}$.\n\\item[{\\rm (b)}] $(z_{1},z_{5})$, $(z_{5},z_{4})\\in E(D)$ with $z_{5} \\in V^{+}$. \n\\end{enumerate}\nthen there is a strong vertex cover $\\tilde{\\mathcal{C}}$ such that $|\\tilde{\\mathcal{C}}\\cap V(C)|=4$.\n\\end{lemma}\n\\begin{proof}\nWe take $\\mathcal{C}=\\big( \\mathcal{C}_0\\setminus V(C)\\big) \\cup N_D (z_1) \\cup N_{D}^{+}(z_2,x)$ where $\\mathcal{C}_0$ is a vertex cover and $x=z_3$ if $C$ satisfies {\\rm (a)} or $x=z_5$ if $C$ satisfies {\\rm (b)}. Thus, $x\\in V^{+}$. Furthermore, $z_2,z_3,z_5\\in N_D(z_1)\\cup N_{D}^{+}(z_2)$ and $z_4\\in N_{D}^{+}(z_3)$ if $C$ satisfies {\\rm (a)} or $z_4\\in N_{D}^{+}(z_5)$ if $C$ satisfies {\\rm (b)}. Hence, $\\{ z_{2}, z_{3}, z_{4}, z_{5} \\} \\subseteq N_D (z_1)\\cup N_{D}^{+}(z_2,x)$. Consequently, $\\{ z_2,z_3,z_4,z_5\\} \\subseteq \\mathcal{C}$, implying $\\mathcal{C}$ is a vertex cover, since $\\mathcal{C}_0$ is vertex cover and $N_D(z_1)\\subseteq \\mathcal{C}$. Also, $z_1 \\notin \\mathcal{C}$, since $z_1 \\notin N_D (z_1)\\cup N_{D}^{+}(z_2 , z_3)$ and $z_1\\notin N_{D}^{+}(z_5)$ if $C$ satisfies {\\rm (b)}. By Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}^{\\prime}$ such that $N_{D}^{+}(z_2,x)\\subseteq \\mathcal{C}^{\\prime}\\subseteq \\mathcal{C}$, since $\\{ z_2,x\\} \\subseteq V^{+}$. So, $z_1\\notin \\mathcal{C}^{\\prime}$, since $z_1\\notin \\mathcal{C}$. Then, by Remark \\ref{einC}, $N_D(z_1)\\subseteq \\mathcal{C}^{\\prime}$. Hence, $\\{ z_2,z_3,z_4,z_5\\} \\subseteq N_D(z_1)\\cup N_{D}^{+}(z_2,x)\\subseteq \\mathcal{C}^{\\prime}$. Therefore, $|\\mathcal{C}^{\\prime} \\cap V(C)|=4$, since $z_1\\notin \\mathcal{C}^{\\prime}$. \\qed\n\\end{proof}\n\n\\begin{definition}\\rm\nLet $C$ be an induced $5$-cycle, we say that $C$ has the {\\it $\\star$-property\\\/} if for each $(a,b)\\in E(C)$ where $a\\in V^{+}$, then $C=(a^{\\prime},a,b,b^{\\prime},c,a^{\\prime})$ with the following properties:\n\\begin{enumerate}[noitemsep]\n\\item[$(\\star .1)$] $(a^{\\prime},a)\\in E(D)$ and $w(a^{\\prime})=1$.\n\\item[$(\\star .2)$] $N_{D}^{-}(a)\\subseteq N_{D}(c)$ and $N_{D}^{-}(a)\\cap V^{+}\\subseteq N_{D}^{-}(c)$.\n\\item[$(\\star .3)$] $N_D(b^{\\prime})\\subseteq N_D(a^{\\prime}) \\cup N_{D}^{+}(a)$ and $N_{D}^{-}(b^{\\prime})\\cap V^{+}\\subseteq N_{D}^{-}(a^{\\prime})$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{lemma}\\rm\\label{NotStarCover}\nLet $C=(a_{1}^{\\prime},a_1,b_1,b_{1}^{\\prime},c_1,a_{1}^{\\prime})$ be a basic $5$-cycle of $D$, such that $(a_{1}^{\\prime},a_1)\\in E(D)$, $deg_D(a_1)\\geqslant 3$, $deg_D(c_1)\\geqslant 3$ and $w(b_1)=1$. If there is a strong vertex cover $\\mathcal{C}$ of $D$, such that $V(C)\\subseteq \\mathcal{C}$, then $C$ has no the $\\star$-property.\n\\end{lemma}\n\\begin{proof}\nBy contradiction, suppose $C$ has the $\\star$-property and there is a strong vertex cover $\\mathcal{C}$, such that $V(C)\\subseteq \\mathcal{C}$. Then, $deg_D(a_{1}^{\\prime})=deg_D(b_{1}^{\\prime})=2$, since $C$ is a basic cycle, $deg_D(a_1)\\geqslant 3$ and $deg_D(c_1)\\geqslant 3$. Hence, $a_{1}^{\\prime}, b_{1}^{\\prime} \\in L_3(\\mathcal{C})$, since $V(C)\\subseteq \\mathcal{C}$. Thus, $(c_1,a_{1}^{\\prime})\\in E(D)$ and $w(c_1)\\neq 1$, since $a_{1}^{\\prime}\\in L_3(\\mathcal{C})$, $deg_D(a_{1}^{\\prime})=2$, $(a_{1}^{\\prime},a_1)\\in E(D)$ and $\\mathcal{C}$ is strong. By ($\\star .1$) with $(a,b)=(c_1,a_{1}^{\\prime})$, we have that $(b_{1}^{\\prime},c_1)\\in E(D)$. Hence, $N_{D}^{-}(b_{1}^{\\prime})\\subseteq \\{ b_1\\}$, since $deg_D(b_{1}^{\\prime})=2$. This is a contradiction, since $b_{1}^{\\prime}\\in L_3(\\mathcal{C})$ and $w(b_1)=1$. \\qed\n\\end{proof}\n\n\\begin{proposition}\\label{Basic5Cycle-Equ}\nLet $C$ be a basic $5$-cycle, then $C$ has the $\\star$-property if and only if $|\\mathcal{C} \\cap V(C)|=3$ for each strong vertex cover $\\mathcal{C}$ of $D$.\n\\end{proposition}\n\\begin{proof}\n\\noindent\n$\\Rightarrow )$\nBy contradiction, we suppose there is a strong vertex cover $\\mathcal{C}$ such that $|\\mathcal{C}\\cap V(C)|\\geqslant 4$. Thus, there is a path $L=(d_1,d_2,d_3,d_4)\\subseteq C$ such that $V(L)\\subseteq \\mathcal{C}$. Then, $deg_{D}(d_2)=2$ or $deg_{D}(d_3)=2$, since $C$ is basic. We can suppose $deg_D (d_2)=2$, then $N_D (d_2)\\subseteq \\mathcal{C}$. This implies $b_1 :=d_2 \\in L_3 (\\mathcal{C})$. So, there is $(a_1,b_1)\\in E(D)$ with $a_1\\in \\big( \\mathcal{C} \\setminus L_{1}(\\mathcal{C})\\big) \\cap V^{+}$, since $\\mathcal{C}$ is strong. Since, $N_D(b_1)\\subseteq C$, we can set $C=(a_{1}^{\\prime},a_1,b_1,b_{1}^{\\prime},c_1,a_{1}^{\\prime})$. Consequently, $\\{ a_1,b_{1}^{\\prime}\\}=N_D(b_1)=N_D(d_2)=\\{ d_1,d_3 \\} \\subseteq \\mathcal{C}$. By ($\\star .1$), $(a_{1}^{\\prime},a_1)\\in E(D)$ and $w(a_{1}^{\\prime})=1$. If $b_1\\in V^{+}$, then by Remark \\ref{rem-V-R-P}, $b_1$ is not a sink. This implies, $(b_1,b_{1}^{\\prime})\\in E(D)$. Then, by ($\\star .1$) with $(a,b)=(b_1,b_{1}^{\\prime})$, $w(a_1)=1$. A contradiction, since $a_1\\in V^{+}$. Hence, $w(b_1)=1$.\n\n\\noindent \nWe prove $a_{1}^{\\prime}\\in \\mathcal{C}$. By contradiction assume $a_{1}^{\\prime}\\not\\in \\mathcal{C}$, then $\\{b_1,a_1,c_1,b_{1}^{\\prime}\\} \\subseteq \\mathcal{C}$, since $|\\mathcal{C}\\cap V(C)|\\geqslant 4$. Suppose $b_{1}^{\\prime} \\in L_3(\\mathcal{C})$, then there is $y\\in \\big( N_{D}^{-}(b_{1}^{\\prime})\\cap V^{+} \\big) \\setminus L_1(\\mathcal{C})$. Then, by ($\\star .3$) with $(a,b)=(a_1,b_1)$, $y\\in N_{D}^{-}(a_{1}^{\\prime})$, i.e. $(y,a_{1}^{\\prime})\\in E(D)$. Consequently, $y\\in L_1(\\mathcal{C})$, since $a_{1}^{\\prime}\\notin \\mathcal{C}$. This is a contradiction. Hence, $b_{1}^{\\prime} \\notin L_3(\\mathcal{C})$, i.e. there is $y^{\\prime}\\in N_D(b_{1}^{\\prime})\\setminus \\mathcal{C}$, since $b_{1}^{\\prime} \\in \\mathcal{C}$. By ($\\star .3$), $y^{\\prime} \\in N_D(a_{1}^{\\prime})\\cup N_{D}^{+}(a_1)$. Furthermore, $a_{1}^{\\prime} \\notin \\mathcal{C}$, then $N_D(a_{1}^{\\prime})\\subseteq \\mathcal{C}$ and $y^{\\prime} \\notin N_D(a_{1}^{\\prime})$, since $\\mathcal{C}$ is a vertex cover and $y^{\\prime}\\notin \\mathcal{C}$. This implies $y^{\\prime} \\in N_{D}^{+}(a_1)$, then $a_1 \\in L_1(\\mathcal{C})$, since $a_1\\in \\mathcal{C}$ and $y^{\\prime} \\notin \\mathcal{C}$. A contradiction, since $a_1 \\notin L_1(\\mathcal{C})$. Therefore, $a_{1}^{\\prime}\\in \\mathcal{C}$.\n\n\\noindent\nThus, $\\{b_1,a_1,a_{1}^{\\prime},b_{1}^{\\prime}\\} \\subseteq \\mathcal{C}$. Now, we prove $c_1\\in \\mathcal{C}$, $deg_D(a_1)\\geqslant 3$ and $deg_D(c_1)\\geqslant 3$.\n\n\\noindent\n{\\bf Case (1)} $a_1\\in L_3(\\mathcal{C})$. Consequently, there is $z\\in N_{D}^{-}(a_1)\\cap V^{+}$ such that $z\\in \\mathcal{C} \\setminus L_1(\\mathcal{C})$. Then, $z\\notin V(C)$, since $N_{D}^{-}(a_1)\\cap V(C)=\\{ a_{1}^{\\prime} \\}$ and $w(a_{1}^{\\prime})=1$. By ($\\star .2$), $z\\in N_{D}^{-}(c_1)$. Thus, $(z,c_1)\\in E(D)$. Consequently, $c_1\\in \\mathcal{C}$, $deg_{D}(a_1)\\geqslant 3$ and $deg_{D}(c_1)\\geqslant 3$, since $z\\in \\mathcal{C} \\setminus L_1(\\mathcal{C})$ and $z\\in N_D(a_1)\\cap N_D(c_1)$. \n\n\\noindent\n{\\bf Case (2)} $a_1\\notin L_3(\\mathcal{C})$. This implies, there is $z^{\\prime}\\in N_D(a_1)$ such that $z^{\\prime}\\notin \\mathcal{C}$. Then, $z^{\\prime}\\notin V(C)$, since $N_D(a_1)\\cap V(C)=\\{ a_{1}^{\\prime}, b_1\\} \\subseteq \\mathcal{C}$. Consequently, $z^{\\prime}\\in N_{D}^{-}(a_1)$, since $a_1\\in \\mathcal{C}\\setminus L_1(\\mathcal{C})$. By ($\\star .2$), we have $z^{\\prime}\\in N_{D}^{-}(a_1)\\subseteq N_D(c_1)$. Hence, $c_1\\in \\mathcal{C}$, $deg_D(a_1)\\geqslant 3$ and $deg_D(c_1)\\geqslant 3$, since $z^{\\prime}\\notin \\mathcal{C}$ and $z^{\\prime}\\in N_D(a_1)\\cap N_D(c_1)$. \n\n\\noindent\nThis implies, $V(C)\\subseteq \\mathcal{C}$. A contradiction, by Lemma \\ref{NotStarCover}, since $C$ has the $\\star$-property. \\medskip\n\n\\noindent\n$\\Leftarrow )$\nAssume $C=(a^{\\prime},a,b,b^{\\prime},c,a^{\\prime})$ with $(a,b)\\in E(C)$ such that $w(a)\\neq 1$. We take a minimal vertex cover $\\mathcal{C}$ of $D$. We will prove ($\\star .1$), ($\\star .2$) and ($\\star .3$). \\smallskip \n\n\\noindent\n$\\mathbf{(\\star .1)}$ First we will prove $(a^{\\prime},a)\\in E(D)$. By contradiction, suppose $(a,a^{\\prime})\\in E(D)$. By Remark \\ref{rem-V-R-P}, there is $y\\in N_{D}^{-}(a)$, since $a\\in V^{+}$. Thus, $y\\notin V(C)$ and $deg_{D}(a)\\geq 3$. Consequently, $deg_{D}(a^{\\prime})=deg_{D}(b)=2$, since $C$ is basic. Also, $deg_{D}(b^{\\prime})=2$ or $deg_{D}(c)=2$, since $C$ is basic. We can assume $deg_{D}(c)=2$, then $N_D(c)=\\{ a^{\\prime},b^{\\prime}\\}$. So, by Remark \\ref{Compl-einC}, $\\mathcal{C}_1=\\big(\\mathcal{C}\\setminus \\{y,c\\}\\big) \\cup N_{D}(y,b)\\cup N_{D}^{+}(a)$ is a vertex cover, since $\\mathcal{C}$ is a vertex cover, $\\{ a^{\\prime},b^{\\prime}\\} \\subseteq N_D(b)\\cup N_{D}^{+}(a)\\subseteq \\mathcal{C}_1$. Since $deg_{D}(c)=2$, we have $c\\not\\in N_{D}^{}(y)$. Furthermore, $c\\notin N_D(b)\\cup N_{D}^{+}(a)$, since $C$ is induced. Then, $c\\not\\in \\mathcal{C}_1$. Also, $N_D(b)=\\{ b^{\\prime},a\\}$, implies $y\\not\\in \\mathcal{C}_1$, since $y\\not\\in N_{D}^{+}(a)$. By Proposition \\ref{GeneratinAStrongVC} there is a strong vertex cover $\\mathcal{C}_{1}^{\\prime}$ such that $N_{D}^{+}(a)\\subseteq \\mathcal{C}_{1}^{\\prime}\\subseteq \\mathcal{C}_{1}$, since $a\\in V^{+}$. Thus, $c,y\\notin \\mathcal{C}_{1}^{\\prime}$, since $c,y\\notin\\mathcal{C}_1$. By Remark \\ref{einC}, $a^{\\prime},b^{\\prime},a\\in N_D (c)\\cup N_D (y)\\subseteq \\mathcal{C}_{1}^{\\prime}$. Furthermore, $b\\in N_{D}^{+}(a)\\subseteq \\mathcal{C}_{1}^{\\prime}$. Hence, $|\\mathcal{C}_{1}^{\\prime}\\cap V(C)|=4$. A contradiction.\n\n\\noindent\nNow, we prove $w(a^{\\prime})=1$. By contradiction, assume $w(a^{\\prime})\\neq 1$. By the last argument, $(c,a^{\\prime})\\in E(D)$, since $(a^{\\prime},a)\\in E(D)$ and $a\\in V^{+}$. A contradiction, by {\\rm (a)} in Lemma \\ref{lemma-sep11}. \\smallskip \n\n\\noindent\n$\\mathbf{(\\star .2)}$ We will prove $N_{D}^{-}(a)\\subseteq N_{D}(c)$. By contradiction, suppose there is $y\\in N_{D}^{-}(a)\\setminus N_{D}(c)$. Also, $N_{D}^{-}(a)\\cap V(C)\\subseteq \\{a^{\\prime}\\} \\subseteq N_{D}(c)$, since $b\\in N_{D}^{+}(a)$. Hence, $y\\notin V(C)$. By Remark \\ref{Compl-einC}, $\\mathcal{C}_2=\\big( \\mathcal{C} \\setminus \\{y,c\\}\\big) \\cup N_{D}(y,c)\\cup N_{D}^{+}(a)$ is a vertex cover. Furthermore, $y,c\\notin \\mathcal{C}_2$, since $y\\in N_{D}^{-}(a)\\setminus N_D(c)$ and $c\\notin N_D(a,y)$. By Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}_{2}^{\\prime}$ such that $N_{D}^{+}(a)\\subseteq \\mathcal{C}_{2}^{\\prime} \\subseteq \\mathcal{C}_2$, since $a\\in V^{+}$. Thus, $y,c\\notin \\mathcal{C}_{2}^{\\prime}$ since $y,c\\notin \\mathcal{C}_2$. By Remark \\ref{einC}, $a,a^{\\prime},b^{\\prime}\\in N_D(y,c)\\subseteq \\mathcal{C}_{2}^{\\prime}$. Hence, $|\\mathcal{C}_{2}^{\\prime}\\cap V(C)|=4$, since $b\\in N_{D}^{+}(a)\\subseteq \\mathcal{C}_{2}^{\\prime}$. A contradiction. \n\n\\noindent\nNow, we prove $N_{D}^{-}(a) \\cap V^{+} \\subseteq N_{D}^{-}(c)$. By contradiction, suppose there is $y\\in N_{D}^{-}(a)\\cap V^{+}\\setminus N_{D}^{-}(c)$. By Remark \\ref{Compl-einC}, $\\mathcal{C}_3=(\\mathcal{C}\\setminus \\{c\\})\\cup N_{D}(c)\\cup N_{D}^{+}(a,y)$ is a vertex cover. Furthermore, $c\\notin N_{D}^{+}(a,y)$, then $c\\notin \\mathcal{C}_3$. By Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}_{3}^{\\prime}$ such that $N_{D}^{+}(a,y)\\subseteq \\mathcal{C}_{3}^{\\prime}\\subseteq \\mathcal{C}_3$ since $\\{ a,y\\} \\subseteq V^{+}$. So, $c\\notin \\mathcal{C}_{3}^{\\prime}$, since $c\\notin \\mathcal{C}_3$. Thus, by Remark \\ref{einC} $a^{\\prime},b^{\\prime}\\in N_D (c)\\subseteq \\mathcal{C}_{3}^{\\prime}$. Also, $a,b\\in N_{D}^{+}(a,y)\\subseteq \\mathcal{C}_{3}^{\\prime}$. Hence, $|\\mathcal{C}^{\\prime}_{3}\\cap V(C)|= 4$, a contradiction. \\smallskip\n\n\\noindent\n$\\mathbf{(\\star .3)}$ We prove $N_D(b^{\\prime})\\subseteq N_D(a^{\\prime})\\cup N_{D}^{+}(a)$. By contradiction, we suppose there is $y\\in N_D(b^{\\prime})\\setminus \\big( N_D(a^{\\prime})\\cup N_{D}^{+}(a)\\big)$. Thus, $y\\notin C$, since $N_D (b^{\\prime})\\cap V(C)=\\{ c,b\\} \\subseteq N_D (a^{\\prime}) \\cup N_{D}^{+}(a)$. By Remark \\ref{Compl-einC}, $\\mathcal{C}_4 =\\big( \\mathcal{C}\\setminus \\{ y,a^{\\prime}\\})\\big) \\cup N_D(y,a^{\\prime})\\cup N_{D}^{+} (a)$. Furthermore, $y\\notin \\mathcal{C}_4$, since $y\\notin N_D(a^{\\prime})\\cup N_{D}^{+}(a)$. By ($\\star .1$), $(a^{\\prime},a)\\in E(D)$, then $a^{\\prime} \\notin \\mathcal{C}_4$, since $a^{\\prime} \\notin N_D(y) \\cup N_{D}^{+}(a)$. By Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}_{4}^{\\prime}$ such that $N_{D}^{+}(a)\\subseteq \\mathcal{C}_{4}^{\\prime} \\subseteq \\mathcal{C}_4$, since $a\\in V^{+}$. So, $y,a^{\\prime} \\notin \\mathcal{C}_{4}^{\\prime}$, since $y,a^{\\prime} \\notin \\mathcal{C}_4$. Thus, by Remark \\ref{einC} $b^{\\prime},a,c\\in N_D(y)\\cup N_D(a^{\\prime})\\subseteq \\mathcal{C}_{4}^{\\prime}$. Also, $b\\in N_{D}^{+} (a) \\subseteq \\mathcal{C}_{4}^{\\prime}$. Hence, $|\\mathcal{C}_{4}^{\\prime} \\cap V(C)|=4$, a contradiction. \\smallskip\n\n\\noindent\nFinally, we prove $N_{D}^{-}(b^{\\prime})\\cap V^{+} \\subseteq N_{D}^{-}(a^{\\prime})$. By contradiction, we suppose there is $y\\in \\big( N_{D}^{-}(b^{\\prime})\\cap V^{+}\\big) \\setminus N_{D}^{-}(a^{\\prime})$. By $(\\star .1)$, $a^{\\prime} \\in N_{D}^{-}(a)$. Furthermore, by {\\rm (a)} in Lemma \\ref{lemma-sep11}, $y\\neq b$, since $y\\in V^{+}$. If $y=c$, then $(c,b^{\\prime})\\in E(D)$. Thus, by $(\\star .1)$, with the edge $(a^{\\prime},c)\\in E(D)$. A contradiction by {\\rm (b)} in Lemma \\ref{lemma-sep11}, since $c=y\\in V^{+}$. Hence, $y\\notin V(C)$. By Remark \\ref{Compl-einC}, $\\mathcal{C}_5=\\big( \\mathcal{C}\\setminus \\{ a^{\\prime}\\}\\big) \\cup N_{D}(a^{\\prime})\\cup N_{D}^{+}(y,a)$ is a vertex cover. By Remark \\ref{Compl-einC}, $a^{\\prime} \\notin N_{D}^{+}(y,a)$, since $(a^{\\prime},a)\\in E(D)$ and $y\\notin N_{D}^{-}(a^{\\prime})$. Consequently, $a^{\\prime} \\notin \\mathcal{C}_5$. By Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}_{5}^{\\prime}$ such that $N_{D}^{+} (a,y)\\subseteq \\mathcal{C}_{5}^{\\prime} \\subseteq \\mathcal{C}_5$, since $\\{ a,y\\} \\subseteq V^{+}$. So, $a^{\\prime} \\notin \\mathcal{C}_{5}^{\\prime}$, since $a^{\\prime} \\notin \\mathcal{C}_{5}$. Then, by Remark \\ref{einC} $a,c\\in N_{D}(a^{\\prime}) \\subseteq \\mathcal{C}_{5}^{\\prime}$. Furthermore, $b,b^{\\prime} \\in N_{D}^{+}(a,y) \\subseteq \\mathcal{C}_{5}^{\\prime}$. Hence, $|\\mathcal{C}_{5}^{\\prime} \\cap V(C)|=4$, a contradiction. \\qed \\smallskip\n\\end{proof}\n\n\\begin{lemma}\\label{lemma-08ene}\nLet $\\mathcal{C}$ be a vertex cover of $D$ where $G$ is an $SCQ$ graph. Hence, $|\\mathcal{C}|=\\tau(G)$ if and only if $|\\mathcal{C}\\cap V(K)|=|V(K)|-1$, $|\\mathcal{C}\\cap V(C)|=3$ and $|\\mathcal{C}\\cap e|=1$ for each $K\\in S_G$, $C\\in C_G$ and $e\\in Q_G$, respectively.\n\\end{lemma}\n\\begin{proof}\nWe set $\\mathcal{C}$ a vertex cover of $D$, $K\\in S_G$, $C\\in C_G$ and $e\\in Q_G$. Then, there are $y\\in V(G)$ and $a,a^{\\prime}\\in V(C)$ such that $K=G[N_G[y]]$, $deg_G(a)=deg_G(a^{\\prime})=2$ and $\\{ a,a^{\\prime}\\} \\notin E(G)$. We set $A_K:=V(K)\\setminus \\{ y\\}$ and $B_C:=V(C)\\setminus \\{ a,a^{\\prime}\\}$. Also, $\\mathcal{C}\\cap V(K)$ is a vertex cover of $K$, so $|\\mathcal{C}\\cap V(K)|\\geqslant\\tau(K)=|V(K)|-1$. Similarly, $|\\mathcal{C}\\cap V(C)|\\geqslant\\tau(C)=3$ and $|\\mathcal{C}\\cap e|\\geqslant\\tau(e)=1$. Thus, \n\\begin{equation}\\label{tau-SCQ}\n|\\mathcal{C}|=\\sum\\limits_{K\\in S_G} |\\mathcal{C}\\cap V(K)|+\\sum\\limits_{C\\in C_G} |\\mathcal{C}\\cap V(C)|+\\sum\\limits_{e\\in Q_G} |\\mathcal{C}\\cap e|\\geqslant \\sum\\limits_{K\\in S_G} \\big( |V(K)|-1\\big)+3|C_G|+|Q_G|,\n\\end{equation} \nsince $\\mathcal{H}=\\{ V(H)\\mid H\\in S_G\\cup C_G\\cup Q_G\\}$ is a partition of $V(G)$. Now, we take a maximal stable set $S$ contained in $V(Q_G):=\\{ x\\in V(G)\\mid x\\in e \\ {\\rm and} \\ e\\in Q_G\\}$. Then, $|S\\cap e|\\leqslant 1$ for each $e\\in Q_G$, since $S$ is stable. If $S\\cap e=\\emptyset$ for some $e=\\{ x_1,x_2\\} \\in Q_G$, then there are $y_1,y_2\\in S$ such that $\\{ x_1,y_1\\}$, $\\{ x_2,y_2\\} \\in E(G)$, since $S$ is maximal. But $Q_G$ satisfies the property {\\bf (P)}, then $\\{ y_1,y_2\\}\\in E(G)$. A contradiction, since $S$ is stable. Hence, $|S\\cap e|=1$ for each $e\\in Q_G$. Consequently, $|S|=|Q_G|$ and $|S^{\\prime}|=|Q_G|$, where $S^{\\prime}=V(Q_G)\\setminus S$. Now, we take \n\\begin{center}\n$\\mathcal{C}(S^{\\prime})= \\Big( \\bigcup\\limits_{K\\in S_G} A_K \\Big) \\bigcup \\Big( \\bigcup\\limits_{C\\in C_G} B_C \\Big) \\bigcup S^{\\prime}$.\n\\end{center}\nWe prove $\\mathcal{C}(S^{\\prime})$ is a vertex cover of $D$. By contradiction, suppose there is $\\hat{e}\\in E(G)$ such that $\\hat{e}\\cap \\mathcal{C}(S^{\\prime})=\\emptyset$. We set $z\\in \\hat{e}$, then $\\hat{e}=\\{ z,z^{\\prime}\\}$. If $z\\in V(\\tilde{K})$ for some $\\tilde{K}\\in S_G$, \\linebreak then $\\tilde{K}=G[N_G[z]]$, since $A_{\\tilde{K}}\\subseteq \\mathcal{C}(S^{\\prime})$ and $z\\notin \\mathcal{C}(S^{\\prime})$. So, $z^{\\prime}\\in N_G(z)\\subseteq \n\\tilde{K}\\setminus \\{ z\\}=A_{\\tilde{K}}\\subseteq \\mathcal{C}(S^{\\prime})$. A contradiction, since $\\hat{e}\\cap \\mathcal{C}(S^{\\prime})=\\emptyset$. Now, if $z\\in V(\\tilde{C})$ for some $\\tilde{C}\\in C_G$, then $z\\notin B_{\\tilde{C}}$. Thus, $deg_G(z)=2$ implying $z^{\\prime}\\in B_{\\tilde{C}}\\subseteq \\mathcal{C}(S^{\\prime})$, since $\\{ z,z^{\\prime}\\}\\in E(G)$. A contradiction. Then, $\\hat{e}\\subseteq V(Q_G)$, since $\\mathcal{H}$ is a partition of $V(G)$. Also, $\\hat{e}\\cap S^{\\prime}=\\emptyset$, this implies $\\hat{e}\\subseteq V(Q_G)\\setminus S^{\\prime}=S$. But $S$ is stable. This is a contradiction. Hence, $\\mathcal{C}(S^{\\prime})$ is a vertex cover of $D$. Furthermore,\n\\begin{center}\n$|\\mathcal{C}(S^{\\prime})|=\\sum\\limits_{K\\in S_G} |A_K|+\\sum\\limits_{C\\in C_G} |B_C|+|S^{\\prime}|=\\sum\\limits_{K\\in S_G} \\big( |V(K)|-1\\big)+3|C_G|+|Q_G|$.\n\\end{center}\nThus, $\\tau(G)=\\sum_{K\\in S_G} \\big( |V(K)|-1\\big)+3|C_G|+|Q_G|$. Therefore, by (\\ref{tau-SCQ}), $|\\mathcal{C}|=\\tau(G)$ if and only if $|\\mathcal{C}\\cap V(K)|=|K|-1$, $|\\mathcal{C}\\cap V(C)|=3$ and $|\\mathcal{C}\\cap e|=1$ for each $K\\in S_G$, $C\\in C_G$ and $e\\in Q_G$, respectively.\n\\qed\n\\end{proof}\n\n\\begin{theorem}\\label{SCQ-char}\nLet $D=(G,\\mathcal{O},w)$ be a weighted oriented graph where $G$ is an $SCQ$ graph. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following conditions:\n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (a)}] Each basic $5$-cycle of $G$ has the $\\star$-property.\n\\item[{\\rm (b)}] Each simplex of $D$ has no generating $\\star$-semi-forests.\n\\item[{\\rm (c)}] $N_D(b)\\subseteq N_{D}^{+}(a)$ when $a\\in V^{+}$, $\\{ b,b^{\\prime} \\} \\in Q_G$ and $b^{\\prime} \\in N_{D}^{+}(a)$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n$\\Rightarrow )$\nWe take a strong vertex cover $\\mathcal{C}$ of $D$, then by Remark \\ref{MinimalStrongProp}, $|\\mathcal{C}|=\\tau (G)$. Consequently, by Lemma \\ref{lemma-08ene}, $|\\mathcal{C}\\cap V(K)|=|V(K)|-1$, $|\\mathcal{C}\\cap V(C)|=3$ and $|\\mathcal{C}\\cap e|=1$ for each $K\\in S_G$, $C\\in C_G$ and $e\\in Q_G$. Thus, $V(K)\\not\\subseteq \\mathcal{C}$. Consequently, by Theorem \\ref{theorem-oct29}, $D$ satisfies {\\rm (b)}. Furthermore, by Propositions \\ref{Basic5Cycle-Equ} and \\ref{prop-unmixed}, $D$ satisfies {\\rm (a)} and {\\rm (c)}. \\medskip\n\n\\noindent\n$\\Leftarrow )$\nLet $\\mathcal{C}$ be a strong vertex cover of $D$. By {\\rm (a)} and Proposition \\ref{Basic5Cycle-Equ}, we have $|\\mathcal{C}\\cap V(C)|=3$ for each $C\\in C_G$. Furthermore, by {\\rm (b)} and Theorem \\ref{theorem-oct29}, $V(K)\\not\\subseteq \\mathcal{C}$ for each $K\\in S_G$. Consequently, $|V(K)|>|\\mathcal{C} \\cap V(K)|\\geqslant \\tau (K)=|V(K)|-1$. So, $|\\mathcal{C}\\cap V(K)|=|V(K)|-1$. Now, if $e\\in Q_G$, then $e$ has the property {\\bf (P)}, since $Q_G$ has the property {\\bf (P)}. Thus, by {\\rm (c)} and Proposition \\ref{prop-unmixed}, $|\\mathcal{C} \\cap e|=1$. Hence, by Lemma \\ref{lemma-08ene}, $|\\mathcal{C}|=\\tau (G)$. Therefore $I(D)$ is unmixed, by $(2)$ in Theorem \\ref{theorem42}. \\qed\n\\end{proof}\n\n\\begin{corollary}\\label{Simp-Chor-Unm}\nLet $D=(G,\\mathcal{O},w)$ be a weighted oriented graph where $G$ is a simplicial or chordal graph. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following conditions:\n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (a)}] Each vertex is in exactly one simplex of $D$.\n\\item[{\\rm (b)}] Each simplex of $D$ has not a generating $\\star$-semi-forest.\n\\end{enumerate} \n\\end{corollary}\n\\begin{proof}\n$\\Rightarrow )$ By $(3)$ in Theorem \\ref{theorem42} and Remark \\ref{1star}, $G$ is well-covered. Thus, by Theorem \\ref{T1-T2Prisner}, $G$ sa\\-tis\\-fies {\\rm (a)}. Furthermore, by Remark \\ref{rem-nov13}, $G$ is an $SCQ$ graph with $C_G=Q_G=\\emptyset$. Hence, by Theorem \\ref{SCQ-char}, $D$ satifies {\\rm (b)}. \\medskip\n\n\\noindent\n$\\Leftarrow )$ By {\\rm (a)}, $\\{ V(H)\\mid H\\in S_G\\}$ is a partition of $V(G)$. Hence, $G$ is an $SCQ$ graph with $C_G=\\emptyset$ and $Q_{G}=\\emptyset$. Therefore, by {\\rm (b)} and Theorem \\ref{SCQ-char}, $I(D)$ is unmixed. \\qed\n\\end{proof}\n\n\\section{Unmixedness of weighted oriented graphs without some small cycles}\nLet $D=(G,\\mathcal{O},w)$ be a weighted oriented graph. In this Section, we study and cha\\-rac\\-te\\-ri\\-ze the unmixed property of $I(D)$ when $G$ has no $3$- or $5$- cycles (Theorem \\ref{theorem-oct31}), or $G$ is a graph without $4$- or $5$-cycles (Theorem \\ref{No4,5Cyc-Unmix}), or $girth(G)\\geqslant 5$ (Theorem \\ref{No3,4,5Cyc-Unmix}). In other words, in this Section, we characterize the unmixed property of $I(D)$ when $G$ has at most one of the following types of cycles: $3$-cycles, $4$-cycles and $5$-cycles.\n\n\n\\begin{proposition}\\label{prop-oct31}\nIf for each $(y,x)\\in E(D)$ with $y\\in V^{+}$, we have that $N_D(y^{\\prime})\\subseteq N_{D}^{+}(y)$ for some $y^{\\prime}\\in N_D(x)\\setminus y$, then $L_3(\\mathcal{C})=\\emptyset$ for each strong vertex cover $\\mathcal{C}$ of $D$.\n\\end{proposition}\n\\begin{proof}\nBy contradiction, suppose there is a strong vertex cover $\\mathcal{C}$ of $D$ and $x\\in L_3(\\mathcal{C})$. Hence, there is $y\\in \\big( \\mathcal{C}\\setminus L_1(\\mathcal{C})\\big) \\cap V^{+}$ with $(y,x)\\in E(D)$. Then, $N_D(x)\\subseteq \\mathcal{C}$ and $N_{D}^{+}(y)\\subseteq \\mathcal{C}$, since $x\\in L_3(\\mathcal{C})$ and $y\\in \\mathcal{C}\\setminus L_1(\\mathcal{C})$. By hypothesis, there is a vertex $y^{\\prime}\\in N_D(x)\\setminus y\\subseteq \\mathcal{C}$ such that $N_D(y^{\\prime})\\subseteq N_{D}^{+}(y)\\subseteq \\mathcal{C}$. Thus, $y^{\\prime}\\in L_3(\\mathcal{C})$. Since $\\mathcal{C}$ is strong, there is $(y_1,y^{\\prime})\\in E(D)$ with $y_1\\in V^{+}$. So, $y_1\\in N_D(y^{\\prime})\\subseteq N_{D}^{+}(y)$. On the other hand, $(y_1,x_1)\\in E(D)$ where $x_1:=y^{\\prime}$ and $y_1\\in V^{+}$, then by hypothesis, there is $y_{1}^{\\prime}\\in N_D(x_1)\\setminus y_1$ such that $N_D(y_{1}^{\\prime})\\subseteq N_{D}^{+}(y_1)$. Hence, $y_{1}^{\\prime}\\in N_D(x_1)=N_D(y^{\\prime})\\subseteq N_{D}^{+}(y)$. Consequently, $y\\in N_D(y_{1}^{\\prime})\\subseteq N_{D}^{+}(y_1)$. A contradiction, since $y_1\\in N_{D}^{+}(y)$. \\qed\n\\end{proof}\n\n\\begin{corollary}\\label{cor-oct30}\nIf $G$ is well-covered and $V^{+}$ is a subset of sinks, then $I(D)$ is unmixed.\n\\end{corollary}\n\\begin{proof}\nIf $y\\in V^{+}$, then $y$ is a sink. Thus, $(y,x)\\notin E(D)$ for each $x\\in V(D)$. Hence, by Proposition \\ref{prop-oct31}, $L_3(\\mathcal{C})=\\emptyset$, for each strong vertex cover $\\mathcal{C}$ of $D$. Furthermore, by Remark \\ref{1star}, $I(G)$ is unmixed. Therefore $I(D)$ is unmixed, by $(3)$ in Theorem \\ref{theorem42}. \\qed\n\\end{proof}\n\n\\begin{lemma}\\label{StableSet}\nLet $(z,y), (y,x)$ be edges of $D$ with $y\\in V^{+}$ and $N_D(x)=\\{ y,x_1,\\ldots ,x_s\\}$. If there are $z_i\\in N_D(x_i)\\setminus N_{D}^{+}(y)$ such that $\\{ z,x,z_1,\\ldots ,z_s\\}$ is a stable set, then $I(D)$ is mixed.\n\\end{lemma}\n\\begin{proof}\nWe take $A:=\\{ z,z_1,\\ldots ,z_s\\}$, then $A\\cup \\{ x\\}$ is a stable set. We can take a maximal stable set $S$ of $V(G)$, such that $A\\cup \\{ x\\}\\subseteq S$. So, $\\tilde{\\mathcal{C}}=V(G)\\setminus S$ is a minimal vertex cover of $D$. Hence, $\\mathcal{C}=\\tilde{\\mathcal{C}} \\cup N_{D}^{+}(y)$ is a vertex cover of $D$. Also $A\\cap \\mathcal{C}=\\emptyset$, since $A\\subseteq S$, $z\\in N_{D}^{-}(y)$ and $z_i\\notin N_{D}^{+}(y)$. By Proposition \\ref{GeneratinAStrongVC}, there is a strong vertex cover $\\mathcal{C}^{\\prime}$ of $D$ such that $N_{D}^{+}(y)\\subseteq \\mathcal{C}^{\\prime} \\subseteq \\mathcal{C}$, since $y\\in V^{+}$. Thus, $A\\cap \\mathcal{C}^{\\prime}=\\emptyset$, since $A\\cap \\mathcal{C}=\\emptyset$. Then, by Remark \\ref{einC}, $N_D(A)\\subseteq \\mathcal{C}^{\\prime}$. Furthermore $N_D(x)=\\{ y,x_1,\\ldots ,x_s\\} \\subseteq N_D(A)$. Consequently, $N_D(x)\\subseteq \\mathcal{C}^{\\prime}$. Hence, $x\\in L_3(\\mathcal{C}^{\\prime})$, since $x\\in N_{D}^{+}(y)\\subseteq \\mathcal{C}^{\\prime}$. Therefore, by {\\rm (3)} in Theorem \\ref{theorem42}, $I(D)$ is mixed. \\qed\n\\end{proof}\n\n\\begin{theorem}\\label{theorem-oct31}\nLet $D=(G,\\mathcal{O},w)$ be a weighted oriented graph such that $G$ has no $3$- or $5$-cycles. Hence, $I(D)$ is unmixed if and only if $D$ satisfies the following conditions:\n\\begin{enumerate}[noitemsep]\n\\item[{\\rm (a)}] $G$ is well-covered.\n\\item[{\\rm (b)}] If $(y,x)\\in E(D)$ with $y\\in V^{+}$, then $N_D(y^{\\prime})\\subseteq N_{D}^{+}(y)$ for some $y^{\\prime}\\in N_D(x)\\setminus y$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n$\\Leftarrow )$ By Proposition \\ref{prop-oct31} and {\\rm (b)}, we have that $L_3(\\mathcal{C})=\\emptyset$ for each strong vertex cover $\\mathcal{C}$ of $D$. Furthermore, by {\\rm (a)} and Remark \\ref{1star}, $I(G)$ is unmixed. Therefore, by $(3)$ in Theorem \\ref{theorem42}, $I(D)$ is unmixed.\\medskip\n\n\\noindent\n$\\Rightarrow )$ By {\\rm (3)} in Theorem \\ref{theorem42} and Remark \\ref{1star}, $D$ satisfies {\\rm (a)}. Now, we take $(y,x)\\in E(D)$ with $y\\in V^{+}$. Then, by Remark \\ref{rem-V-R-P}, there is $z\\in N_{D}^{-}(y)$. Furthermore $z \\notin N_D(x)$, since $G$ has no $3$-cycles. We set $N_D(x)\\setminus y=\\{ x_1,\\ldots ,x_s\\}$. We will prove {\\rm (b)}. By contradiction, suppose there is $z_i\\in N_D(x_i)\\setminus N_{D}^{+}(y)$ for each $i=1,\\ldots , s$. If $\\{ z_i,z_j\\} \\in E(G)$ for some $1\\leqslant i1$}};\n\\node[draw,fill,circle,inner sep=1pt] (7) at (2,-6){};\n\\node (17) at (2.35,-6){{\\small $y_{\\tiny 2}$}};\n\\node[draw,fill,circle,inner sep=1pt] (8) at (1,-6){};\n\\node (18) at (1,-6.3){{\\small $x_{\\tiny 2}$}};\n\\node (118) at (1,-6.65){{\\small $w>1$}};\n\\node[draw,fill,circle,inner sep=1pt] (9) at (0,-4){};\n\\node (19) at (0,-3.7){{\\small $y_{\\tiny 3}$}};\n\\node[draw,fill,circle,inner sep=1pt] (10) at (0,-5){};\n\\node (20) at (-.3,-5){{\\small $x_{\\tiny 3}$}};\n\\node (120) at (0.7,-5){{\\small $w>1$}};\n\\node (0) at (0,-5.7){$H$};\n\n\\path [draw,postaction={on each segment={mid arrow=thick}}] \n(6) to (10)\n(10) to (8)\n(8) to (6)\n(6) to (5)\n(8) to (7)\n(9) to (10);\n\n\\draw[-] (2)-- (5);\n\\draw[-] (3)--(7);\n\\draw[-] (4)--(9);\n\n\n\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[c]{8.6cm}\n\\centering\n\n\\begin{itemize}[noitemsep]\n\\item If we take $\\mathcal{C}_1=\\{ x_1,x_2,y_1,y_2,y_3\\}$ as a minimal vertex cover of $D_2$, then $\\mathcal{C}_{1}^{\\prime}=\\{ x_1,x_2,x_3,y_1,y_2,y_3\\}$ and $L_3(\\mathcal{C}_{1}^{\\prime})\\setminus N_{D}^{+}(\\tilde{H}) =\\emptyset$. Thus, by Proposition \\ref{GeneratinAStrongVC}, $\\mathcal{C}=\\mathcal{C}_{1}^{\\prime}=\\{ x_1,x_2,x_3,y_1,y_2,y_3\\}$ is a strong vertex cover such that $V(K^{4})\\subseteq \\mathcal{C}$ and it is no minimal. Furthermore, in this case it is enough to know the orientation of $E(K^{4})$, since $L_3(\\mathcal{C})=\\{ x_1,x_2,x_3\\}$ and $N_{D_2}(x_i)\\subseteq \\mathcal{C}$ for $i=1,2,3$.\n\\item Now, if we take $\\mathcal{C}_2=\\{ x_1,x_2,x_3,z_1,z_2,z_3\\}$ as a minimal vertex cover of $D_2$, then $\\mathcal{C}_{2}^{\\prime}=V(D_2)\\setminus \\{ y_3\\}$ and $L_3(\\mathcal{C}_{2}^{\\prime})\\setminus N_{D}^{+}(\\tilde{H}) =\\{ z_1,z_2\\}$. Thus, by the algorithm in the proof of Proposition \\ref{GeneratinAStrongVC}, $\\mathcal{C}^{\\prime}=\\mathcal{C}_{4}^{\\prime}=\\{ x_1,x_2,x_3,y_1,y_2,z_3\\}$ is a strong vertex cover such that $V(K^{4})\\subseteq \\mathcal{C}^{\\prime}$. \\medskip\n\\end{itemize}\n\\end{minipage}\n\n\\noindent\nSince $\\mathcal{C}$ and $\\mathcal{C}^{\\prime}$ are no minimal with $L_3(\\mathcal{C})=\\{ x_1,x_2,x_3\\}$ and $L_3(\\mathcal{C}^{\\prime})=\\{ x_1,x_2\\}$, then $D_2$ is mixed. Furthermore, $V(D_2)$ has a partition in complete graphs: $K^{i}=D_2[y_i,z_i]$ for $i=1,2,3$ and $K^{4}=D_2[x_1,x_2,x_3]$ . \n\n\\begin{example}\\rm\nLet $D_3=(G,\\mathcal{O},w)$ be the following oriented graph. Hence, \n\\end{example}\n\\begin{minipage}[c]{4.7cm} \n\\centering\n\\begin{tikzpicture}[dot\/.style={draw,fill,circle,inner sep=1pt},scale=.97]\n\n\\node[draw,fill,circle,inner sep=1pt] (2) at (-5,.5){};\n\\node (12) at (-5,.7){{\\tiny $w>1$}};\n\\node (22) at (-5,.2){{\\small $b^{\\prime}$}};\n\\node[draw,fill,circle,inner sep=1pt] (5) at (-6.5,-1.5) {};\n\\node (15) at (-6.8,-1.5){{\\small $d_{\\small 1}^{\\prime}$}};\n\\node[draw,fill,circle,inner sep=1pt](3) at (-4,.5) {};\n\\node (23) at (-4,.2){{\\small $b$}};\n\\node[draw,fill,circle,inner sep=1pt] (6) at (-2.5,-1.5){};\n\\node (16) at (-2.2,-1.5){{\\small $d_{\\small 2}^{\\prime}$}};\n\\node[draw,fill,circle,inner sep=1pt] (1) at (-6,.5){};\n\\node (11) at (-6.2,.65){{\\small $1$}};\n\\node (21) at (-6.2,.25){{\\small $c$}};\n\\node[draw,fill,circle,inner sep=1pt] (4) at (-5.5,-1.5) {};\n\\node (14) at (-5.2,-1.5){{\\small $d_{\\small 1}$}};\n\\node (24) at (-5.5,-1.75){{\\tiny $w>1$}};\n\\node[draw,fill,circle,inner sep=1pt] (7) at (-3,.5){};\n\\node (11) at (-2.7,.65){{\\tiny $w>1$}};\n\\node (21) at (-2.8,.25){{\\small $a$}};\n\\node[draw,fill,circle,inner sep=1pt] (8) at (-3.5,-1.5){};\n\\node (18) at (-3.8,-1.5){{\\small $d_{\\small 2}$}};\n\\node (28) at (-3.5,-1.75){{\\tiny $w>1$}};\n\\node[draw,fill,circle,inner sep=1pt] (9) at (-4.5,1.75){};\n\\node (19) at (-4.2,1.8){{\\small $a^{\\prime}$}};\n\\node (29) at (-4.5,2){{\\small $1$}};\n\n\\path [draw,postaction={on each segment={mid arrow=thick}}]\n(9) to (7)\n(7) to (3)\n(2) to (3)\n(1) to (2)\n(7) to (8)\n(4) to (1)\n(4) to (7)\n(5) to (4)\n(8) to (6);\n\n\\path [draw]\n(1) to (9)\n(1) to (8);\n\n\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[c]{8.6cm}\n\\centering\n\\begin{itemize}[noitemsep]\n\\item $G$ has no $3$- and $4$-cycles.\n\\item The basic $5$-cycle $C=(a^{\\prime},a,b,b^{\\prime},c,a^{\\prime})$ satisfies the $\\star$-property. \n\\item $D_3[\\{ d_1,d_{1}^{\\prime}\\}]$ has not a generating $\\star$-semi-forest. But $T_1=\\tilde{e}=(d_2,d_{2}^{\\prime})$ is a ROT with parent $v_1=d_2$ and $W=W_2=\\{ w_1=a\\}$. Furthermore, $H=T_1$ is a $\\star$-semi-forest with $V(H)=V(K)$, where $K:=D_3[\\{ d_2,d_{2}^{\\prime}\\}]$. Therefore, $H$ is a ge\\-ne\\-ra\\-ting $\\star$-semi-forest of $K$ and $I(D_3)$ is mixed.\n \\end{itemize}\n\\end{minipage}\n\n\\begin{example}\\rm\nLet $D_4=(G,\\mathcal{O},w)$ be the following oriented weighted graph. Hence,\n\\end{example}\n\\begin{minipage}[c]{4cm} \n\\centering\n\\begin{tikzpicture}[dot\/.style={draw,fill,circle,inner sep=1pt},scale=.97]\n\n\\node[draw,fill,circle,inner sep=1pt] (2) at (-5,.5){};\n\\node (12) at (-4.7,.7){{\\tiny $w>1$}};\n\\node (22) at (-5.25,.45){{\\small $d$}};\n\\node[draw,fill,circle,inner sep=1pt] (5) at (-5,-.5) {};\n\\node (15) at (-5.1,-.7){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt](3) at (-4,.5) {};\n\\node (13) at (-4,.8){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt] (6) at (-4,-.5){};\n\\node (16) at (-3.7,-.5){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt] (1) at (-6,.5){};\n\\node (11) at (-6.3,.5){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt] (4) at (-6,-.5) {};\n\\node (14) at (-6.3,-.5){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt] (7) at (-3,1.5){};\n\\node (17) at (-2.7,1.5){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt] (8) at (-3,-1.5){};\n\\node (18) at (-2.7,-1.5){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt] (10) at (-4.5,-1.5){};\n\\node (20) at (-4.8,-1.5){{\\small $1$}};\n\\node[draw,fill,circle,inner sep=1pt] (9) at (-5.5,1.5){};\n\\node (19) at (-5.8,1.5){{\\small $1$}};\n\n\\path [draw,postaction={on each segment={mid arrow=thick}}]\n(2) to (3)\n(2) to (9)\n(5) to (2)\n(10) to (8)\n(8) to (7)\n(7)to (9)\n(1) to (9)\n(1) to (4)\n(4) to (5)\n(10) to (5)\n(10) to (6)\n(3) to (6);\n\n\\end{tikzpicture}\n\\end{minipage}\n\\begin{minipage}[c]{9.3cm}\n\\centering\n\\begin{itemize}[noitemsep]\n\\item $G=P_{10}$, then by Theorem \\ref{wellcovered-characterization2} and Remark \\ref{1star}, $G$ is well-covered and $I(G)$ is unmixed.\n\\item $d$ is not a sink and $d\\in V^{+}$.\n\\item By Proposition \\ref{P10-Unmix}, $I(D_4)$ is unmixed. \n\\end{itemize}\n\\end{minipage}\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet $D$ be a quaternion division algebra over a totally real number field $F$.\nWe assume that $D$ splits at exactly one infinite place\n$\\chi_0: F \\rightarrow \\mathbb{R}$. We fix an isomorphism\n$D \\otimes_{F, \\chi_0} \\mathbb{R} \\cong {\\rm M}_2(\\mathbb{R})$. Let $h_D$ be the\nhomomorphism \n\\begin{equation}\\label{defhD}\n \\mathbb{C}^{\\times} \\rightarrow (D \\otimes_{F, \\chi_0} \\mathbb{R})^{\\times} \n \\subset (D \\otimes \\mathbb{R})^{\\times} , \\quad \nz = a + b \\mathbf{i} \\mapsto \\left(\n \\begin{array}{rr} \n a & -b\\\\\n b & a\n \\end{array} \\right) .\n\\end{equation} \n\nWe regard $D^\\times$ as an algebraic group over $\\mathbb{Q}$. Then\n$(D^\\times, h_D)$ is a Shimura datum of a Shimura curve ${\\mathrm{Sh}}(D^{\\times}, h_D)$.\nWe prove\nthe $p$-adic uniformization of these curves under certain conditions, discovered by Cherednik \\cite{Ch}. Let us describe our main result.\n\nWe fix a prime number $p$ and choose a diagram of field embeddings,\n\\begin{equation}\\label{LDiagramm} \n \\mathbb{C} \\leftarrow \\bar{\\mathbb{Q}} \\rightarrow \\bar{\\mathbb{Q}}_p.\n\\end{equation}\nThis gives a bijection \n\\begin{displaymath}\n \\Hom_{\\mathbb{Q}{\\rm -Alg}}(F, \\mathbb{C}) = \n \\Hom_{\\mathbb{Q}{\\rm -Alg}}(F, \\bar{\\mathbb{Q}}_p) .\n \\end{displaymath}\nLet $\\mathfrak{p}_0, \\ldots, \\mathfrak{p}_s$ be the prime ideals of $F$ over\n$p$. We assume that $\\mathfrak{p}_0$ is induced by\n$\\chi_0: F \\rightarrow \\bar{\\mathbb{Q}}_p$ and that\n$D_{\\mathfrak{p}_0} = D \\otimes_{F} F_{\\mathfrak{p}_0}$ is a division algebra.\nLet $O_{D_{\\mathfrak{p}_0}}$\nbe the maximal order of $D_{\\mathfrak{p}_0}$. We choose an open and compact\nsubgroup $\\mathbf{K} \\subset (D \\otimes_{\\mathbb{Q}} \\mathbb{A}_f)^{\\times}$\nas follows. We set $\\mathbf{K}_{\\mathfrak{p}_0} = O_{D_{\\mathfrak{p}_0}}^{\\times}$. For\n$i = 1, \\ldots, s$ we choose arbitrarily open and compact subgroups\n$\\mathbf{K}_{\\mathfrak{p}_i} \\subset D_{\\mathfrak{p}_i}^{\\times}$. We set\n\\begin{equation}\\label{defKp}\n \\mathbf{K}_p = \\prod_{i=0}^{s} \\mathbf{K}_{\\mathfrak{p}_i} \\subset\n (D \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p)^{\\times}. \n \\end{equation}\nWe also choose a sufficiently small open compact subgroup\n$\\mathbf{K}^p \\subset (D \\otimes_{\\mathbb{Q}} \\mathbb{A}^p_f)^{\\times}$ and set\n\\begin{displaymath}\n\\mathbf{K} = \\mathbf{K}_p \\mathbf{K}^p.\n\\end{displaymath}\nThe Shimura field $E(D^{\\times}, h_{D})$ is $\\chi_0(F)$. The diagram\n(\\ref{LDiagramm}) induces a $p$-adic place $\\nu$ of the Shimura field, and \n$\\chi_0$ gives an identification $E(D^{\\times}, h_{D})_{\\nu} \\cong F_{\\mathfrak{p}_0}$.\nAs an abbreviation we write $E_{\\nu} = E(D^{\\times}, h_{D})_{\\nu}$. We denote by\n$\\breve{E}_{\\nu}$ the completion of the maximal unramified extension of\n$E_{\\nu}$. \n\nWe will prove (see Corollary \\ref{remstab}) that the curve ${\\mathrm{Sh}}_{\\mathbf{K}}(D^{\\times}, h_D)$ has stable reduction over\n$\\Spec O_{E_{\\nu}}$, i.e., ${\\mathrm{Sh}}_{\\mathbf{K}}(D^{\\times}, h_D)$ extends to a stable curve $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(D^{\\times}, h_D)$ over $O_{E_{\\nu}}$, in the sense of Deligne-Mumford \\cite{DM}. By \\cite[Lem. 1.12]{DM}, this extension is unique up to unique isomorphism. The action of \n$(D \\otimes \\mathbb{A}_f)^{\\times}$ on the tower ${\\mathrm{Sh}}_{\\mathbf{K}}(D^{\\times}, h_D)$ for\nvarying $\\mathbf{K}$ as above extends to the stable model. \n\nLet $\\check{D}$ be a \\emph{Cherednik twist} of $D$. This is a quaternion\nalgebra over $F$ such that\n\\begin{equation}\\label{CherednikUnif1e}\n \\check{D} \\otimes_{F} \\mathbb{A}_{F,f}^{\\mathfrak{p}_0} \\cong\n D \\otimes_{F} \\mathbb{A}_{F,f}^{\\mathfrak{p}_0} \n \\end{equation} \nand such that\n$\\check{D} \\otimes_{F} F_{\\mathfrak{p}_0} \\cong \\rm{M}_2 (F_{\\mathfrak{p}_0})$ \nand such that $\\check{D}$ is non-split at all infinite places of $F$.\nFor a more canonical definition of $\\check{D}$ see (\\ref{Cheredniktwist}).\n\nLet $\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2$ be the \\emph{integral model of the Drinfeld\n halfplane} for the local field $F_{\\mathfrak{p}_0}$, cf. \\cite{Dr}. \nIt is a $p$-adic formal scheme over $\\Spf O_{F_{\\mathfrak{p}_0}}$ with an action\nof the group\n$\\check{D}_{\\mathfrak{p}_0}^{\\times} = (\\check{D} \\otimes_{F} F_{\\mathfrak{p}_0})^{\\times} \\cong \\mathrm{GL}_2(F_{\\mathfrak{p}_0})$, cf. \\cite{Dr}, (\\ref{PGL-O1e}). \nThis action factors through an action of ${\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$. \nWe consider on \n\\begin{equation}\\label{CherednikUnif2e} \n (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0}\n \\Spf O_{\\breve{E}_\\nu})\\times (D \\otimes_{F} F_{\\mathfrak{p}_0})^{\\times}\/\n \\mathbf{K}_{\\mathfrak{p}_0} =\n (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0} \\Spf O_{\\breve{E}_\\nu}) \\times \\mathbb{Z}, \n\\end{equation}\nthe action of $\\check{D}_{\\mathfrak{p}_0}$ which is on the first factor on the\nright hand side obtained from the action introduced above and which acts on\n$\\mathbb{Z}$ by translation with\n$\\ord_{F_{\\mathfrak{p}_0}} \\det_{\\check{D}_{\\mathfrak{p}_0}\/F_{\\mathfrak{p}_0}}$, cf. Proposition\n\\ref{RZ7p}. \nWe formulate our main result as follows. \n\\begin{theorem}\\label{MainIntro}\nLet \n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(D^{\\times}, h_{D})^\\wedge_{ \\, \/\\Spf O_{\\breve{E}_{\\nu}}}$ be the\ncompletion of the scheme\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(D^{\\times},h_D)\\times_{\\Spec O_{E_{\\nu}}}\\Spec O_{\\breve{E}_{\\nu}}$\nalong the special fiber. Then there is an isomorphism of formal schemes\n \\begin{equation}\\label{CherednikUnif3e} \n \\check{D}^{\\times} \\backslash \\big((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0} \\Spf O_{\\breve{E}_\\nu})\\times\n (D \\otimes \\mathbb{A}_f)^{\\times}\/\\mathbf{K})\\big) \n\\overset{\\sim}{\\longrightarrow}\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(D^{\\times}, h_{D})^\\wedge_{\\, \/ \\Spf O_{\\breve{E}_{\\nu}}} .\n \\end{equation}\n The action of $\\check{D}^{\\times}$ is given by (\\ref{CherednikUnif1e}) and\n (\\ref{CherednikUnif2e}). \n For varying $\\mathbf{K}$ this uniformization isomorphism is compatible with \n the action of Hecke correspondences in $(D\\otimes \\mathbb{A}_f)^{\\times}$ on both sides. \n \n Let $\\Pi \\in D_{\\mathfrak{p}_0}$ be a prime element in this division algebra\n over $F_{\\mathfrak{p}_0}$. We denote also by $\\Pi$ the image by the canonical\n embedding $D_{\\mathfrak{p}_0}^\\times \\subset (D \\otimes \\mathbb{A}_f)^{\\times}$. \n Let $\\tau \\in \\Gal(\\breve{E}_{\\nu}\/ E_{\\nu})$ be the Frobenius automorphism and \n $\\tau_c = \\Spf\\tau^{-1}: \\Spf O_{\\breve{E}_{\\nu}}\\rightarrow\\Spf O_{\\breve{E}_{\\nu}}$. \n The natural Weil descent datum with respect to \n $O_{\\breve{E}_\\nu}\/O_{E_{\\nu}}$ on the right hand side of (\\ref{CherednikUnif3e})\n induces on the\n left hand side the Weil descent datum given by the following diagram\n \\begin{displaymath}\n\\xymatrix{\n \\check{D}^{\\times} \\backslash ((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0} \\Spf O_{\\breve{E}_\\nu})\\times\n D^{\\times}(\\mathbb{A}_f)\/\\mathbf{K})\n \\ar[d]_{ \\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c \\times \\Pi^{-1}} \\ar[r] & \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(D^{\\times}, h_{D})^\\wedge_{\\,\/ \\Spf O_{\\breve{E}_{\\nu}}}\n \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\\\\\n \\check{D}^{\\times} \\backslash ((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0} \\Spf O_{\\breve{E}_\\nu})\\times\n D^{\\times}(\\mathbb{A}_f)\/\\mathbf{K})\n \\ar[r] & \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(D^{\\times}, h_{D})^\\wedge_{\\,\/ \\Spf O_{\\breve{E}_{\\nu}}}. \n }\n \\end{displaymath}\n\\end{theorem}\nThe left hand side of (\\ref{CherednikUnif3e}) can be written in more concrete\nterms as follows. We write\n$\\mathbf{K} = \\mathbf{K}_{\\mathfrak{p}_0} \\mathbf{K}^{\\mathfrak{p}_0}$ where\n$\\mathbf{K}^{\\mathfrak{p}_0} \\subset (\\check{D}\\otimes_{F}\\mathbb{A}_{F,f}^{\\mathfrak{p}_0})^{\\times} = (D \\otimes_{F} \\mathbb{A}_{F,f}^{\\mathfrak{p}_0})^{\\times}$. For \n$g \\in (D \\otimes_{F} \\mathbb{A}_{F,f}^{\\mathfrak{p}_0})^{\\times}$, let\n$$\\Gamma_g = \\{d \\in \\check{D}^\\times \\cap g\\mathbf{K}^{\\mathfrak{p}_0}g^{-1} \\mid \\ord_{F_{\\mathfrak{p}_0}} \\det d = 0\\} .$$ \nLet $\\bar{\\Gamma}_g$ be the image of $\\Gamma_g$ by the natural map\n$\\check{D}^{\\times} \\rightarrow \\check{D}^{\\times}_{\\mathfrak{p}_0}\\rightarrow {\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$. \nThen $\\bar{\\Gamma}_g$ is a discrete cocompact subgroup of\n${\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$, comp. the proof of Proposition\n\\ref{uniform4l}. It acts properly discontinuously on the formal scheme $\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0} \\Spf O_{\\breve{E}_\\nu}$, and the quotients \n$\\mathfrak{X}_{\\Gamma_g} := \\bar{\\Gamma}_g \\backslash (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0} \\Spf O_{\\breve{E}_\\nu})$ are exactly the connected components of the formal scheme on the LHS of (\\ref{CherednikUnif3e}), for varying $g$. By \\cite{Mum}, $\\mathfrak{X}_{\\Gamma_g}$ \nis algebraizable, i.e. it is the formal scheme associated to a proper scheme\n$\\mathfrak{X}_{\\Gamma_g}^{\\mathrm{alg}}$ over $O_{\\breve{E}_\\nu}$. The general fibers\nof these schemes for varying $g$ give back the\nconnected components of ${\\mathrm{Sh}}(D^{\\times}, h_D)_{\\breve{E}_{\\nu}}$. \n\n\n\\begin{comment}\nWe reformulate the Theorem in a less precise form in the style of the classical \nuniformization theorem of Koebe, compare Cherednik \\cite{Ch}. We write\n$\\mathbf{K} = \\mathbf{K}_{\\mathfrak{p}_0} \\mathbf{K}^{\\mathfrak{p}_0}$ where\n$\\mathbf{K}^{\\mathfrak{p}_0} \\subset (\\check{D}\\otimes_{F}\\mathbb{A}_{F,f}^{\\mathfrak{p}_0})^{\\times} = (D \\otimes_{F} \\mathbb{A}_{F,f}^{\\mathfrak{p}_0})^{\\times}$. Let \n$g \\in (D \\otimes_{F} \\mathbb{A}_{F,f}^{\\mathfrak{p}_0})^{\\times}$ an arbitrary\nelement. We define the group\n$\\Gamma_g = \\{d \\in \\check{D} \\cap g\\mathbf{K}^{\\mathfrak{p}_0}g^{-1} \\; | \\; \\ord_{F_{\\mathfrak{p}_0}} \\det d = 0\\}$. \nLet $\\bar{\\Gamma}_g$ be the image of $\\Gamma_g$ by the natural map\n$\\check{D}^{\\times} \\rightarrow \\check{D}^{\\times}_{\\mathfrak{p}_0}\\rightarrow {\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$. \nThe group $\\bar{\\Gamma}_g$ is a discrete cocompact subgroup of\n${\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$. One can find in the proof of Proposition\n\\ref{uniform4l} some explanations. By \\cite{Mum} the formal scheme\n$\\mathfrak{X}_{\\Gamma_g} := \\bar{\\Gamma}_g \\backslash (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\chi_0} \\Spf O_{\\breve{E}_\\nu})$\nis algebraizable, i.e. it is the formal scheme associated to a proper scheme\n$\\mathfrak{X}_{\\Gamma_g}^{\\mathrm{alg}}$ over $O_{\\breve{E}_\\nu}$. The general fibers\nof these schemes over $\\Spec \\breve{E}_{\\nu}$ for varying $g$ are exactly the\nconnected components of ${\\mathrm{Sh}}(D^{\\times}, h_D)_{\\breve{E}_{\\nu}}$. \n\\end{comment}\nWe prove Theorem \\ref{MainIntro} using the method which Drinfeld \\cite{Dr}\nused in the case\n$F = \\mathbb{Q}$. The case $F \\neq \\mathbb{Q}$ becomes more difficult\nbecause in this case the Shimura curve is not described by a PEL-moduli \nproblem. In fact, the Shimura curve is then a Shimura variety of abelian type which is not of Hodge type. Also, the weight homomorphism $w\\colon \\ensuremath{\\mathbb {G}}\\xspace_m\\to D^\\times_\\ensuremath{\\mathbb {R}}\\xspace$ is not defined over $\\ensuremath{\\mathbb {Q}}\\xspace$. The existence of a canonical model is proved by the method of Shimura and Deligne \\cite[\\S 6]{D-TS}, by embedding this Shimura variety into one of PEL type (\\emph{m\\'ethode des mod\\`eles \\'etranges}). We use here a variant of this method to construct integral models over $O_{\\breve{E}_{\\nu}}$ of the Shimura curve. A similar approach was used by Carayol \\cite{C}. More precisely, we show that the Shimura curve\n${\\mathrm{Sh}}(D^{\\times}, h_D)$ can be embedded as an open and closed subscheme in a Shimura variety \n which is an unramified twist of a PEL-moduli scheme which has a natural integral model. This PEL-moduli scheme can be so chosen that it \nhas a $p$-adic uniformization by \\cite[\\S 6]{RZ}. In this way, we obtain the isomorphism \\eqref{CherednikUnif3e}. Finally we must determine the\ndescent datum to obtain the result over $E_{\\nu}$. Let us explain our strategy\nin more detail. \n\nLet $K\/F$ be a CM-field and assume that each $\\mathfrak{p}_i$ is split in\n$K$, i.e. $\\mathfrak{p}_i O_K = \\mathfrak{q}_i \\bar{\\mathfrak{q}_i}$.\nBy (\\ref{LDiagramm}) we write \n\\begin{equation}\\label{CherednikUnif4e}\n \\begin{array}{l} \n \\Phi := \\Hom_{\\mathbb{Q}{\\rm -Alg}}(K, \\mathbb{C}) = \n \\Hom_{\\mathbb{Q}{\\rm -Alg}}(K, \\bar{\\mathbb{Q}}_p) = \\\\[2mm] \n \\big(\\coprod_{i=0}^s \\Hom_{\\mathbb{Q}{\\rm -Alg}}(K_{\\mathfrak{q}_i}, \\bar{\\mathbb{Q}}_p)\\big)\n \\; \\coprod \\; \n \\big(\\coprod_{i=0}^s \\Hom_{\\mathbb{Q}{\\rm -Alg}}(K_{\\bar{\\mathfrak{q}}_i}, \\bar{\\mathbb{Q}}_p)\\big). \n \\end{array}\n\\end{equation}\nWe denote by $\\varphi_0 \\in \\Phi$ the extension of $\\chi_0$ which on the right\nhand side of (\\ref{CherednikUnif4e}) lies in the first summand. We define a\nfunction $r: \\Phi \\rightarrow \\{0,1,2\\}$ as follows. We set\n$r_{\\varphi_0} = r_{\\bar{\\varphi}_0} = 1$. If the restriction of $\\varphi \\in \\Phi$\nto $F$ is not $\\chi_0$ we set $r_{\\varphi} = 0$ if $\\varphi$ is in the first $s$\nsummands on the right hand side and $r_{\\varphi} = 2$ if $\\varphi$ is in the last\n$s$ summands. If $\\chi\\neq\\chi_0$ the extension $\\varphi$ of $\\chi$ such that\n$r_{\\varphi} = 2$ defines an isomorphism\n$K \\otimes_{F, \\chi} \\mathbb{R} \\cong \\mathbb{C}$. We define the group\nhomomorphism\n\\begin{displaymath}\n \\begin{array}{rcr}\n h_K: \\mathbb{C}^{\\times} & \\rightarrow & (K \\otimes \\mathbb{R})^{\\times} \\cong\n \\prod_{\\chi} (K \\otimes_{F, \\chi} \\mathbb{R})^{\\times} =\n (K \\otimes_{F, \\chi_0}\\mathbb{R})^{\\times} \\times \\prod_{\\chi \\neq \\chi_0}\n \\mathbb{C}^{\\times}.\n \\\\[2mm]\n 1 & \\mapsto & (1,z,z,\\ldots,z) \\qquad\\\\\n \\end{array}\n\\end{displaymath}\nLet $B = D^{{\\rm opp}} \\otimes_{F} K$. We denote by $d \\mapsto d^{\\iota}$ the main \ninvolution of $D$ and by $a \\mapsto \\bar{a}$ the conjugation of $K\/F$.\nWe denote by $b \\mapsto b'$ the involution of the second kind on $B\/K$\nwhich is defined by $d \\otimes a \\mapsto d^{\\iota} \\otimes \\bar{a}$. \nLet $V = B$ considered as a $B$-left module. Multiplication from the right\ndefines a ring homomorphism\n\\begin{displaymath}\nD \\otimes_F K \\rightarrow \\End_B V. \n\\end{displaymath}\nIn particular the group $D^\\times \\times K^\\times$ acts on $V$. \nBy (\\ref{defhD}) we obtain a ring homomorphism \n\\begin{displaymath}\n \\mathbb{C} \\rightarrow D \\otimes_{F, \\chi_0} \\mathbb{R} \\rightarrow\n (D \\otimes_F K) \\otimes_{F, \\chi_0} \\mathbb{R}\n \\end{displaymath}\nand by the isomorphisms $K \\otimes_{F, \\chi} \\mathbb{R} \\cong \\mathbb{C}$\nchosen above for $\\chi \\neq \\chi_0$ we obtain ring homomorphisms \n\\begin{displaymath}\n \\mathbb{C} \\rightarrow K \\otimes_{F, \\chi} \\mathbb{R} \\rightarrow\n (D \\otimes_F K) \\otimes_{F, \\chi} \\mathbb{R}. \n\\end{displaymath}\nTaking the product of these ring homomorphisms over all \n$\\chi: F \\rightarrow \\mathbb{R}$ we obtain a ring homomorphism\n\\begin{displaymath}\n\\mathbb{C} \\rightarrow (D \\otimes_F K) \\otimes_{\\mathbb{Q}} \\mathbb{R} \n\\end{displaymath}\nand therefore a complex structure on the real vector space\n$V \\otimes \\mathbb{R}$. \nAlternatively, this complex structure is given by the group homomorphism \n\\begin{displaymath}\n h = h_D \\times h_K: \\mathbb{S} \\rightarrow\n \\prod_{\\chi\\in \\Hom_{\\mathbb{Q}{\\rm -Alg}}(F,\\mathbb{C})} \\big((D\\otimes_{F,\\chi} \\mathbb{R})^{\\times}\n \\times (K \\otimes_{F, \\chi} \\mathbb{R})^{\\times}\\big) ,\n\\end{displaymath}\nwhere the group on the right hand side acts on $V \\otimes \\mathbb{R}$ by\nthe action of $D^\\times \\times K^\\times$ on $V$. \n\nWe consider\n$\\mathbb{Q}$-bilinear forms $\\psi: V \\times V \\rightarrow \\mathbb{Q}$\nsuch that\n\\begin{displaymath}\n\\psi(x b , y) = \\psi(x , y b'), \\quad \\text{for}\\; x,y \\in V, \\; b \\in B. \n\\end{displaymath}\nBy \\cite{D-TS} one can choose $\\psi$ in such a way that the complex structure\n$h$ satisfies the Riemann period relations. We consider \n$G^{\\bullet} = \\{b \\in B^{{\\rm opp}} \\mid b'b \\in F^{\\times}\\}$ as an algebraic\ngroup over $\\mathbb{Q}$. The right multiplication by elements $d \\otimes 1$ and\n$1 \\otimes a$ define elements of $G^{\\bullet}$. This gives a homomorphism of algebraic groups,\n\\begin{equation}\\label{DmalK-G.1e}\n D^{\\times} \\times K^{\\times} \\overset{\\kappa}{\\rightarrow}\n G^{\\bullet} .\n \\end{equation}\nThen $(G^{\\bullet}, h)$ is the Shimura datum for a Shimura variety of PEL-type. \nBy (\\ref{DmalK-G.1e}) we have an embedding $D^\\times \\rightarrow G^{\\bullet}$. The decomposition\n$B\\otimes\\mathbb{Q}_p=\\prod_{i=0}^s (B_{\\mathfrak{q}_i}\\times B_{\\bar{\\mathfrak{q}}_i})$\ninduces a similiar decomposition of $V \\otimes \\mathbb{Q}_p$.\n We choose maximal orders $O_{D_{\\mathfrak{p}_i}} \\subset D_{\\mathfrak{p}_i}$ and hence\n maximal orders $O_{B_{\\mathfrak{q}_i}} \\subset B_{\\mathfrak{q}_i}$. \n We assume in the definition \\eqref{defKp} that\n $\\mathbf{K}_{\\mathfrak{p}_i} \\subset O^{\\times}_{D_{\\mathfrak{p}_i}}$.\n There is a natural isomorphism\n$D_{\\mathfrak{p}_i}^{\\times} \\cong (B^{{\\rm opp}}_{\\mathfrak{q}_i})^{\\times}$. The image\n$\\mathbf{K}_{\\mathfrak{p}_i}$ by this isomorphism will be denoted by\n$\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i}$. From these last groups we define a subgroup\n$\\mathbf{K}^{\\bullet}_p \\subset G^{\\bullet}(\\mathbb{Q}_p)$, cf. (\\ref{BZKpPkt1e})\nwith $\\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} = O_{F_{\\mathfrak{p}_i}}^{\\times}$. This\nsubgroup satisfies\n$\\mathbf{K}_p = D^\\times(\\mathbb{Q}_p) \\cap \\mathbf{K}^{\\bullet}_p$. Moreover \nwe choose $\\mathbf{K}^{\\bullet,p}$ such that\n$\\mathbf{K}^p = (D\\otimes\\mathbb{A}_f^p)^\\times \\cap \\mathbf{K}^{\\bullet,p}$. \n\nThe form $\\psi$ induces an involution $\\star$ of the second kind on $B$,\n \\begin{displaymath}\n\\psi(bx, y) = \\psi(b^{\\star}x, y), \\quad x,y \\in V, \\; b \\in B. \n \\end{displaymath}\n We denote by $O_{B_{\\bar{\\mathfrak{q}}_i}} \\subset B_{\\bar{\\mathfrak{q}}_i}$ the image\n of $O_{B_{\\mathfrak{q}_i}}$ by $\\star$. Let $O_{B, (p)}$ be the set of elements of $B$\n whose images in $B_{\\mathfrak{q}_i}$ and $B_{\\bar{\\mathfrak{q}}_i}$ lie in the\n chosen maximal orders. \n We obtain the lattice $\\Lambda_{\\mathfrak{q}_i} = O_{B_{\\mathfrak{q}_i}} \\subset V_{\\mathfrak{q}_i}$. \n\n Let $U_p(F) \\subset F^{\\times}$ be the subgroup of elements which are units in\n each $F_{\\mathfrak{p}_i}$. We\n define the following functor on the category of $O_{E_{\\nu}}$-schemes $S$. The upper index $t$ is referring to the fact that $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$, when restricted to the category of $E_\\nu$-schemes, is a twisted form of another functor ${\\mathcal{A}}^{\\bullet }_{\\mathbf{K}^{\\bullet}}$.\n\n \\begin{definition}\nLet $S$ be an $O_{E_{\\nu}}$-scheme. \nA point of $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(S)$ consists of the \nfollowing data: \n\\begin{enumerate} \n\\item[(a)] An abelian scheme $A$ over $S$ up to isogeny prime to $p$ with an\n action \n $\\iota: O_{B,(p)} \\rightarrow \\End A \\otimes_{\\mathbb{Z}} \\mathbb{Z}_{(p)}$.\n\n\\item[(b)] \n An $U_p(F)$-homogeneous polarization $\\bar{\\lambda}$ of $A$ which is\n principal in $p$. \n\n \n\\item[(c)]\n A class $\\bar{\\eta}^p$ modulo $\\mathbf{K}^{\\bullet, p}$ of\n $B \\otimes \\mathbb{A}^p_f$-module isomorphisms\n \\begin{displaymath}\n \\eta^p: V \\otimes \\mathbb{A}^p_f \\isoarrow \\mathrm{V}^p_f(A) ,\n \\end{displaymath}\n such that \n \\begin{displaymath}\n\\psi(\\xi^{(p)}(\\lambda) v_1, v_2) = E^{\\lambda}(\\eta^p(v_1), \\eta^p(v_2))\n \\end{displaymath}\n for some function $\\xi^{(p)}(\\lambda) \\in (F \\otimes \\mathbb{A}^p_f)^{\\times}(1)$\n on $\\bar{\\lambda}$. \n\\item[(e)]\n A class $\\bar{\\eta}_{\\mathfrak{q}_i}$ modulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i}$\n of $O_{B_{\\mathfrak{q}_i}}$-module isomorphisms for each $i = 1, \\ldots s$, \n \\begin{displaymath}\n \\eta_{\\mathfrak{q}_i}: \\Lambda_{\\mathfrak{q}_i} \\isoarrow T_{\\mathfrak{q}_i}(A) .\n \\end{displaymath} \n \\end{enumerate}\nWe require that the following Kottwitz condition ${\\rm (KC)}$ holds,\n \\begin{equation}\n {\\rm char}(T, \\iota(b) \\mid \\Lie A) = \\prod_{\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}}\n \\varphi(\\Nm^o_{B\/K} (T -b))^{r_{\\varphi}} . \n \\end{equation}\n\n\\end{definition}\n The general fiber over $E_{\\nu}$ of this functor is a Galois form of \n ${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h)_{E_{\\nu}}$ but this is irrelevant for\n this Introduction. \n We prove that the \\'etale sheafification of \n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ is representable, cf. \n Proposition \\ref{BZ8p}. We also show that (cf. (\\ref{Hecke4e})) \n \\begin{equation}\\label{Intro1e}\n (K \\otimes \\mathbb{Q}_p)^{\\times} = \\prod_{i=0}^{s} K_{\\mathfrak{q}_i}^{\\times} \\times\n \\prod_{i=0}^{s} K_{\\bar{\\mathfrak{q}}_i}^{\\times} \n \\end{equation} \n acts by Hecke operators on the functor\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$.\n \n We consider the homomorphism\n \\begin{equation}\\label{def:bulletshim}\n h_D^{\\bullet} = h_D \\times 1: \\mathbb{C}^{\\times} \\rightarrow\n (D \\otimes \\mathbb{R})^{\\ast} \\times (K \\otimes \\mathbb{R})^{\\ast}\n \\rightarrow G^{\\bullet}_{\\mathbb{R}}. \n \\end{equation} \nThe Shimura variety ${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ \nis defined over $E_{\\nu}$. It is a Galois form of\n${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h)$. \n\nWe find a model\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ over $O_{E_{\\nu}}$\nof this Shimura variety and a commutative diagram\n \\begin{equation}\\label{Intro2e}\n \\begin{aligned}\\xymatrix{\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \n \\ar[d]_{\\dot{z} \\times \\tau_c} \\ar[r]\n & \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \n \\times_{\\Spec O_{E_{\\nu}}} \\Spec O_{E^{nr}_{\\nu}} \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\\\\ \n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \\ar[r] &\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \n \\times_{\\Spec O_{E_{\\nu}}} \\Spec O_{E^{nr}_{\\nu}} ,\\\\ \n } \n \\end{aligned}\n \\end{equation}\n cf. Proposition \\ref{Sh_D1p} and (\\ref{tildeSh_D1e}). \n Here $E_{\\nu}^{nr}$ is the maximal unramified extension of $E_{\\nu}$, and \n $\\tau \\in \\Gal(E_{\\nu}^{nr}\/E_{\\nu})$ is the Frobenius automorphism, and \n $\\tau_c = \\Spec \\tau^{-1}$. We denote by\n $\\pi_{\\mathfrak{p}_0} \\in F_{\\mathfrak{p}_0} = K_{\\bar{\\mathfrak{q}}_0}$ a prime\n element and by $f_{\\nu}$ the inertia index of $E_{\\nu}\/\\mathbb{Q}_p$. \n The element \n \\begin{displaymath}\n \\dot{z} = (1, \\ldots, 1) \\times\n (\\pi_{\\mathfrak{p}_0}^{-1} p^{f_{\\nu}}, p^{f_{\\nu}}, \\ldots p^{f_{\\nu}})\n \\end{displaymath}\n from the right hand side of (\\ref{Intro1e}) acts as an Hecke operator.\n \n The horizontal arrow in the diagram (\\ref{Intro2e}) is the \\'etale\n sheafification. It follows from \\cite{RZ} that the \\'etale sheafification of\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ has a $p$-adic uniformization\n by the formal scheme $\\hat{\\Omega}^2_{E_\\nu}$, cf. Theorem \\ref{4epeg1t}. \n This gives a uniformization of the model \n $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$. The embedding of Shimura data $(D^{\\times}, h_{D}) \\subset\n (G^{\\bullet}, h^{\\bullet}_{D})$ and the fact that $\\mathbf{K}\\subset \\mathbf{K}^\\bullet$ define a morphism of Shimura varieties $ {\\mathrm{Sh}}_{\\mathbf{K}}(D^{\\times}, h_{D})_{E_{\\nu}} \\to\n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}} $.\nBy a theorem of Chevalley, for suitable\n$\\mathbf{K}^{\\bullet,p} \\in G^{\\bullet}(\\mathbb{A}^p_f)$ of the type considered\nabove, this morphism induces an open and closed embedding,\n \\begin{equation}\n {\\mathrm{Sh}}_{\\mathbf{K}}(D^{\\times}, h_{D})_{E_{\\nu}} \\subset\n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}} .\n \\end{equation}\n The closure of the left hand side in\n $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ gives the stable \n model $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(D^{\\times}, h_D)$ whose formal scheme\n inherits a uniformization by $\\hat{\\Omega}^2_{E_{\\nu}}$, proving the main theorem. \n \nSo far, we have only mentioned the Shimura pairs $(G^\\bullet, h)$ and\n$(G^\\bullet, h^\\bullet_D)$. However, in the body of the paper, also Shimura pairs\n$(G, h)$ and $(G, h\\delta)$ play an important role. Here $G\\subset G^\\bullet$ is the subgroup where the similitude factor lies in $\\ensuremath{\\mathbb {Q}}\\xspace$, and $\\delta$ is a central character of $G$. The Shimura variety for $(G, h)$ is of PEL-type and has the key property that it is a fine moduli scheme for a moduli problem $\\ensuremath{\\mathcal {A}}\\xspace_{\\mathbf K}$, for small enough level ${\\mathbf K}$. Similarly, the Shimura variety for $(G, h\\delta)$ is the unramified twist of the fine moduli scheme for a moduli problem $\\ensuremath{\\mathcal {A}}\\xspace^t_{\\mathbf K}$, which, furthermore, has a natural extension $\\tilde\\ensuremath{\\mathcal {A}}\\xspace^t_{\\mathbf K}$ over $\\Spec O_{E_\\nu}$. The fine moduli scheme for $\\tilde\\ensuremath{\\mathcal {A}}\\xspace^t_{\\mathbf K}$ is then used to show that the horizontal arrow in the diagram (\\ref{Intro2e}) is the \\'etale sheafification. \n \n The lay-out of the paper is as follows. In \\S \\ref{s:LA} we explain the linear algebra behind the formation of the Shimura pairs $(G^\\bullet, h)$ and $(G, h)$. In \\S \\ref{s:AG}, we explain the Shimura varieties for $(G, h)$ and $(G, h\\delta)$ and the corresponding moduli problems $\\ensuremath{\\mathcal {A}}\\xspace_{\\mathbf K}$ and $\\ensuremath{\\mathcal {A}}\\xspace^t_{\\mathbf K}$ and the integral extension $\\tilde\\ensuremath{\\mathcal {A}}\\xspace^t_{\\mathbf K}$ of the latter. In \\S \\ref{s:AGbullet} we explain the Shimura varieties for $(G^\\bullet, h)$ and $(G^\\bullet, h_D^\\bullet)$ and the corresponding moduli problems $\\ensuremath{\\mathcal {A}}\\xspace^\\bullet_{\\mathbf K}$ and $\\ensuremath{\\mathcal {A}}\\xspace^{\\bullet, t}_{\\mathbf K}$ and the integral extension $\\tilde\\ensuremath{\\mathcal {A}}\\xspace^t_{\\mathbf K}$. Furthermore, we establish a relation between the integral extensions $\\tilde\\ensuremath{\\mathcal {A}}\\xspace^t_{\\mathbf K}$ and the integral extension $\\tilde\\ensuremath{\\mathcal {A}}\\xspace^{\\bullet, t}_{\\mathbf K}$ and use this to show that the horizontal arrow in the diagram (\\ref{Intro2e}) is the \\'etale sheafification. In \\S \\ref{s:RZ} we explain the Rapoport-Zink spaces relevant to these moduli problems. In \\S \\ref{s:uniform} we prove the $p$-adic uniformization of the integral models of the Shimura varieties for the pairs $(G^\\bullet, h^\\bullet_D)$ and $(D^\\times, h_D)$. The last two sections are really appendices. In \\S \\ref{s:desc}, we clarify our conventions about Galois descent, and in \\S \\ref{s:shimvar} we make precise our sign conventions for Shimura varieties.\n We formulate a result of Kisin \\cite{KisinJAMS} on embeddings of Shimura\n varieties in the form needed here. \n\n \n The present paper is an improved version of parts of the preprint \\cite{BZ}.\n The strategy here is the same but some serious gaps in the arguments are\n repaired. However, not all results of \\cite{BZ} are covered. \n\n We thank M. Rapoport for his many useful suggestions which helped to improve\n our work.\n \n \\section{The Shimura data}\\label{s:LA}\n In this section, we introduce the linear algebra which leads to the\n definition of the Shimura pairs $(G^\\bullet, h^\\bullet_D)$ and $(G^\\bullet, h)$\n and $(G, h)$.\n \n Let $K\/F$ be a CM-field.\nLet $a \\mapsto \\bar{a}$ be the conjugation of $K\/F$. We\nconsider a quaternion algebra $D$ over $F$. Let $d \\mapsto d^{\\iota}$ be the\nmain involution of $D$. We set $B = D^{{\\rm opp}} \\otimes_F K$. We extend the map\n$d \\mapsto d^{\\iota}$ $K$-linearily to $B$. Then we obtain the main involution\n$b \\mapsto b^{\\iota}$ of $B\/K$. The conjugation acts via the second factor\non $B = D^{{\\rm opp}} \\otimes_F K$. We set $b' = \\bar{b}^{\\iota}$. We consider the\nsesquilinear form \n\\begin{displaymath}\n \\varkappa_0: B \\times B \\rightarrow K, \\quad\n \\varkappa_0(b_1, b_2) = {\\rm Tr}^{o}_{B\/K} b_2 b'_1. \n\\end{displaymath}\nIt is $K$-linear in the second variable and antilinear in the first and\nit is hermitian\n\\begin{displaymath}\n \\varkappa_0(b_1, b_2) = \\overline{\\varkappa_0(b_2, b_1)}.\n \\end{displaymath}\nMoreover we obtain \n\\begin{equation}\\label{BZ1e} \n \\varkappa_0(x b, y) = \\varkappa_0(x, y b'), \\quad \n \\varkappa_0(bx, y) = \\varkappa_0(x, b'y),\\quad x, y, b \\in B. \n\\end{equation}\nWe set\n\\begin{equation}\\label{Gpunkt2e} \nG^{\\bullet} = \\{b \\in B^{{\\rm opp}} \\; | \\; b'b \\in F^{\\times} \\}, \n\\end{equation}\nand consider it as an algebraic group over $\\mathbb{Q}$. We write\n$\\tilde{G}^{\\bullet}$ if we consider it as an algebraic group over $F$, i.e.\n$\\Res_{F\/\\mathbb{Q}} \\tilde{G}^{\\bullet} = G^{\\bullet}$. \n\nWe will write $V = B$ considered as a $B$-left module. The right multiplication\nby an element of $B$ gives an isomorphism $\\End_B V = B^{{\\rm opp}} = D \\otimes_F K$.\nTherefore we can write \n\\begin{equation}\\label{Gpunkt1e} \n G^{\\bullet} = \\{g \\in GL_B (V) \\mid \\varkappa_0(g v_1, g v_2) =\n \\mu(g) \\varkappa_0 (v_1, v_2), \\mu(g) \\in F^{\\times} \\} . \n \\end{equation}\n\n\n\\begin{lemma}\\label{BZ1l} \n There is an exact sequence of algebraic groups over $\\ensuremath{\\mathbb {Q}}\\xspace$,\n \\begin{displaymath}\n 0 \\rightarrow F^{\\times} \\rightarrow D^{\\times} \\times K^{\\times}\n \\overset{\\kappa}{\\rightarrow}\n G^{\\bullet} \\rightarrow 0. \n \\end{displaymath}\n The map $\\kappa$ maps $(d, k)$ to $d \\otimes k$. \\qed\n \\end{lemma} \nWe set $\\Phi = \\Hom_{\\mathbb{Q}{\\rm -Alg}}(K, \\mathbb{C})$. We assume that there is\na unique embedding $\\chi_0: F \\rightarrow \\mathbb{R}$ such that the quaternion\nalgebra $D \\otimes_{F, \\chi_0} \\mathbb{R}$ splits. We consider a generalized\nCM-type of rank $2$ in the sense of \\cite{KRnew}, comp. \\cite{KRZ},\n\\begin{equation}\\label{BZgCM1e} \nr: \\Phi \\rightarrow \\mathbb{Z}_{\\geq 0}, \n \\end{equation}\nsuch that $r_{\\varphi_0} = r_{\\bar{\\varphi}_0} = 1$ for the extensions\n$\\varphi_0, \\bar{\\varphi}_0: K \\rightarrow \\mathbb{C}$ of $\\chi_0$ \nand such that $r_{\\varphi} = 0, 2$ for all other $\\varphi \\in \\Phi$. \n\nWe will define a complex structure on the $\\mathbb{R}$-vector space\n$V \\otimes \\mathbb{R}=B \\otimes \\mathbb{R}$. For this we consider the decomposition\n\\begin{displaymath}\n B \\otimes \\mathbb{R} = \\bigoplus_{\\chi: F \\rightarrow \\mathbb{R}} B \\otimes_{F, \\chi}\n \\mathbb{R} = \\bigoplus_{\\chi}\\big( (D^{{\\rm opp}} \\otimes_{F, \\chi} \\mathbb{R})\n \\otimes_{\\mathbb{R}} (K \\otimes_{F, \\chi} \\mathbb{R})\\big). \n\\end{displaymath}\nWe define the complex structure on each summand on the right hand side.\nLet $\\chi \\neq \\chi_0$ and let\n$\\varphi : K \\rightarrow \\mathbb{C}$ be the\nextension of $\\chi$ such that $r_{\\varphi} = 2$. Then $\\varphi$ defines an\nisomorphism \n$K \\otimes_{F, \\chi} \\mathbb{R} \\cong \\mathbb{C}$. This induces a complex\nstructure on the summand belonging to $\\chi$ via the second factor of the\ntensor product. For $\\chi_0$ we have\n$D \\otimes_{F, \\chi_0} \\mathbb{R} \\cong {\\rm M}_2(\\mathbb{R})$. We endow the\n$\\mathbb{R}$-vector space $D^{{\\rm opp}} \\otimes_{F, \\chi_0} \\mathbb{R}$ with the\ncomplex structure $J_{\\chi_0}$ given by right multiplication by \n\\begin{displaymath}\n J_{\\chi_0} =\n\\left(\n \\begin{array}{rr} \n 0 & -1\\\\\n 1 & 0\n \\end{array}\n \\right). \n\\end{displaymath}\nThis induces a complex structure on \n$(D^{{\\rm opp}}\\otimes_{F,\\chi_0}\\mathbb{R})\\otimes_{\\mathbb{R}}(K\\otimes_{F,\\chi_0}\\mathbb{R})$\nvia the first factor. Together we obtain a complex structure $J$ on\n$B \\otimes_{\\mathbb{Q}} \\mathbb{R}$ such that \n\\begin{displaymath}\n \\Trace_{\\mathbb{C}} (k | (B \\otimes_{\\mathbb{Q}} \\mathbb{R}, J)) =\n \\sum_{\\varphi \\in \\Phi} 2 r_{\\varphi} \\varphi(k), \\quad k\\in K. \n \\end{displaymath}\nThis complex structure on $V \\otimes_{\\mathbb{Q}} \\mathbb{R}$ commutes with the\n$B \\otimes_{\\mathbb{Q}} \\mathbb{R}$-module structure and defines therefore a\nhomomorphism $\\mathbb{C} \\rightarrow B^{{\\rm opp}} \\otimes_{\\mathbb{Q}} \\mathbb{R}$.\nThis homomorphism induces a homomorphism of groups \n\\begin{equation}\\label{BZh1e}\n h: \\mathbb{S} \\rightarrow \\prod_{\\chi \\in \\Hom_{\\mathbb{Q}{\\rm -Alg}}(F, \\mathbb{C})}\n \\big((D \\otimes_{F, \\chi} \\mathbb{R})^{\\times}\n \\times (K \\otimes_{F, \\chi} \\mathbb{R})^{\\times}\\big). \n\\end{equation}\nLet $z \\in \\mathbb{C}^\\times = \\mathbb{S}(\\mathbb{R})$. Then the $\\chi_0$-component\n$h_{\\chi_0}(z)$ is\n\\begin{displaymath}\n\\left(\n \\begin{array}{rr} \n a & -b\\\\\n b & a\n \\end{array}\n \\right) \\times 1, \\quad z = a + b{\\bf i} , \n\\end{displaymath}\nand for $\\chi \\neq \\chi_0$ the component $h_{\\chi}(z)$ is\n$1 \\times 1\\otimes z \\in (D \\otimes_{F,\\chi}\\mathbb{R})^{\\times}\\times (K \\otimes_{K, \\varphi} \\mathbb{C})^{\\times}$.\nHere $\\varphi \\in \\Phi$ is the extension of $\\chi$ with $r_{\\varphi} = 2$.\nWe have used the natural isomorphism\n$K \\otimes_{F, \\chi} \\mathbb{R} = K \\otimes_{K, \\varphi} \\mathbb{C}$.\nWe can write $h({\\bf i}) = J$. The composite with the projection to\n$G^{\\bullet}_{\\mathbb{R}}$ given by Lemma \\ref{BZ1l} is also denoted by $h$,\n\\begin{equation}\\label{hforbul}\nh\\colon \\ensuremath{\\mathbb {S}}\\xspace\\to G^{\\bullet}_{\\mathbb{R}} .\n\\end{equation} \n\\begin{lemma}\n There exist elements $\\gamma \\in B$ such that \n $\\mathfrak{h}(x,y) = \\varkappa_0(\\gamma x, y J)$ is hermitian and positive\n definite on $B \\otimes \\mathbb{R}$. More precisely, this means that for\n each $\\varphi$ the form\n $$\\mathfrak{h}_{\\varphi}: B \\otimes_{K, \\varphi} \\mathbb{C}\n \\times B \\otimes_{K, \\varphi} \\mathbb{C} \\rightarrow K \\otimes_{K, \\varphi}\n \\mathbb{C}$$\n is hermitian and positive definite. \n\\end{lemma}\nThis follows as in \\cite{D-TS}. Note that alternatively we can say that\n$\\Trace_{K\/F} \\mathfrak{h}$ is symmetric and positive definite on\n$B \\otimes \\mathbb{R}$. Let\n\\begin{equation}\\label{defG}\nG = \\{b \\in B^{{\\rm opp}} \\; | \\; b'b \\in \\mathbb{Q}^{\\times} \\} \\subset G^{\\bullet}. \n\\end{equation}\nSince $h(z)'h(z) = \\bar{z} z \\in \\mathbb{R}^{\\times}$ for $z \\in \\mathbb{C}^{\\times}=\\ensuremath{\\mathcal {S}}\\xspace(\\ensuremath{\\mathbb {R}}\\xspace)$,\nthe morphism $h$ factors through $G_{\\mathbb{R}}$. We define\n\\begin{displaymath}\n\\varkappa: V \\times V \\rightarrow K , \\quad \\quad\\varkappa(x, y) = \\varkappa_0(\\gamma x, y), \\quad x, y \\in V = B.\n \\end{displaymath}\n The first \nequation of (\\ref{BZ1e}) continues to hold for $\\varkappa$. We have\n$\\mathfrak{h}(x, y) = \\varkappa(x, y J)$. We note that $\\varkappa$ is an\nantihermitian form: \n\\begin{displaymath}\n \\overline{\\varkappa(y, x)} = - \\overline{\\varkappa(y, x J^{2})} =\n - \\overline{\\mathfrak{h}(y, x J)} = - \\mathfrak{h}(x J, y) =\n - \\varkappa(x J, y J) = - \\varkappa(x, y). \n\\end{displaymath}\nIt is easily seen that $\\gamma' = -\\gamma$ is equivalent with the property\nthat $\\varkappa$ is antihermitian or that $\\mathfrak{h}$ is hermitian. Equivalently\none could use the alternating\nform \n\\begin{equation}\\label{BZpsi1e}\n\\psi : V \\times V \\rightarrow \\mathbb{Q}, \\quad\\psi(x, y) = \\Trace_{K\/\\mathbb{Q}} \\varkappa (x, y), \\quad x,y \\in B ,\n \\end{equation} \nwhich satisfies\n\\begin{displaymath}\n\\psi(k x, y) = \\psi (x, \\bar{k} y), \\quad k \\in K. \n\\end{displaymath}\nThen $\\psi (x, y J)$ is symmetric and positive definite. We define an involution $b \\mapsto b^{\\star}$ on $B$ by\n\\begin{equation}\\label{defstar}\n\\varkappa(b x, y) = \\varkappa(x , b^{\\star} y). \n\\end{equation}\nBecause the same equation holds for $\\mathfrak{h}$, the involution\n$b \\mapsto b^{\\star}$ is positive. From the definition we obtain\n$b = \\gamma^{-1} b' \\gamma$. We obtain\n\\begin{equation}\\label{BZpsi3e}\n \\psi(b x, y) = \\psi(x , b^{\\star} y), \\quad \\psi(x b , y) = \\psi(x , y b'),\n \\quad x,y,b \\in B. \n \\end{equation}\nWe can also write \n\\begin{equation}\\label{BZh3e}\n G = \\{g \\in \\End_B(V) \\; | \\; \\psi(g x, g y) = \\mu(g) \\psi(x, y), \\;\n \\text{for} \\; \\mu(g) \\in \\mathbb{Q}^{\\times} \\}. \n\\end{equation}\nWe also obtain $G$ if we replace on the right hand side $\\psi$ by $\\varkappa$. \n\nThe action of $ g = (d, k) \\in D^{\\times} \\times K^{\\times}$ on $B$ is by\ndefinition\n\\begin{displaymath}\n(d,k) (u \\otimes a) = ud \\otimes ak, \\quad u \\otimes a \\in D^{{\\rm opp}} \\otimes_{F} K \n= B. \n\\end{displaymath}\nThe product $ud$ is taken in $D^{{\\rm opp}}$.\n\nThe homomorphism induced by (\\ref{BZh1e}) \n\\begin{equation}\\label{BZh2e} \nh: \\mathbb{S} \\rightarrow G_{\\mathbb{R}} \n \\end{equation}\ngives a Shimura datum in the sense of \\cite{D-TS}, except that we denote by\n$h$ what is $h^{-1}$ in Deligne's normalization. The Hodge structure on $V$ is therefore\nin this paper of type $(1,0), (0,1)$. \n\n\nWe fix a prime number $p$ and we choose a diagram\n\\begin{equation}\\label{BZ2e}\n\\mathbb{C} \\leftarrow \\bar{\\mathbb{Q}} \\rightarrow \\bar{\\mathbb{Q}}_p.\n\\end{equation} \nBy this diagram we obtain $\\Phi = \\Hom_{\\mathbb{Q}{\\rm -Alg}}(K, \\bar{\\mathbb{Q}}_p)$. \nWe assume that all prime ideals of $O_F$ containing $pO_F$ are split in $K\/F$. \nWe denote these prime ideals of $O_F$ by\n\\begin{equation}\\label{BZsplit1e}\n\\mathfrak{p}_0, \\ldots, \\mathfrak{p}_s.\n\\end{equation}\nLet $\\mathfrak{q}_i, \\bar{\\mathfrak{q}}_i$ the two prime ideals of $O_K$ over\n$\\mathfrak{p}_i$. We obtain\n\\begin{displaymath}\n\\mathfrak{p}_i O_K = \\mathfrak{q}_i \\bar{\\mathfrak{q}}_i. \n \\end{displaymath}\nWe obtain a decomposition \n\\begin{equation}\\label{p-zerlegt} \n \\begin{array}{l} \n \\Hom_{\\mathbb{Q}{\\rm -Alg}}(K, \\bar{\\mathbb{Q}}_p) = \\\\[2mm] \n \\quad (\\coprod_{i=0}^{s}\n \\Hom_{\\mathbb{Q}_p{\\rm -Alg}}(K_{\\mathfrak{q}_i}, \\bar{\\mathbb{Q}}_p))\n \\; \\amalg\\;\n (\\coprod_{i=0}^{s}\n \\Hom_{\\mathbb{Q}_p{\\rm -Alg}}(K_{\\bar{\\mathfrak{q}}_i},\\bar{\\mathbb{Q}}_p)). \n \\end{array}\n\\end{equation} \n\n\nWe denote the components of this disjoint sum by\n$\\Phi_{\\mathfrak{q}_i}$, resp. $\\Phi_{\\bar{\\mathfrak{q}}_i}$. We assume that\n$\\varphi_0 \\in \\Phi_{\\mathfrak{q}_0}$ and $ \\bar{\\varphi}_0 \\in \\Phi_{\\bar{\\mathfrak{q}}_0}$.\nFor all other $\\varphi$ we require that\n\\begin{equation}\\label{BZEnu1e} \n \\begin{array}{ll}\n r_{\\varphi} = 0 & \\text{if} \\; \\varphi \\in \\Phi_{\\mathfrak{q}_i} \\;\n \\text{for some} \\; i = 0, \\ldots, s,\n \\; \\text{and} \\, \\varphi \\neq \\varphi_0 \\\\\n r_{\\varphi} = 2 & \\text{if} \\; \\varphi \\in \\Phi_{\\bar{\\mathfrak{q}}_i} \\; \n \\text{for some} \\; i = 0, \\ldots, s,\n \\; \\text{and} \\, \\varphi \\neq \\bar{\\varphi}_0 . \n \\end{array}\n \\end{equation}\nLet $E = E(G,h)$ be the reflex field, i.e.,\n\\begin{equation}\\label{BZE1e}\n \\Gal(\\bar{\\mathbb{Q}}\/E) = \\{\\sigma \\in \\Gal(\\bar{\\mathbb{Q}}\/\\mathbb{Q})\n \\; | \\; r_{\\sigma\\varphi} = r_{\\varphi}, \\; \\text{for all}\\; \\varphi \\in \\Phi\\}. \n\\end{equation}\nThe embedding $E \\rightarrow \\bar{\\mathbb{Q}}_p$ in the sense of diagram\n(\\ref{BZ2e}) defines a place $E_{\\nu} \\subset \\bar{\\mathbb{Q}}_p$. We call this\nthe\\emph{ local Shimura field}. If $\\varphi \\neq \\varphi_0, \\bar{\\varphi}_0$, the\nnumber $r_{\\varphi}$ depends only on the place $\\mathfrak{q}_i$ of $K$ which is\ninduced by $\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}_p$. We conclude that\n\\begin{equation}\\label{BZ3e} \n \\Gal(\\bar{\\mathbb{Q}}_p\/E_{\\nu}) = \\{\\sigma \\in\n \\Gal(\\bar{\\mathbb{Q}}_p\/\\mathbb{Q}_p) \\; | \\;\n r_{\\sigma \\varphi_0} = r_{\\varphi_0} \\}. \n\\end{equation}\n The condition (\\ref{BZ3e})\non $\\sigma$ signifies that $\\sigma$ fixes the embedding\n$F_{\\mathfrak{p}_0} \\rightarrow \\bar{\\mathbb{Q}}_p$ induced by $\\varphi_0$.\nWe obtain that\n\\begin{equation}\\label{BZ4e}\n E_{\\nu} = \\varphi_0(F_{\\mathfrak{p}_0}). \n\\end{equation}\nWe remark that $E_\\nu$ coincides with the localization of the Shimura field\n$\\chi_0(F)$ of the Shimura curve we have chosen, cf. the beginning of this\nsection. \n\n\\bigskip \n\n\n\\section{The moduli problem for ${\\mathrm{Sh}}(G, h)$ and a reduction modulo $p$}\\label{s:AG}\n \n\n\nWe consider the alternating $\\mathbb{Q}$-bilinear form $\\psi$ on the\n$B$-module $V$ (\\ref{BZpsi1e}). It satisfies \n\\begin{displaymath}\n\\psi(b v_1, v_2) = \\psi(v_1, b^{\\star} v_2) \\quad v_1, v_2 \\in V. \n\\end{displaymath}\nWe state the moduli problem associated to the $B$-module $V$ and the alternating form $\\psi$, cf. \\cite[4.10]{D-TS}. Recall \n$(G, h)$ from (\\ref{BZh3e}), (\\ref{BZh2e}).\n\nLet $\\mathbf{K} \\subset G(\\mathbb{A}_f)$ be an open compact subgroup. \nThe Shimura variety ${\\mathrm{Sh}}(G,h)_{\\mathbf{K}}$ is the coarse moduli scheme of the\nfollowing functor $\\mathcal{A}_{\\mathbf{K}}$ on the category of schemes over\n$E=E(G, h)$. If $\\mathbf{K}$ satisfies the condition (\\ref{BZneat1e})\nbelow, the functor $\\mathcal{A}_{\\mathbf{K}}$ is representable. \n\\begin{definition}\\label{BZAK1d} \nLet $S$ be a scheme over $E$. \nA point of $\\mathcal{A}_{\\mathbf{K}}(S)$ is given by the following data:\n\\begin{enumerate} \n\\item[(a)] An abelian scheme $A$ over $S$ up to isogeny with an action\n $\\iota: B \\rightarrow \\End^o A$.\n\\item[(b)] A $\\mathbb{Q}$-homogeneous polarization $\\bar{\\lambda}$ of $A$\n which induces on $B$ the involution $b \\mapsto b^{\\star}$.\n\\item[(c)] A class $\\bar{\\eta}\\; \\text{modulo}\\; \\mathbf{K}$ of\n $B \\otimes \\mathbb{A}_f$-module isomorphisms\n \\begin{displaymath}\n\\eta: V \\otimes \\mathbb{A}_f \\isoarrow V_f(A) \n \\end{displaymath}\n such that for each $\\lambda \\in \\bar{\\lambda}$ there is locally\n for the Zariski topology on $S$ a constant\n $\\xi \\in \\mathbb{A}_f^{\\times}(1)$ with\n \\begin{equation}\\label{BZsimilis1e} \n\\xi \\psi(v_1,, v_1) = E^{\\lambda}(\\eta(v_1), \\eta(v_2)). \n \\end{equation}\n \\end{enumerate}\n We require that the following condition $ {\\rm (KC)}$ holds,\n \\begin{equation}\n {\\rm char}(T, \\iota(b) \\mid \\Lie A) = \\prod_{\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}}\n \\varphi(\\Nm^o_{B\/K} (T -b))^{r_{\\varphi}} . \n \\end{equation}\n \n \\end{definition} \nA more precise formulation of the datum $(c)$ is as follows. We\nassume that $S$ is connected and we choose a geometric point $\\bar{s}$ of\n$S$. Then we may regard $V_f(A)$ resp. $V \\otimes \\mathbb{A}_f$ as continuous\nrepresentation of the fundamental group $\\pi_1(\\bar{s}, S)$. We denote by\n$\\mathbb{A}_f(1)$ the group $\\mathbb{A}_f$ endowed with the action of\n$\\pi_1(\\bar{s}, S)$ via the cyclotomic character\n\\begin{displaymath}\n \\varsigma: \\pi_1(\\bar{s}, S) \\rightarrow \\hat{\\mathbb{Z}}^{\\times} \\subset\n \\mathbb{A}_f^{\\times}. \n \\end{displaymath}\n\n\nThen $\\bar{\\eta}$ is determined by a\n$B \\otimes \\mathbb{A}_f$-linear symplectic similitude $\\eta$, i.e. \n(\\ref{BZsimilis1e}) holds with $\\xi \\in \\mathbb{A}_f$. The class $\\bar{\\eta}$\nmust be invariant by the action of $\\pi_1(\\bar{s}, S)$, i.e., for each\n$\\gamma \\in \\pi_1(\\bar{s}, S)$ there is $k(\\gamma) \\in \\mathbf{K}$ such that\n\\begin{equation}\\label{BZeta1e}\n\\gamma \\eta(v) = \\eta (k(\\gamma) v), \\quad v \\in V \\otimes \\mathbb{A}_f. \n\\end{equation}\nSince the polarization $\\lambda$ is defined over $S$ the form $E^{\\lambda}$\nsatisfies\n\\begin{displaymath}\n E^{\\lambda}(\\gamma \\eta(v_1), \\gamma \\eta(v_2)) = \\gamma (E^{\\lambda}(v_1, v_2)) =\n \\varsigma(\\gamma) E^{\\lambda}(v_1, v_2) .\n\\end{displaymath}\n When we apply the symplectic similitude $\\eta$, this translates into\n\\begin{equation}\\label{BZeta2e}\n\\varsigma(\\gamma) = \\mu(k(\\gamma)), \n \\end{equation}\nwhere $\\mu$ is the multiplicator (\\ref{BZh3e}). The datum $\\bar{\\eta}$\nof $(c)$ for a connected scheme $S$ is now equivalently a class\nmodulo $\\mathbf{K}$ of symplectic similitudes $\\eta$ of\n$B \\otimes \\mathbb{A}_f$-modules\n$V \\otimes \\mathbb{A}_f \\isoarrow V_f(A_{\\bar{s}})$ such that (\\ref{BZeta1e})\nand (\\ref{BZeta2e}) hold. \n\n\nAn alternative way to describe the functor $\\mathcal{A}_{\\mathbf{K}}$ is as\nfollows, cf. \\cite[4.12]{D-TS}. We fix a $\\mathbb{Z}$-lattice $\\Gamma \\subset V$\nsuch that $\\psi(\\Gamma \\times \\Gamma) \\subset \\mathbb{Z}$. Let $m > 0$ an\ninteger and assume that $\\mathbf{K} = \\mathbf{K}_m$ is the subgroup of all\n$g \\in G(\\mathbb{A}_f)$, such that $g \\hat\\Gamma = \\hat\\Gamma$ and $g \\equiv \\ensuremath{\\mathrm{id}}\\xspace_{\\hat\\Gamma}$\nmodulo $m\\hat\\Gamma$. Let $O_B \\subset B$ the order of all elements $b$ such that\n$b \\Gamma \\subset \\Gamma$. Then for a connected scheme $S$ over $E$ a point of\n$\\mathcal{A}_{\\mathbf{K}_m}(S)$ consists of \n\\begin{enumerate} \n\\item[(a)] An abelian scheme $A_0$ over $S$ with an action\n $\\iota: O_B \\rightarrow \\End A_0$.\n\\item[(b)] A polarization $\\lambda$ of $A_0$ \n which induces on $B$ the involution $b \\mapsto b^{\\star}$.\n\\item[(c)] An isomorphism of \\'etale sheaves on $S$ \n \\begin{displaymath}\n\\eta_m: \\Gamma\/m \\Gamma \\rightarrow A_0[m] \n \\end{displaymath}\nsuch that $\\eta_m$ lifts to an isomorphism of $O_B$-modules \n \\begin{displaymath}\n\\eta: \\Gamma \\otimes \\hat{\\mathbb{Z}} \\rightarrow \\hat{T}(A_{0, {\\bar{s}}}) \n \\end{displaymath}\nand such that there is $\\xi \\in \\hat{\\mathbb{Z}}^{\\times}(1)$ with \n \\begin{displaymath}\n \\xi \\psi(v_1,, v_1) = E^{\\lambda}(\\eta(v_1), \\eta(v_2)), \\quad\n v_1, v_2 \\in \\Gamma. \n \\end{displaymath}\n \\end{enumerate}\nWe require that the following condition ${\\rm (KC)}$ holds,\n \\begin{equation}\n {\\rm char} (T, \\iota(b) \\mid \\Lie A_0) = \\prod_{\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}}\n \\varphi(\\Nm^o_{B\/K} (T -b))^{r_{\\varphi}} , \\quad b \\in O_B. \n \\end{equation}\n \n\nIf we start with a point $(A, \\iota, \\bar{\\lambda}, \\bar{\\eta})$ of the first\ndescription of $\\mathcal{A}_{\\mathbf{K}}(S)$, we construct a point of the second\ndescription as follows. We choose $\\eta \\in \\bar{\\eta}$. Then there is\nan abelian variety $A_0 \\in A$ such that\n\\begin{displaymath}\n \\eta: \\Gamma \\otimes \\hat{\\mathbb{Z}} \\overset{\\sim}{\\longrightarrow}\n \\hat{T}(A_0). \n \\end{displaymath}\nThen $A_0$ is independent of the choice of $\\eta$. There exists a unique\n$\\lambda \\in \\bar{\\lambda}$ such that the equation (\\ref{BZsimilis1e})\nholds with $\\xi \\in \\hat{\\mathbb{Z}}^{\\times}(1)$. Modulo $m$ we obtain an\nisomorphism\n\\begin{displaymath}\n \\eta_m: \\Gamma\/ m\\Gamma \\overset{\\sim}{\\longrightarrow}\n \\hat{T}(A_0) \/ m\\hat{T}(A_0) \\cong A_0[m], \n\\end{displaymath}\nwhich is also independent of the choice of $\\eta$.\nConversely, to produce from a point of the second decription a point of the first\ndescription is even more obvious and we omit it. \n\nIt follows from these considerations that the functor $\\mathcal{A}_{\\mathbf{K}}$\nis representable if $\\mathbf{K}$ satisfies the following condition: \n\\begin{equation}\\label{BZneat1e} \n \\begin{aligned}\n \\text{\\emph{There is a $\\mathbb{Z}$-lattice $\\Gamma \\subset V$ and an integer $m \\geq 3$\n such that}}\\\\\n \\mathbf{K}\\subset \\{g\\in G(\\mathbb{A}_f)\\mid g (\\Gamma\\otimes \\hat{\\mathbb{Z}})\\subset \\Gamma\\otimes \\hat{\\mathbb{Z}}, \\, g\\equiv\\ensuremath{\\mathrm{id}}\\xspace \\mod m(\\Gamma\\otimes \\hat{\\mathbb{Z}})\\}.\n \\end{aligned}\n \\end{equation}\nAs above (\\ref{BZsplit1e}) we will assume that all prime ideals of $O_F$\nover $p$ are split in $K\/F$. Let $K \\rightarrow F_{\\mathfrak{p}_i}$ be the embedding\nover $F$ which induces the prime ideal $\\mathfrak{q}_i$ of $K$. It we compose \nthe embedding with the conjugation on $K$, the induced prime ideal is\n$\\bar{\\mathfrak{q}}_i$. We write\n \\begin{displaymath}\n \\begin{aligned} \n K \\otimes_{F} F_{\\mathfrak{p}_i} & \\overset{\\sim}{\\longrightarrow} &\n F_{\\mathfrak{p}_i} \\times F_{\\mathfrak{p}_i} =\n K_{\\mathfrak{q}_i} \\times K_{\\bar{\\mathfrak{q}}_i}\\\\\n x \\otimes f & \\longmapsto & (xf, \\bar{x} f)\\\\\n \\end{aligned}\n \\end{displaymath}\nWe will from now on always assume that the\nfunction $r_{\\varphi}$ satifies (\\ref{BZEnu1e}). We consider the moduli problem\n$\\mathcal{A}_{\\mathbf{K}}$ over the local reflex field $E_{\\nu}$ (\\ref{BZ3e}). We will extend it to a\nmoduli problem over $O_{E_{\\nu}}$. For this, we need to impose some restrictions on $\\mathbf{K}$.\n\n We set \n\\begin{displaymath}\n V_{\\mathfrak{q}_i} = V \\otimes_K K_{\\mathfrak{q}_i}, \\quad \n V_{\\bar{\\mathfrak{q}}_i} = V \\otimes_K K_{\\bar{\\mathfrak{q}}_i}, \\quad \n V_{\\mathfrak{p}_i} = V \\otimes_F F_{\\mathfrak{p}_i} = V_{\\mathfrak{q}_i} \\oplus\n V_{\\bar{\\mathfrak{q}}_i}. \n \\end{displaymath}\nWe use the decompositions \n\\begin{equation}\\label{p-zerlegt3e}\n B \\otimes \\mathbb{Q}_p = \\prod_{i=0}^s \n (B_{\\mathfrak{q}_i} \\times B_{\\bar{\\mathfrak{q}}_i}), \\quad \n V \\otimes \\mathbb{Q}_p = \\bigoplus_{i=0}^s V_{\\mathfrak{p}_i} = \\bigoplus_{i=0}^s\n (V_{\\mathfrak{q}_i} \\oplus V_{\\bar{\\mathfrak{q}}_i}), \n\\end{equation} \ncf. (\\ref{p-zerlegt}). All $V_{\\mathfrak{p}_i} $ in the last decomposition are orthogonal with respect to\n$\\psi_p: V\\otimes\\mathbb{Q}_p\\times V\\otimes\\mathbb{Q}_p\\rightarrow\\mathbb{Q}_p$.\n\nAn element $g \\in G(\\mathbb{Q}_p)$ has the form\n$g = (\\ldots, g_{\\mathfrak{q}_i}, g_{\\bar{\\mathfrak{q}}_i}, \\ldots)$, where\n$g_{\\mathfrak{q}_i} \\in \\End_{B_{\\mathfrak{q}_i}} V_{\\mathfrak{q}_i}$ and\n$g_{\\bar{\\mathfrak{q}}_i} \\in \\End_{B_{\\bar{\\mathfrak{q}}_i}} V_{\\bar{\\mathfrak{q}}_i}$.\nWe define $g'_{\\mathfrak{q}_i} \\in \\End_{B_{\\bar{\\mathfrak{q}}_i}} V_{\\bar{\\mathfrak{q}}_i}$\nby\n\\begin{equation}\\label{BZGQ_p4e} \n \\psi_p(g_{\\mathfrak{q}_i} v, w) = \\psi_p(v, g'_{\\mathfrak{q}_i} w), \\quad v \\in\n V_{\\mathfrak{q}_i}, \\; w \\in V_{\\bar{\\mathfrak{q}}_i}.\n\\end{equation}\nWe see that $g \\in G(\\mathbb{Q}_p)$ if and only if \n\\begin{equation}\\label{BZGQ_p1e}\ng'_{\\mathfrak{q}_i} g_{\\bar{\\mathfrak{q}}_i} \\in \\mathbb{Q}_p^{\\times}\n \\end{equation}\nand if this value is independent of $i$. \nWe set\n\\begin{equation}\\label{BZGQ_p5e}\nG_{\\mathfrak{q}_i} = \\Aut_{B_{\\mathfrak{q}_i}} \\! V_{\\mathfrak{q}_i}\n \\end{equation}\nBy (\\ref{BZGQ_p1e}) we obtain a canonical isomorphism\n\\begin{equation}\\label{BZGQ_p2e}\n G(\\mathbb{Q}_p) \\cong\n G_{\\mathfrak{q}_0} \\times \\ldots \\times G_{\\mathfrak{q}_s} \\times \\mathbb{Q}_p^{\\times}.\n\\end{equation}\nThe multiplier homomorphism $\\mu: G(\\mathbb{Q}_p) \\rightarrow \\mathbb{Q}_p^{\\times}$\ncorresponds on the right hand side to the projection on the factor\n$\\mathbb{Q}_p^{\\times}$. \n\nWe are only interested in the case where \n\\begin{equation}\\label{Dsplit-in0} \n D_{\\mathfrak{p}_0}^{{\\rm opp}} \\cong B_{\\mathfrak{q}_0} \\cong B_{\\bar{\\mathfrak{q}}_0} \\quad\n \\text{\\emph{is a quaternion division algebra over}} \\; F_{\\mathfrak{p}_0} .\n \\end{equation}\n\nFor \neach prime $\\mathfrak{q}_i$, $i = 0, 1, \\ldots, s$ we choose a maximal order\n$O_{B_{\\mathfrak{q}_i}} \\subset B_{\\mathfrak{q}_i}$. The image of $O_{B_{\\mathfrak{q}_i}}$\nby the involution $\\star : B_{\\mathfrak{q}_i} \\rightarrow B_{\\bar{\\mathfrak{q}}_i}$\nwill be denoted by $O_{B_{\\bar{\\mathfrak{q}}_i}}$. \n\nWe set\n$\\Lambda_{\\mathfrak{q}_i} = O_{B_{\\mathfrak{q}_i}} \\subset V_{\\mathfrak{q}_i}$.\nMoreover we set\n\\begin{displaymath}\n \\Lambda_{\\bar{\\mathfrak{q}}_i} = \\{u \\in V_{\\bar{\\mathfrak{q}}_i} \\; | \\; \n \\psi_{\\mathfrak{p}_i}(x, u) \\in \\mathbb{Z}_p, \\; \\text{for all}\\;\n x \\in \\Lambda_{\\mathfrak{q}_i} \\}, \n\\end{displaymath}\nwhere $\\psi_{\\mathfrak{p}_i}$ is the restriction of $\\psi_p$ to $V_{\\mathfrak{p}_i}$. \nThen $\\Lambda_{\\bar{\\mathfrak{q}}_i}$ is an $O_{B_{\\bar{\\mathfrak{q}}_i}}$-module and the\npairings \n\\begin{equation}\\label{BZLambda1e}\n \\psi_{\\mathfrak{p}_i}: \\Lambda_{\\mathfrak{q}_i} \\times \\Lambda_{\\bar{\\mathfrak{q}}_i}\n \\rightarrow \\mathbb{Z}_p \n \\end{equation}\nare perfect. We write\n$\\Lambda_{\\mathfrak{p}_i} =\\Lambda_{\\mathfrak{q}_i}\\oplus\\Lambda_{\\bar{\\mathfrak{q}}_i}$\nand $\\Lambda_p = \\oplus_{i=0}^{s} \\Lambda_{\\mathfrak{p}_i}$. \n\nWe choose an open subgroup $\\mathbf{M} \\subset \\mathbb{Z}_p^{\\times}$. \nWe set $\\mathbf{K}_{\\mathfrak{q}_0} = \\Aut_{O_{B_{\\mathfrak{q}_0}}} \\Lambda_{\\mathfrak{q}_0}$. \nFor $i > 0$ we choose arbitrarily open and compact subgroups \n$\\mathbf{K}_{\\mathfrak{q}_i} \\subset \\Aut_{O_{B_{\\mathfrak{q}_i}}} \\Lambda_{\\mathfrak{q}_i}$. We set \n\\begin{equation}\n G_{\\mathfrak{p}_i} = \\{g \\in \\Aut_{B_{\\mathfrak{p}_i}} \\! V_{\\mathfrak{p}_i} \\; | \\;\n \\psi_{\\mathfrak{p}_i}(g v, g w) = \\mu_{\\mathfrak{p}_i}(g)\\psi_{\\mathfrak{p}_i}(v, w) \\;\n \\text{for} \\; \\mu_{\\mathfrak{p}_i}(g) \\in \\mathbb{Q}_p^{\\times} \\}. \n\\end{equation}\nWe define\n$\\mathbf{K}_{\\mathfrak{p}_i} \\subset G_{\\mathfrak{p}_i}$ \nas the group of all pairs $g=(c_1, c_2)$ of automorphisms\n\\begin{displaymath}\n c_1 \\in \\mathbf{K}_{\\mathfrak{q}_i}, \\quad c_2 \\in\n \\Aut_{B_{\\bar{\\mathfrak{q}}_i}} V_{\\bar{\\mathfrak{q}}_i}\n\\end{displaymath}\nsuch that for some $m \\in \\mathbf{M}$ \n\\begin{displaymath}\n \\psi(c_1 v, c_2 w) = m \\psi(v, w), \\quad \\text{for all} \\;\n v \\in V_{\\mathfrak{q}_i}, w \\in V_{\\bar{\\mathfrak{q}}_i}. \n\\end{displaymath}\nSince $c_1 (\\Lambda_{\\mathfrak{q}_i}) \\subset \\Lambda_{\\mathfrak{q}_i}$ it follows\nfrom (\\ref{BZGQ_p4e}) that\n$c'_1 (\\Lambda_{\\bar{\\mathfrak{q}}_i}) \\subset \\Lambda_{\\bar{\\mathfrak{q}}_i}$.\nSince $c'_1c_2 = m$, this implies that \n$c_2 \\in \\Aut_{O_{B_{\\bar{\\mathfrak{q}}_i}}} \\Lambda_{\\bar{\\mathfrak{q}}_i}$.\n\nWe obtain an isomorphism \n\\begin{displaymath}\n \\begin{array}{ccc}\n \\mathbf{K}_{\\mathfrak{p}_i} & \\cong &\\mathbf{K}_{\\mathfrak{q}_i}\\times \\mathbf{M}\\\\\n (c_1,c_2) & \\mapsto & c_1 \\times m .\n \\end{array}\n\\end{displaymath}\nWe define the subgroup $\\mathbf{K}_p \\subset G(\\mathbb{Q}_p)$ as\n\\begin{equation}\\label{BZKp1e}\n \\begin{array}{rl}\n \\mathbf{K}_p = & \\{g = (g_{\\mathfrak{p}_i})\\in\\prod_i \\mathbf{K}_{\\mathfrak{p}_i} \\;\n | \\; \\mu (g_{\\mathfrak{p}_0}) =\\ldots= \\mu(g_{\\mathfrak{p}_s})\n \\in \\mathbf{M} \\}\\\\[2mm] \n \\cong & \\mathbf{K}_{\\mathfrak{q}_0} \\times \\ldots \\times \\mathbf{K}_{\\mathfrak{q}_s}\n \\times \\mathbf{M}.\n \\end{array}\n\\end{equation}\nThe last equation follows from (\\ref{BZGQ_p2e}). \nWe choose an arbitrary open compact subgroup\n$\\mathbf{K}^p \\subset G(\\mathbb{A}^p_f)$ and define\n\\begin{equation}\\label{BZ7e} \n\\mathbf{K} = \\mathbf{K}_p \\mathbf{K}^p \\subset G(\\mathbb{A}_f). \n \\end{equation} \nThis concludes the description of the class of open compact subgroups $\\mathbf{K}$ for which we will extend $\\mathcal{A}_{\\mathbf{K}}$ to a moduli problem over $\\Spec O_{E_\\nu}$. For these $\\mathbf{K}$ we may reformulate the Definition \\ref{BZAK1d} of\nthe functor $\\mathcal{A}_{\\mathbf{K}}$. The datum $\\bar{\\eta}$ is then the\nproduct of two classes $\\bar{\\eta}^p$ modulo $\\mathbf{K}^p$, resp. \n$\\bar{\\eta}_p$ modulo $\\mathbf{K}_p$, of isomorphisms \n\\begin{displaymath}\n \\eta^p: V \\otimes \\mathbb{A}^p_f \\isoarrow V^p_f(A), \\quad \\text{resp.} \n \\quad \\eta_p: V \\otimes \\mathbb{Q}_p \\isoarrow V_p(A), \n\\end{displaymath}\nwhich respect the bilinear forms on both sides up to a constant in\n$(\\mathbb{A}_f^p)^{\\times}(1)$, resp. $\\mathbb{Q}_p^{\\times}(1)$. In particular there\nis for each $\\lambda \\in \\bar{\\lambda}$ locally on $S$ a constant\n$\\xi_p(\\lambda) \\in \\mathbb{Q}_p^{\\times}(1)$ such that for the Riemann form\n$E^{\\lambda}$ \n\\begin{equation}\\label{BZ21e}\n E^{\\lambda}(\\eta_p(v), \\eta_p(w)) = \\xi_p(\\lambda) \\psi(v,w), \\quad v,w \\in\n V \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p. \n\\end{equation}\nIf we change $\\eta_p$ in its class by an element $g \\in \\mathbf{K}_p$ we find\n\\begin{displaymath}\n E^{\\lambda}(\\eta_p(gv), \\eta_p(gw)) = \\xi_p(\\lambda) \\psi(gv,gw) =\n \\xi_p(\\lambda) \\mu(g) \\psi(v,w). \n \\end{displaymath}\nSince $\\mu(g) \\in \\mathbf{M}$, the class of\n$\\xi_p(\\lambda) \\in \\mathbb{Q}_p^{\\times}(1)\/\\mathbf{M}$ is well-defined by the\nclass $\\bar{\\eta}_p$. If we change $\\lambda$ into $u\\lambda$ for\n$u \\in \\mathbb{Q}^{\\times}$, we obtain\n\\begin{displaymath}\n\\xi_p(u \\lambda) = u \\xi_p(\\lambda). \n\\end{displaymath}\nBy (\\ref{p-zerlegt3e}) $\\eta_p$ decomposes into isomorphisms\n\\begin{displaymath}\n \\eta_{\\mathfrak{q}_i}: V \\otimes_K K_{\\mathfrak{q}_i} \\isoarrow V_{\\mathfrak{q}_i}(A),\n \\quad \\eta_{\\bar{\\mathfrak{q}}_i}: V \\otimes_K K_{\\bar{\\mathfrak{q}}_i}\n \\isoarrow V_{\\bar{\\mathfrak{q}}_i}(A), \\quad \\text{for} \\; i= 0, \\ldots, s.\n\\end{displaymath}\nThe equation (\\ref{BZ21e}) becomes equivalent to the equations for $i = 0, \\ldots, s$,\n\\begin{equation}\n E^{\\lambda}(\\eta_{\\mathfrak{q}_i}(v_i), \\eta_{\\bar{\\mathfrak{q}}_i}(w_i) =\n \\xi_p(\\lambda) \\psi(v_i, w_i), \\quad v_i\\in V\\otimes_{K} K_{\\mathfrak{q}_i},\\, w_i\\in V\\otimes_{K} K_{\\bar{\\mathfrak{q}}_i} .\n\\end{equation}\nFrom these equations it is clear that the set of data\n$\\eta_{\\mathfrak{q}_i}, \\eta_{\\bar{\\mathfrak{q}}_i}$ is determined by\n$\\eta_{\\mathfrak{q}_i}, \\xi_p(\\lambda)$.\n\nWe obtain the following reformulation of Definition \\ref{BZAK1d}. \n\\begin{definition}\\label{BZAK1altd} (alternative of Definition \\ref{BZAK1d} for\n $\\mathcal{A}_{\\mathbf{K}}$) \nLet $\\mathbf{K}=\\mathbf{K}_p\\mathbf{K}^p \\subset G(\\mathbb{A}_f)$, where $\\mathbf{K}_p$ is defined as in \\eqref{BZKp1e}.\n Then we can replace the datum $(c)$ of Definition \\ref{BZAK1d} \nby the following data\n \\begin{enumerate}\n \\item[($c^p$)] A class $\\bar{\\eta}^p$ modulo $\\mathbf{K}^p$ of\n $B \\otimes \\mathbb{A}^p_f$-module isomorphisms \n \\begin{displaymath}\n \\eta^p: V \\otimes \\mathbb{A}^p_f \\isoarrow \\mathrm{V}^p_f(A) \n \\end{displaymath}\n such that for each $\\lambda \\in \\bar{\\lambda}$ there is a constant\n $\\xi^{(p)}(\\lambda) \\in \\mathbb{A}^p_f(1)$\n with\n \\begin{displaymath}\n\\xi^{(p)}(\\lambda) \\psi(v_1, v_2) = E^{\\lambda}(\\eta^p(v_1), \\eta^p(v_2)). \n \\end{displaymath}\n\\item[($c_p$)] For each $i = 0, \\ldots s$ a class $\\bar{\\eta}_{\\mathfrak{q}_i}$\n modulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i}$ of $B_{\\mathfrak{q}_i}$-module\n isomorphisms \n \\begin{displaymath}\n \\eta_{\\mathfrak{q}_i}: V \\otimes_K K_{\\mathfrak{q}_i} \\isoarrow\n V_{\\mathfrak{q}_i}(A). \n \\end{displaymath}\n\\item[($\\xi_p$)] A function\n $$\\xi_p: \\bar{\\lambda} \\rightarrow \\mathbb{Q}_p^{\\times}(1)\/\\mathbf{M}$$\n such that\n $\\xi_p(u \\lambda) = u \\xi_p(\\lambda)$ for each $u \\in \\mathbb{Q}^{\\times}$. \n \\end{enumerate}\n \\end{definition}\nLet $g \\in G(\\mathbb{Q}_p) \\subset G(\\mathbb{A}_f)$ and consider the Hecke\noperator\n\\begin{equation}\\label{Heckeoperator1e}\n g: \\mathcal{A}_\\mathbf{\\mathbf{K}} \\rightarrow\n \\mathcal{A}_{g^{-1}\\mathbf{K}g} \n\\end{equation}\nwhich maps a point $(A, \\iota, \\bar{\\lambda}, \\bar{\\eta})$ to \n$(A, \\iota, \\bar{\\lambda}, \\overline{\\eta g})$. This makes sense, since with\n$\\mathbf{K}$ also $g^{-1} \\mathbf{K}g$ satisfies the conditions imposed in\n\\eqref{BZ7e}. According to the isomorphism (\\ref{BZGQ_p2e}) we represent $g$\nin the form\n$(\\ldots, g_{\\mathfrak{q}_i}, g_{\\bar{\\mathfrak{q}}_i}, \\ldots, \\mu(g))$. \nIn terms of the Definition \\ref{BZAK1altd}, the point is represented in the form\n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, \\xi_p)$. \nThe Hecke operator (\\ref{Heckeoperator1e}) maps it to the point\n$(A,\\iota,\\bar{\\lambda},\\bar{\\eta}^p,(\\overline{\\eta_{\\mathfrak{q}_i}g_{\\mathfrak{q}_i}})_i,\\mu(g)\\xi_p)$.\n\nLet $u \\in \\mathbb{Q}_p^{\\times}$. The action which maps\n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, \\xi_p)$\nto\n$(A,\\iota,\\bar{\\lambda},\\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, u\\xi_p)$\nis denoted by \n\\begin{equation}\\label{xioperator1e}\nu_{\\mid \\xi}: \\mathcal{A}_\\mathbf{\\mathbf{K}} \\rightarrow\n \\mathcal{A}_{\\mathbf{K}} .\n \\end{equation}\nThe element\n\\begin{displaymath}\n s_u := (1, \\ldots 1) \\times (u, \\ldots, u)\\in (K \\otimes \\mathbb{Q}_p)^{\\times}\n = (\\prod_{i=0}^{s} K^{\\times}_{\\mathfrak{q}_i}) \\times\n (\\prod_{i=0}^{s} K^{\\times}_{\\bar{\\mathfrak{q}}_i}) \n \\end{displaymath}\nlies in $G(\\mathbb{Q}_p)$. The action of the Hecke operator $s_u$ coincides\nwith the action of $u_{\\mid \\xi}$. The notation $u_{\\mid \\xi}$ is a reminder that the action of $s_u$ only changes the last entry in a tuple $(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, \\xi_p)$. \n\nWe set\n\\begin{equation}\\label{BZG1e}\nO_{B,p} = \\prod_{i=0}^s\n (O_{B_{\\mathfrak{q}_i}} \\times O_{B_{\\bar{\\mathfrak{q}}_i}}) \\subset B \\otimes \\mathbb{Q}_p. \n \\end{equation}\nLet $O_{B,(p)} \\subset B$ the subring of elements which lie in $O_{B,p}$. This subring is invariant under the involution $\\star$, cf. \\eqref{defstar}. \nWe will consider abelian schemes $A$ with an action\n\\begin{displaymath}\n O_{K,(p)} = O_K \\otimes_{\\mathbb{Z}} \\mathbb{Z}_{(p)} \\rightarrow\n\\End A \\otimes_{\\mathbb{Z}} \\mathbb{Z}_{(p)}. \n\\end{displaymath}\nThen $O_K \\otimes_{\\mathbb{Z}} \\mathbb{Z}_p$ acts on the $p$-divisible group $X$\nof $A$. Therefore we obtain a decomposition\n\\begin{displaymath}\n X = (\\prod_{i=0}^{s} X_{\\mathfrak{q}_i}) \\times\n (\\prod_{i=0}^{s} X_{\\bar{\\mathfrak{q}}_i}). \n \\end{displaymath}\n We will write $X_{\\mathfrak{p}_i} = X_{\\mathfrak{q}_i} \\times X_{\\bar{\\mathfrak{q}}_i}$.\nThis continues to make sense if $A$ is an abelian scheme up to isogeny of order\nprime to $p$.\nWe set\n\\begin{displaymath}\n U_p(\\mathbb{Q}) = \\{d \\in \\mathbb{Q}^{\\times} \\; | \\; \\ord_p d = 0 \\} =\n \\mathbb{Z}_{(p)}^{\\times}. \n \\end{displaymath}\n\\begin{definition}\\label{BZsAK2d} \nLet $\\mathbf{K}=\\mathbf{K}_p\\mathbf{K}^p \\subset G(\\mathbb{A}_f)$, where $\\mathbf{K}_p$ is defined as in \\eqref{BZKp1e}. \nWe define a variant $\\mathcal{A}^{ bis}_{\\mathbf{K}}$ of the functor\n$\\mathcal{A}_{\\mathbf{K}}$. A point of $\\mathcal{A}^{bis}_{\\mathbf{K}}$ with\nvalues in an $E$-scheme $S$ consists of: \n\\begin{enumerate} \n\\item[(a)] An abelian scheme $A$ over $S$ up to isogeny of degree prime to\n $p$ with an action\n $\\iota: O_{B,(p)} \\rightarrow \\End A \\otimes_{\\mathbb{Z}} \\mathbb{Z}_{(p)}$, \n\n\\item[(b)] A $U_p(\\mathbb{Q})$-homogeneous polarization $\\bar{\\lambda}$ of $A$\n which induces on $B$ the involution $b \\mapsto b^{\\star}$ and which is\n principal in $p$, i.e. of degree prime to $p$. \n \n\\item[(c)] A class $\\bar{\\eta}^p$ modulo $\\mathbf{K}^p$ of\n $B \\otimes \\mathbb{A}^p_f$-module isomorphisms \n \\begin{displaymath}\n \\eta^p: V \\otimes \\mathbb{A}^p_f \\isoarrow \\mathrm{V}^p_f(A) .\n \\end{displaymath}\n such that for each $\\lambda \\in \\bar{\\lambda}$ there is a constant\n $\\xi^{(p)}(\\lambda) \\in \\mathbb{A}^p_f(1)$\n with\n \\begin{displaymath}\n\\xi^{(p)}(\\lambda) \\psi(v_1, v_2) = E^{\\lambda}(\\eta^p(v_1), \\eta^p(v_2)). \n \\end{displaymath}\n\\item[(d)]\n For each polarization $\\lambda \\in \\bar{\\lambda}$ a section \n $\\xi_p(\\lambda) \\in \\mathbb{Z}_p^{\\times}(1)\/\\mathbf{M}$ such that\n $\\xi_p(u \\lambda) = u \\xi_p(\\lambda)$ for each $u \\in U_p(\\mathbb{Q})$. \n\\item[(e)]\n A class $\\bar{\\eta}_{\\mathfrak{q}_j}$ modulo $\\mathbf{K}_{\\mathfrak{q}_i}$ of\n $O_{B_{\\mathfrak{q}_j}}$-module isomorphisms for each $j = 1, \\ldots s$,\n \\begin{displaymath}\n \\eta_{\\mathfrak{q}_j}: \\Lambda_{\\mathfrak{q}_j} \\isoarrow T_{\\mathfrak{q}_j}(A) .\n \\end{displaymath} \n \\end{enumerate}\nWe require that the condition ${\\rm (KC)}$ holds\n \\begin{equation}\n {\\rm char}(T, \\iota(b) \\mid \\Lie A) = \\prod_{\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}}\n \\varphi(\\Nm^o_{B\/K} (T -b))^{r_{\\varphi}} . \n \\end{equation}\n \n\\end{definition}\nHere and in the sequel, the index $i$ will run through $0,\\ldots, s$ and the index $j$ through $1,\\cdots, s$. \n\\begin{proposition}\\label{BZ1p}\n The functors $\\mathcal{A}_{\\mathbf{K}}$ and $\\mathcal{A}^{bis}_{\\mathbf{K}}$ \n on the category of $E$-schemes are isomorphic.\n \\end{proposition}\n\\begin{proof}\n We begin with a point $(A, \\iota, \\bar{\\lambda}, \\bar{\\eta})$ of\n $\\mathcal{A}_{\\mathbf{K}}(S)$ and construct a point of\n $\\mathcal{A}^{bis}_{\\mathbf{K}}(S)$.\n We choose an element $\\eta \\in \\bar{\\eta}$ and consider the component \n $\\eta_p: V \\otimes \\mathbb{Z}_p \\isoarrow \\mathrm{V}_p(A)$. By the choice\n of $\\mathbf{K}_{p}$, the image of $\\Lambda_p$ by this morphism is independent\n of the choice of $\\eta_p$. Therefore we find an abelian variety $A_0 \\in A$\n up to isogeny prime to $p$ such that $\\eta_p$ induces an isomorphism \n \\begin{displaymath}\n\\eta_p: \\Lambda_{p} \\isoarrow T_p(A_0). \n \\end{displaymath}\n We choose a polarization $\\lambda_0 \\in \\bar{\\lambda}$. Then we obtain \n \\begin{displaymath}\n E_p^{\\lambda_0}(\\eta_p(v_1), \\eta_p(v_2)) = \\xi \\psi_p(v_1, v_2), \\quad\n v_1, v_2 \\in \\Lambda_p, \\; \\xi \\in \\mathbb{Q}_p^{\\times}(1). \n \\end{displaymath}\n After multiplying $\\lambda_0$ by a power of $p$ we may assume that \n $\\xi \\in \\mathbb{Z}_p^{\\times}(1)$. This determines an \n $U_p(\\mathbb{Q})$-homogeneous polarization $\\bar{\\lambda}_0$ on $A_0$.\n We remark that the class of $\\xi$ in $\\mathbb{Z}_p^{\\times}(1)\/\\mathbf{M}$\n is independent of the choice of $\\eta$. \n Finally $\\eta_p$ induces $O_{B_{\\mathfrak{q}_i}}$-module isomorphisms\n$\\eta_{\\mathfrak{q}_i}:\\Lambda_{\\mathfrak{q}_i}\\isoarrow T_{\\mathfrak{q}_i}(A_0)$. \n We have obtained a point\n $(A_0, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p,(\\eta_{\\mathfrak{q_j}})_j, \\xi_p)$ of\n $\\mathcal{A}^{bis}_{\\mathbf{K}}$. \n\n Conversely assume that\n $(A_0, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p,(\\eta_{\\mathfrak{q_j}})_j, \\xi_p) \\in\n \\mathcal{A}^{bis}_{\\mathbf{K}}$ \nis given. Let $\\lambda \\in \\bar{\\lambda}_0$. The Riemann form \n $E^{\\lambda}$ on $T_{\\mathfrak{p}_0}(A_0)$ is by\nassumption perfect. Therefore we find for any given\n$\\xi' \\in \\mathbb{Z}_p^{\\times}(1)$ an isomorphism of $O_{B_{\\mathfrak{p}_0}}$-modules \n \\begin{displaymath}\n\\eta_{\\mathfrak{p}_0}: \\Lambda_{\\mathfrak{p}_0} \\isoarrow T_{\\mathfrak{p}_0}(A_0) \n \\end{displaymath}\nsuch that \n \\begin{displaymath}\n \\xi' \\psi(v_1, v_2) =\n E^{\\lambda}(\\eta_{\\mathfrak{p}_0}(v_1), \\eta_{\\mathfrak{p}_0}(v_2)), \\quad\n \\; v_1, v_2 \\in \\Lambda_{\\mathfrak{p}_0}. \n \\end{displaymath}\n We choose $\\eta_{\\mathfrak{p}_0}$ such that $\\xi' = \\xi_p(\\lambda)$. \n For $i > 0$ the isomorphism $\\eta_{\\mathfrak{q}_i}$ induces by duality\n $\\Hom( ?, \\mathbb{Z}_p)$ an isomorphism\n $T_{\\bar{\\mathfrak{q}}_i}(A_0) \\isoarrow \\Lambda_{\\bar{\\mathfrak{q}}_i}(1)$.\n We multiply the inverse map with $\\xi_p(\\lambda)$ and obtain\n \\begin{displaymath}\n \\eta_{\\bar{\\mathfrak{q}}_i}: \\Lambda_{\\bar{\\mathfrak{q}}_i} \\isoarrow\n T_{\\bar{\\mathfrak{q}}_i}(A_0). \n \\end{displaymath}\n We obtain an isomorphism \n \\begin{displaymath}\n \\eta_{\\mathfrak{p}_i} = \\eta_{\\mathfrak{q}_i} \\oplus \\eta_{\\bar{\\mathfrak{q}}_i}:\n \\Lambda_{\\mathfrak{p}_i} \\isoarrow T_{\\mathfrak{p}_i} (A_0)\n \\end{displaymath}\n which respects the bilinear forms on both sides up to the factor\n $\\xi_p(\\lambda)$. Then $\\eta_p = \\oplus^{s}_{i=0} \\eta_{\\mathfrak{p}_i}$ defines\n an isomorphism $\\eta_p: \\Lambda_p \\rightarrow T_p(A_0)$ such that\n \\begin{displaymath}\n \\xi_p(\\lambda) \\psi(v_1, v_2) = E^{\\lambda}(\\eta_p(v_1), \\eta_p(v_2)), \\quad\n v_1, v_2 \\in \\Lambda_p.\n \\end{displaymath}\n Therefore we have constructed a point of $\\mathcal{A}_{\\mathbf{K}}(S)$. The two procedures are inverses of each other, proving the proposition.\n\\end{proof}\n\nFrom now on we will always assume that $\\mathbf{K}$ satisfies the assumptions\nmade in Definition \\ref{BZsAK2d}. If we write $\\mathcal{A}_{\\mathbf{K}}$,\nwe understand that this functor is given in the form of Definition\n\\ref{BZsAK2d}. To extend \nthe functor $\\mathcal{A}_{\\mathbf{K}}(S)$ to an arbitrary $O_{E_{\\nu}}$-scheme $S$, the main obstacle\nis the datum $(d)$, since we do not have the\n$(\\mathbb{Z}\/p\\mathbb{Z})^{\\times}$-\\'etale torsor of primitive $p$th roots of unity.\nTherefore we define a new functor $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ by\nreplacing the data $\\xi_p(\\lambda) \\in \\mathbb{Z}_p^{\\times}(1)\/\\mathbf{M}$ by\nsections in the constant sheaf\n$\\xi_p(\\lambda) \\in \\mathbb{Z}_p^{\\times}\/\\mathbf{M}$. Here the upper index $t$ in $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ indicates that this functor, when restricted to the category of $E_\\nu$-schemes, is a twisted version of $\\mathcal {A}_{\\mathbf{K}}$. \n\\begin{definition}\\label{BZsAtK2d}\nLet $\\mathbf{K}=\\mathbf{K}_p\\mathbf{K}^p \\subset G(\\mathbb{A}_f)$, where $\\mathbf{K}_p$ is defined as in \\eqref{BZKp1e}. We define a functor $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ on the category of\n$O_{E_{\\nu}}$-schemes $S$. \nAn $S$-valued point consists of the data $(a), (b), (c), (e)$ as in Definition\n\\ref{BZsAK2d}. But we replace $(d)$ by the following datum\n\\begin{enumerate}\n\\item[$(d^t)$]\n A section \n $\\xi_p(\\lambda) \\in \\mathbb{Z}_p^{\\times}\/\\mathbf{M}$ for each polarization $\\lambda \\in \\bar{\\lambda}$ such that\n $$\\xi_p(u \\lambda) = u \\xi_p(\\lambda), \\quad u \\in U_p(\\mathbb{Q}) .\n $$\n \\end{enumerate}\nWe continue to impose the condition ${\\rm (KC)}$. \n\\end{definition}\n The data (a)--(c) continue to make sense over a \n$O_{E_{\\nu}}$-scheme $S$. Since an isogeny of\ndegree prime to $p$ induces an isomorphism on tangent spaces, the condition (KC) also makes\nsense. Since $r_{\\varphi} = 0$ for each\n$\\varphi: K_{\\mathfrak{q}_j} \\rightarrow \\bar{\\mathbb{Q}}_p$ for $j= 1, \\ldots, s$,\nthe $p$-divisible groups $X_{\\mathfrak{q}_j}$ are \\'etale. We note that\n$X_{\\mathfrak{q}_j}$ is a $p$-divisible group of height\n$4[K_{\\mathfrak{q}_j}: \\mathbb{Q}_p]$ and this implies that $T_{\\mathfrak{q}_j}(A)$\nis a free $O_{B_{\\mathfrak{q}_j}}$-module of rank $1$. Therefore the datum \n$(e)$ also continues to make sense. \n\nThe functor $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ is representable if the group\n$\\mathbf{K}$ satisfies the condition (\\ref{BZneat1e}) for some integer\n$m \\geq 3$ which is prime to $p$. A standard argument shows that\n$\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ is proper over $\\Spec O_{E_{\\nu}}$, cf.\n\\cite[Prop. 4.1]{Dr}. If we have another open and compact\nsubgroup $\\tilde{\\mathbf{K}} \\subset \\mathbf{K}$ as in Definition\n\\ref{BZsAtK2d}, we obtain an \\'etale covering\n\\begin{displaymath}\n \\widetilde{\\mathcal{A}}^t_{\\tilde{\\mathbf{K}}} \\rightarrow\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}}. \n\\end{displaymath} \n\nThe general fibre $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}, E_{\\nu}}$ of\n$\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ is a Galois twist of\n$\\mathcal{A}_{\\mathbf{K},E_{\\nu}}=\\mathcal{A}_{\\mathbf{K}}\\times_{\\Spec E}\\Spec E_{\\nu}$. Let us explain this. \nThe definition of \n$\\tilde{\\mathcal{A}}^{t}_{\\mathbf{K}}(S)$ makes sense for any $E$-scheme $S$.\nWe denote this functor on the category of $E$-schemes by \n$\\mathcal{A}^{t}_{\\mathbf{K}}$. (The fact, used above, that the $p$-divisible groups $X_{\\mathfrak{q}_j}$\nare \\'etale is automatic in characteristic $0$, even for $X_{\\mathfrak{q}_0}$.) We may represent a\npoint of $\\mathcal{A}^{t}_{\\mathbf{K}}(S)$ in the same way as in\nDefinition \\ref{BZAK1altd} by\n\\begin{equation}\\label{A^t-points-alt}\n(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, \\xi_p) ,\n\\end{equation}\nexcept that now $\\xi_p$ is a function\n$\\xi_p: \\bar{\\lambda} \\rightarrow \\mathbb{Q}_p^{\\times}\/\\mathbf{M}$. \nThere is the \ncanonical isomorphism\n\\begin{equation}\\label{Gtwist4e}\n \\mathcal{A}^{t}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec E_{\\nu} \\cong\n \\tilde{\\mathcal{A}}^{t}_{\\mathbf{K}} \\times_{\\Spec O_{E_{\\nu}}} \\Spec E_{\\nu}. \n \\end{equation}\n\nWe will identify \n$\\mathcal{A}^{t}_{\\mathbf{K}}$ with a Shimura variety of the form ${\\mathrm{Sh}}(G, h \\cdot c)$ for some\n$c: \\mathbb{S} \\rightarrow G_{\\mathbb{R}}$ which factors through the center\nof $G$. We consider the cyclotomic character \n\\begin{displaymath}\n\\varsigma_{p^{\\infty}}: \\Gal(\\bar{\\ensuremath{\\mathbb {Q}}\\xspace}\/E) \\rightarrow \\mathbb{Z}_p^{\\times}\/\\mathbf{M}. \n\\end{displaymath}\nLet $L \\subset \\bar{\\ensuremath{\\mathbb {Q}}\\xspace}$ be the subfield fixed by the kernel of this\nhomomorphism. Let $\\zeta_{p^{\\infty}} \\in \\mathbb{C}$ be a compatible system of\nprimitive\n$p^n$-th roots of unity. We obtain a natural isomorphism\n\\begin{equation}\\label{Gtwist1e}\n \\begin{array}{ccc}\n \\mathcal{A}^{t}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L & \\isoarrow & \n \\mathcal{A}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L,\\\\\n \\xi_p & \\mapsto & \\zeta_{p^{\\infty}} \\xi_p\n \\end{array}\n\\end{equation}\ni.e., in Definition \\ref{BZsAK2d}\nwe have to change only $(d)$ to pass from one functor to the other. \n\\begin{proposition}\\label{Gtwist1p}\n Let $\\tau \\in \\Gal(L\/E)$ be an automorphism. \n The morphism (\\ref{Gtwist1e}) fits into a commutative diagram\n \\begin{displaymath}\n \\xymatrix{\n \\mathcal{A}^{t}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L \\ar[d]_{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\n \\ar[r] & \\mathcal{A}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L\n \\ar[d]^{\\varsigma_{p^{\\infty}}(\\tau^{-1})_{\\mid \\xi} \\times \\tau_c}\\\\\n \\mathcal{A}^{t}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L \\ar[r] & \n \\mathcal{A}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L .\\\\\n }\n \\end{displaymath}\n {\\rm (see (\\ref{xioperator1e}) for the notation $\\varsigma_{p^{\\infty}}(\\tau^{-1})_{\\mid \\xi}$).} \n\\end{proposition} \n\\begin{proof}\n Let $\\pi\\colon S \\rightarrow \\Spec L$ be an $L$-scheme. Denote by $S_{[\\tau_c]}$ the\n $L$-scheme obtained when the structure morphism is changed to\n $\\tau_c \\circ \\pi$. Our assertion says that the following diagram is\n commutative\n\\begin{equation}\\label{Gtwist2e}\n\\begin{aligned}\n \\xymatrix{\n (\\mathcal{A}^{t}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L)(S)\n \\ar[d]_{{ can}} \\ar[r] & (\\mathcal{A}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L)(S) \n \\ar[d]^{\\varsigma_{p^{\\infty}}(\\tau^{-1})_{\\mid \\xi}\\circ { can}}\\\\\n (\\mathcal{A}^{t}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L)(S_{[\\tau_c]}) \\ar[r] & \n (\\mathcal{A}_{\\mathbf{K}} \\times_{\\Spec E} \\Spec L)(S_{[\\tau_c]}) .\\\\\n }\n \\end{aligned}\n\\end{equation}\nThe morphism ${ can}$ on the left hand side is defined by the identification\n$\\mathcal{A}^{t}_{\\mathbf{K}}(S) = \\mathcal{A}^{t}_{\\mathbf{K}}(S_{[\\tau_c]})$ which\nexists because the functor is defined over $E$. In the same way there is\n${can}$ on the right hand side. To show the commutativity of (\\ref{Gtwist2e}),\nwe start with a point\n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, \\xi_p)$ \nof the upper left corner. The image $\\Theta$ of this point by the left $can$\nin $\\mathcal{A}^{t}_{\\mathbf{K}}(S_{[\\tau_c]})$ is given by the same tuple.\nBy the upper horizontal map the point is mapped to\n\\begin{equation}\\label{Gtwist3e}\n (A,\\iota,\\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i,\n \\pi^*(\\zeta_{p^{\\infty}})\\xi_p)\n \\end{equation}\nwhere $\\pi^*: L \\rightarrow \\Gamma(S,\\mathcal{O}_S)$ is the comorphism of the structure map.\nThe image of (\\ref{Gtwist3e}) by the right $can$-morphism is represented\nby the same tuple but we write the last item as\n\\begin{displaymath}\n \\pi^*( \\tau^{-1} (\\tau (\\zeta_{p^{\\infty}}))) \\xi_p = \\varsigma_{p^{\\infty}}(\\tau)\n \\pi^* (\\tau^{-1} (\\zeta_{p^{\\infty}})) \\xi_p. \n\\end{displaymath}\nOn the other hand the image of $\\Theta$ by the lower horizontal bijection \nis the tuple \n$$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, \\pi^* (\\tau^{-1} (\\zeta_{p^{\\infty}})) \\xi_p).$$ \nThis shows the commutativity of (\\ref{Gtwist2e}). \n\\end{proof}\n\nLet $\\Xi \\subset \\Phi$ be the CM-type of $K$ defined by\n\\begin{equation*}\n\\Xi = \\coprod_{i=0}^{s}\n\\Hom_{\\mathbb{Q}_p{\\rm -Alg}}(K_{\\bar{\\mathfrak{q}}_i},\\bar{\\mathbb{Q}}_p).\n\\end{equation*}\nAs defined by diagram (\\ref{BZ2e}), we can write\n\\begin{equation}\\label{Xi2e}\n\\Xi = \\{ \\bar{\\varphi}_0 \\} \\cup \\{\\varphi \\in \\Phi \\; | \\; r_{\\varphi} =2 \\} .\n \\end{equation}\nWe obtain an isomorphism \n\\begin{equation}\\label{CMXi1e}\n K \\otimes_{\\mathbb{Q}} \\mathbb{R} \\isoarrow \\prod\\nolimits_{\\Xi} \\mathbb{C}, \\quad \n a \\otimes \\rho \\mapsto (\\varphi(a) \\rho)_{\\varphi \\in \\Xi} .\n\\end{equation}\nThis puts a complex structure on the left hand side and hence defines a homomorphism\n$\\mathbb{C}^{\\times} \\rightarrow (K \\otimes_{\\mathbb{Q}} \\mathbb{R})^{\\times}$ which\nwe view as a morphism of algebraic groups\n$\\delta: \\mathbb{S} \\rightarrow (\\Res_{K\/\\mathbb{Q}} \\mathbb{G}_{m,K})_\\ensuremath{\\mathbb {R}}\\xspace$.\nWe consider the algebraic torus over $\\mathbb{Q}$ given by \n\\begin{equation*}\n T(\\mathbb{Q}) = \\{t \\in K^{\\times} \\; | \\; t\\bar{t} \\in \\mathbb{Q}^{\\times} \\}\n \\subset G(\\mathbb{Q}). \n\\end{equation*}\nThis is a central subtorus of $G$. \nClearly $\\delta$ factors through \n\\begin{equation}\\label{delta1e}\n\\delta: \\mathbb{S} \\rightarrow T_{\\mathbb{R}} \\subset G_{\\mathbb{R}}. \n \\end{equation}\nLet ${\\mathrm{Sh}}(G, h \\delta^{-1})$ be the canonical model over $E(G, h \\delta^{-1})$ of the Shimura variety attached to the Shimura datum $(G, h \\delta^{-1})$. Note that $E(G, h \\delta^{-1}) \\subset E_{\\nu}$.\nIndeed, if we restrict\n$\\delta_{\\mathbb{C}}: \\mathbb{G}_{m, \\mathbb{C}} \\times \\mathbb{G}_{m, \\mathbb{C}} \\rightarrow (K \\otimes_{\\mathbb{Q}} \\mathbb{C})^{\\times}$\nto the first factor, we obtain\n\\begin{equation}\\label{zentralerTwist1e}\n \\mu_{\\delta}: \\mathbb{G}_{m, \\mathbb{C}} \\rightarrow\n (K \\otimes_{\\mathbb{Q}} \\mathbb{C})^{\\times} \\cong \\prod_{\\Phi} \\mathbb{C}^{\\times}, \\quad z\\mapsto (1, \\ldots 1, z, \\ldots, z) .\n\\end{equation}\nHere $z$ appears\nexactly at the places $\\Xi \\subset \\Phi$. By our choice of the diagram\n(\\ref{BZ2e}) $\\mu_{\\delta}$ is defined over $\\mathbb{Q}_p$ since it comes\nfrom the canonical embedding\n\\begin{equation*}\n \\mathbb{Q}_p^{\\times} \\rightarrow \\prod_{i=0}^{s} K_{\\bar{\\mathfrak{q}}_i} \\subset\n (K \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p)^{\\times}. \n\\end{equation*}\nTherefore $E(G, \\delta^{-1}) \\subset \\mathbb{Q}_p$, which shows our assertion.\n\\begin{proposition}\\label{Gtwist3p}\n Let \n $\\mathbf{K} \\subset G(\\mathbb{A}_f)$ be as in Definition \\ref{BZsAtK2d}. \n We denote by ${\\mathrm{Sh}}_{\\mathbf{K}}(G, h \\delta^{-1})_{E_{\\nu}}$ the scheme obtained\n by base change via $E(G, h\\delta^{-1}) \\subset E_{\\nu}$ from the canonical\n model. We set \n $\\mathcal{A}^{t}_{\\mathbf{K}, E_{\\nu}} = \\tilde{\\mathcal{A}}^{t}_{\\mathbf{K}} \\times_{\\Spec O_{E_{\\nu}}} \\Spec E_{\\nu}={\\mathcal{A}}^{t}_{\\mathbf{K}} \\times_{\\Spec {E}} \\Spec E_{\\nu}$\n (cf. (\\ref{Gtwist4e})). There exists an isomorphism over the maximal unramified extension\n $E^{nr}_{\\nu}$\n \\begin{equation}\n {\\mathrm{Sh}}_{\\mathbf{K}}(G, h \\delta^{-1})_{E_{\\nu}} \\times_{\\Spec E_{\\nu}} \\Spec E^{nr}_{\\nu} \n \\isoarrow\n \\mathcal{A}^{t}_{\\mathbf{K}, E_{\\nu}} \\times_{\\Spec E_{\\nu}} \\Spec E^{nr}_{\\nu}, \n \\end{equation}\n which for varying $\\mathbf{K}$ is compatible with the Hecke operators in\n $G(\\mathbb{A}_f^p)$.\n \n Let $\\tau \\in \\Gal(E_{\\nu}^{nr}\/E_{\\nu})$ be the Frobenius automorphism.\n Let $f_{\\nu}$ be the inertia index of $E_{\\nu}\/\\mathbb{Q}_p$. Then the\n following diagram is commutative,\n \\begin{displaymath}\n \\xymatrix{\n {\\mathrm{Sh}}_{\\mathbf{K}}(G, h \\delta^{-1})_{E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu} \n \\ar[d]_{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c} \\ar[r]\n & \\mathcal{A}^{t}_{\\mathbf{K}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu}\n \\ar[d]^{{p^{f_{\\nu}}}_{\\mid \\xi} \\times \\tau_c}\\\\\n {\\mathrm{Sh}}_{\\mathbf{K}}(G, h \\delta^{-1})_{E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu} \n \\ar[r] & \n \\mathcal{A}^{t}_{\\mathbf{K}, E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu} .\n }\n \\end{displaymath}\n {\\rm (See (\\ref{xioperator1e}) for the notation ${p^{f_{\\nu}}}_{\\mid \\xi}$).} \n \\end{proposition} \n\\begin{proof}\n Let $e \\in E_{\\nu}^{\\times}$ and let $\\sigma \\in \\Gal(E^{ab}_{\\nu}\/E_{\\nu})$ be the\n automorphism that corresponds to it by local class field theory. We set\n $\\varsigma_{p^{\\infty}}(e) = \\varsigma_{p^{\\infty}}(\\sigma)$. Then we obtain from\n local class field theory \n \\begin{equation}\\label{formLFT}\n\\varsigma_{p^{\\infty}}(e) = (\\Nm_{E_{\\nu}\/\\mathbb{Q}_p} e)^{-1} p^{f_{\\nu} \\ord e} ,\n \\end{equation}\n where $\\ord=\\ord_\\nu\\colon E_{\\nu}^{\\times} \\rightarrow \\mathbb{Z}$ maps a uniformizer\n to $1$. Indeed, this formula makes sense for an arbitrary $p$-adic local\n field $E_{\\nu}$. In the case $E_{\\nu} = \\mathbb{Q}_p$ (\\ref{formLFT}) follows\n from \\cite{CF} Chapt. VI, Thm. 3.2. In the general case the action of\n $\\sigma$ on $\\mu_{p^{\\infty}}$ depends only on the restriction of $\\sigma$ to\n $\\mathbb{Q}^{{\\mathrm{ab}}}$. But this restriction corresponds to\n $\\Nm_{E_{\\nu}\/\\mathbb{Q}_p} e$ by the reciprocity law of the local field\n $\\mathbb{Q}_p$ by the last diagram of \\cite{CF} Chapt. VI, \\S 2.4. This\n shows (\\ref{formLFT}). \n \n\n It is enough to show that the two squares in the following diagram are\n commutative. \n \\begin{displaymath}\n\\xymatrix{\n {\\mathrm{Sh}}_{\\mathbf{K}}(G, h \\delta^{-1}) \\times \\Spec E^{ab}_{\\nu} \n \\ar[d]_{\\ensuremath{\\mathrm{id}}\\xspace \\times \\sigma_c} \\ar[r] &\n \\mathcal{A}_{\\mathbf{K}} \\times \\Spec E^{ab}_{\\nu}\n \\ar[d]_{(\\varsigma_{p^{\\infty}}(e^{-1}) p^{f_{\\nu} \\ord e})_{\\mid \\xi} \\times \\sigma_c} \n & \\mathcal{A}^{t}_{\\mathbf{K}, E_{\\nu}} \\times \\Spec E^{ab}_{\\nu}\n \\ar[d]^{(p^{f_{\\nu} \\ord e})_{\\mid \\xi} \\times \\sigma_c} \\ar[l] \\\\\n {\\mathrm{Sh}}_{\\mathbf{K}}(G, h \\delta^{-1}) \\times \\Spec E^{ab}_{\\nu} \n \\ar[r] & \\mathcal{A}_{\\mathbf{K}} \\times \\Spec E^{ab}_{\\nu}\n & \\mathcal{A}^{t}_{\\mathbf{K}, E_{\\nu}} \\times \\Spec E^{ab}_{\\nu} \\ar[l]\n }\n \\end{displaymath}\n The commutativity of the diagram on the right hand side follows from Proposition\n \\ref{Gtwist1p}. Since $\\mathcal{A}_{\\mathbf{K}} \\cong {\\mathrm{Sh}}_{\\mathbf{K}}(G,h)$, the\n square on the left hand side becomes commutative if we replace the vertical\n arrow in the middle by\n \\begin{equation*} \n r_{\\nu}^{\\rm cft}(T, \\delta^{-1})(\\sigma) \\times \\sigma_c\\colon \n {\\mathrm{Sh}}_{\\mathbf{K}}(G,h)_{E_{\\nu}} \\times_{\\Spec E_{\\nu}} \\Spec E_{\\nu}^{ab} \\rightarrow \n {\\mathrm{Sh}}_{\\mathbf{K}}(G,h)_{E_{\\nu}} \\times_{\\Spec E_{\\nu}} \\Spec E_{\\nu}^{ab}.\n \\end{equation*}\n This is a consequence of Corollary \\ref{zentralerTwist1c}. It remains to\n compute the class field theory version $r_{\\nu}^{\\rm cft}(T, \\delta^{-1})$ of the reciprocity law.\n\n The morphism $\\mu_{\\delta^{-1}}$ is the inverse of (\\ref{zentralerTwist1e}).\n Therefore we find the local reciprocity law\n \\begin{equation}\\label{rec-delta1e}\n \\begin{array}{rcc}\n r_{\\nu}(T, \\delta^{-1}): E_{\\nu}^{\\times} & \\rightarrow & \\hspace{-3cm}\n \\prod_{i=0}^s \n K^\\times_{\\mathfrak{q}_i} \\times \\prod_{i=0}^s K^\\times_{\\bar{\\mathfrak{q}}_i}\\\\[2mm] \n e & \\mapsto & (1, \\ldots, 1) \\times \n (\\Nm_{E_{\\nu}\/\\mathbb{Q}_p} e, \\ldots, \\Nm_{E_{\\nu}\/\\mathbb{Q}_p} e). \n \\end{array} \n \\end{equation}\n We write\n $\\Nm_{E_{\\nu}\/\\mathbb{Q}_p} e = \\varsigma_{p^{\\infty}}(e^{-1}) p^{f \\ord e}$, cf. \\eqref{formLFT}.\n Under the isomorphism (\\ref{BZGQ_p2e}), the element on the right hand side of\n (\\ref{rec-delta1e}) corresponds to\n $(1,\\ldots, 1,\\varsigma_{p^{\\infty}}(e^{-1}) p^{f \\ord e})\\in G_{\\mathfrak{q}_0} \\times \\ldots \\times G_{\\mathfrak{q}_s} \\times \\mathbb{Q}_p^{\\times}$. By the description\n of the Hecke operators (\\ref{Heckeoperator1e}) we obtain the proposition. \n\\end{proof}\nWe will next show how the action of $p^{f_{\\nu}}_{\\mid \\xi}$ on\n$\\mathcal{A}^{t}_{\\mathbf{K}, E_{\\nu}}$ extends naturally to the model\n$\\widetilde{\\mathcal{A}}^t_{\\mathbf{K}}$ over $O_{E_{\\nu}}$. It is enough\nto define $p_{\\mid \\xi}$. Let \n\\begin{equation}\\label{wAt1e} \n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j, \\xi_p) \\in\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}}(S)\n \\end{equation}\nbe a point as in Definition \\ref{BZsAtK2d}. Let\n$\\bar{\\lambda}_{\\mathbb{Q}}$ be the $\\mathbb{Q}$-homogeneous polarization\nwhich contains $\\bar{\\lambda}$. We extend $\\xi_p$ to a map\n$\\xi_p: \\bar{\\lambda}_{\\mathbb{Q}} \\rightarrow \\mathbb{Q}_p^{\\times}\/\\mathbf{M}$ such\nthat $\\xi_p(u\\lambda) = u \\xi_p(\\lambda)$ for $u \\in \\mathbb{Q}^{\\times}$. \nLet $X = \\prod_{i=0}^{s} (X_{\\mathfrak{q}_i} \\times X_{\\bar{\\mathfrak{q}}_i})$ be the\n$p$-divisible group of $A$. We consider the isogeny \n\\begin{equation}\\label{Gtwist6e}\na: \\prod_{i=0}^{s} (X_{\\mathfrak{q}_i} \\times X_{\\bar{\\mathfrak{q}}_i}) \\rightarrow \n\\prod_{i=0}^{s} (X_{\\mathfrak{q}_i} \\times X_{\\bar{\\mathfrak{q}}_i}) \n\\end{equation}\nwhich is the identity on the factors $X_{\\mathfrak{q}_i}$ and multiplication\nby $p$ on the factors $X_{\\bar{\\mathfrak{q}}_i}$. Let us fix a polarization\n$\\lambda \\in \\bar{\\lambda}$. Since $\\lambda$ is principal in $p$ it is given\non $X$ by an isomorphism of $X_{\\bar{\\mathfrak{q}}_i}$ with the dual of\n$X_{\\mathfrak{q}_i}$ for each $i = 0,\\ldots,s$. The inverse\nimage of $\\lambda$ on $X$ by (\\ref{Gtwist6e}) is $p \\lambda$. \nThere is an isogeny of abelian varieties with an $O_{B, (p)}$-action\n\\begin{displaymath}\n\\alpha: A' \\rightarrow A, \n\\end{displaymath}\nof order a power of $p$ such that the induced homomorphism of $p$-divisible\ngroups is isomorphic to (\\ref{Gtwist6e}). A polarization $\\lambda$ of\n$A$ induces a polarization\n$\\lambda' = \\alpha^*(\\lambda) := \\hat{\\alpha} \\lambda \\alpha$ on $A'$.\nThis defines a\n$\\mathbb{Q}$-homogeneous polarization $\\bar{\\lambda'}_{\\mathbb{Q}}$ of $A'$ and\na bijection $\\bar{\\lambda}_{\\mathbb{Q}} \\rightarrow \\bar{\\lambda'}_{\\mathbb{Q}}$.\nBy this bijection $\\xi_p$ induces\n\\begin{displaymath}\n\\xi'_p: \\bar{\\lambda'}_{\\mathbb{Q}} \\rightarrow \\mathbb{Q}_p^{\\times}\/\\mathbf{M}. \n\\end{displaymath}\nWe obtain $\\xi'_p(\\lambda') = \\xi_p(\\lambda) \\in \\mathbb{Z}_p^{\\times}$. \nBut the polarization $\\lambda_1 = (1\/p) \\lambda'$ is principal in $p$, as\nwe see by looking at the $p$-divisible groups (\\ref{Gtwist6e}). Then\n\\begin{equation}\\label{wAt2e}\n(A', \\iota', \\bar{\\lambda_1}, \\bar{\\eta}'^p, (\\bar{\\eta}'_{\\mathfrak{q}_j})_j, p\\xi'_p)\n \\end{equation}\nis a point of $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}(S)$. Here $\\eta'^p$, resp. ${\\eta'_{\\mathfrak{q}_j}}$,\ndenotes the composite\n\\begin{displaymath}\n \\eta'^p: V \\otimes \\mathbb{A}^p_f \\overset{\\eta^p}{\\longrightarrow}\n \\mathrm{V}^p_f(A) \\overset{\\alpha^{-1}}{\\longrightarrow} \\mathrm{V}^p_f(A') ,\n \\end{displaymath}\nresp. \n\\begin{displaymath}\n \\eta'_{\\mathfrak{q}_j}: \\Lambda_{\\mathfrak{q}_j}\n \\overset{\\eta_{\\mathfrak{q}_j}}{\\longrightarrow} T_{\\mathfrak{q}_j}(A) \n \\overset{\\alpha^{-1}}{\\longrightarrow} T_{\\mathfrak{q}_j}(A').\n\\end{displaymath}\nNote that the last arrow is an isomorphism\nby definition of $\\alpha$. \nThe map which assigns (\\ref{wAt2e}) to (\\ref{wAt1e}) defines the desired\nextension of the operator $p_{\\mid \\xi}$ to $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$.\n\n\\begin{corollary}\n The Shimura variety ${\\mathrm{Sh}}_{\\mathbf{K}}(G, h \\delta^{-1})_{E_{\\nu}}$ has \n a unique model $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(G, h \\delta^{-1})$ over\n $O_{E_{\\nu}}$ with the following properties.\nThere is an isomorphism\n \\begin{equation}\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(G, h \\delta^{-1}) \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \n \\isoarrow\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \n \\end{equation}\n which is compatible with the Hecke operators $G(\\mathbb{A}_f^p)$. Let $\\tau \\in \\Gal(E_{\\nu}^{nr}\/E_{\\nu})$ be the Frobenius automorphism.\n Let $f_{\\nu}$ be the inertia index of $E_{\\nu}\/\\mathbb{Q}_p$. Then the\n following diagram is commutative. \n \\begin{displaymath}\n \\xymatrix{\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(G, h \\delta^{-1}) \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \\ar[d]_{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c} \\ar[r]\n & \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \\ar[d]^{(p^{f_{\\nu}})_{\\mid \\xi} \\times \\tau_c}\\\\\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(G, h \\delta^{-1}) \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \\ar[r] & \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}}\n \\times_{\\Spec O_{E_{\\nu}}} \\Spec O_{E^{nr}_{\\nu}} \n }\n \\end{displaymath}\\qed\n \\end{corollary}\n The homomorphism $h \\delta^{-1}$ may be written as follows. It factors as \n \\begin{displaymath}\n \\mathbb{S} \\longrightarrow \n \\prod_{\\chi \\in \\Hom_{\\mathbb{Q}{\\rm -Alg}}(F, \\mathbb{C})}\n (D \\otimes_{F, \\chi} \\mathbb{R})^{\\times}\n \\times (K \\otimes_{F, \\chi} \\mathbb{R})^{\\times}\n \\overset{\\kappa}{\\longrightarrow} G^{\\bullet}(\\mathbb{R}), \n \\end{displaymath}\nThe image of $z = a + b{\\mathbf i} \\in \\mathbb{C} = \\mathbb{S}(\\mathbb{R})$ by the first\narrow is given by\n\\begin{displaymath}\n((\\left(\n \\begin{array}{rr} \n a & -b\\\\\n b & a\n \\end{array}\n \\right) \\times z^{-1}), 1, \\ldots ,1), \n \\end{displaymath}\nwhere the first entry in this vector is at the place $\\chi_0$.\n\n\\begin{comment}\n With the notation of (\\ref{BZh1e}) we consider the homomorphism\n \\mar{What is the rest of this section for? Can it be eliminated?} \n\\begin{displaymath} \n\\mathfrak{c}_0: \\mathbb{S} \\rightarrow \n (D \\otimes_{F, \\chi_0} \\mathbb{R})^{\\times}\n \\times (K \\otimes_{F, \\chi_0} \\mathbb{R})^{\\times}. \n\\end{displaymath}\nwhich for $z = a + bi \\in \\mathbb{C} = \\mathbb{S}(\\mathbb{R})$ is given by \n\\begin{displaymath}\n\\mathfrak{c}_0(z) = (\\left(\n \\begin{array}{rr} \n a & -b\\\\\n b & a\n \\end{array}\n \\right) \\otimes 1) \\times z^{-1}. \n\\end{displaymath}\nThe homomorphism $h \\delta^{-1}$ is given by the composite\n\\begin{displaymath}\n \\begin{array}{l}\n \\mathbb{S} \\overset{\\mathfrak{c}_0}{\\longrightarrow}\n (D \\otimes_{F, \\chi_0} \\mathbb{R})^{\\times}\n \\times (K \\otimes_{F, \\chi_0} \\mathbb{R})^{\\times} \\subset \\\\\n \\hspace{3cm} \\prod_{\\chi \\in \\Hom_{\\mathbb{Q}{\\rm -Alg}}(F, \\mathbb{C})}\n (D \\otimes_{F, \\chi} \\mathbb{R})^{\\times}\n \\times (K \\otimes_{F, \\chi} \\mathbb{R})^{\\times}\n \\overset{\\kappa}{\\longrightarrow} G^{\\bullet}(\\mathbb{R}), \n \\end{array}\n\\end{displaymath}\ncf. Lemma \\ref{BZ1l}. \nIt is clear that the image of the last morphism is in $G(\\mathbb{R})$. \n\n\nRemark: Let us assume that $\\mathbf{K}^p$ is sufficiently small such that\n$\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ is representable. Using deformation theory \none can show that $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ is a regular scheme (see\nbelow). Assume we knew that they are normal. There is the\nfollowing elementary fact: Let $\\mathcal{A}$ be a normal noetherian scheme and\nlet $U \\subset \\mathcal{A}$ be a dense open subset. Let $X_1$ and $X_2$ be\nfinite \\'etale coverings of $\\mathcal{A}$ whose restrictions to $U$ are\nisomorphic. Then this isomorphism extends to an isomorphism\n$X_1 \\rightarrow X_2$ of coverings of $\\mathcal{A}$. Indeed, $X_1$ is the\nnormalization of $\\mathcal{A}$ in $(X_1)_{\\mid U}$. \n\nWe proved the following: Let us fix $\\mathbf{K}^p_o$ sufficiently small an\nlet $\\mathbf{K}_{p,o}$ be arbitrary with the restriction explained above. \nWe set $\\mathbf{K}_o = \\mathbf{K}_{p,o} \\mathbf{K}^p_o$. \nThen there is a unique tower of \\'etale coverings for\n$\\mathbf{K} \\subset \\mathbf{K}_o$ : \n\\begin{displaymath}\nX_{\\mathbf{K}} \\rightarrow \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}_0}\n\\end{displaymath}\nwhich extends the tower\n\\begin{displaymath}\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}, E_{\\nu}} \\rightarrow\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}_{o}, E_{\\nu}} \n \\end{displaymath}\nto a tower over $\\Spec O_{E_{\\nu}}$.\n\\end{comment}\n\n\n\n\\section{The moduli problem for ${\\mathrm{Sh}}(G^{\\bullet}, h)$ and a reduction modulo $p$}\\label{s:AGbullet}\n\nWe recall the groups $G$ of \\eqref{BZh3e} and $G^\\bullet$ of (\\ref{Gpunkt1e}). We consider the map of Shimura data $(G, h) \\rightarrow (G^{\\bullet} , h)$, cf.\n\\eqref{BZh2e} and (\\ref{hforbul}). The Shimura fields coincide, i.e. $E = E(G,h) = E(G^{\\bullet},h)$.\nWe consider a pair of open compact subgroups\n$\\mathbf{K} \\subset G(\\mathbb{A}_f)$ and\n$\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ such that\n$\\mathbf{K} \\subset \\mathbf{K}^{\\bullet}$ and such that the induced map\n${\\mathrm{Sh}}(G,h)_{\\mathbf{K}} \\rightarrow {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}}$ is a closed\nimmersion, cf. \\cite[Prop. 1.15.]{D-TS}. \n\n\nThe Shimura variety ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}}$ is the coarse moduli scheme associated to\nthe following functor. \n\\begin{definition}\\label{BZApkt3d}\n Let $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ be an open compact\n subgroup. We define a functor $\\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet}$ on the\n category of $E$-schemes. A point of\n$\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}(S)$ is given by the following data:\n\\begin{enumerate} \n\\item[(a)] An abelian scheme $A$ over $S$ up to isogeny with an action\n $\\iota: B \\rightarrow \\End^o A$.\n\\item[(b)] A $F$-homogeneous polarization $\\bar{\\lambda}$ of $A$\n which induces on $B$ the involution $b \\mapsto b^{\\star}$.\n\\item[(c)] A class $\\bar{\\eta}\\; \\text{modulo}\\; \\mathbf{K}^{\\bullet}$ of\n $B \\otimes \\mathbb{A}_f$-module isomorphisms\n \\begin{displaymath}\n\\eta: V \\otimes \\mathbb{A}_f \\isoarrow \\mathrm{V}_f(A) \n \\end{displaymath}\n such that for each $\\lambda \\in \\bar{\\lambda}$ there is\n locally for the Zariski-topology on $S$ a constant\n $\\xi \\in (F \\otimes_{\\mathbb{Q}} \\mathbb{A}_f)(1)$ with\n \\begin{displaymath}\n \\psi(\\xi v_1, v_2) = E^{\\lambda}(\\eta(v_1), \\eta(v_2)). \n \\end{displaymath}\n \\end{enumerate}\nWe require that the following condition ${\\rm (KC)}$ holds\n \\begin{displaymath}\n {\\rm char} (T, \\iota(b) \\mid \\Lie A) = \\prod_{\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}}\n \\varphi(\\Nm^o_{B\/K} (T -b))^{r_{\\varphi}} . \n \\end{displaymath}\n\n\\end{definition}\n\nWe will reformulate this moduli problem in a way that makes sense over $O_{E_{\\nu}}$.\nAs in the previous section, we need additional requirements for the group\n$\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$, similar to those discussed \nbefore (\\ref{BZ7e}). We assume that\n$\\mathbf{K}^{\\bullet} = \\mathbf{K}^{\\bullet}_p \\mathbf{K}^{\\bullet, p}$ where\n$\\mathbf{K}^{\\bullet}_p \\subset G(\\mathbb{Q}_p)$. The decomposition\n$V \\otimes \\mathbb{Q}_p = \\oplus_{i=0}^{s} V_{\\mathfrak{p}_i}$ is\northogonal with respect to $\\psi$. \nWe obtain that\n\\begin{displaymath}\nG^{\\bullet}(\\mathbb{Q}_p) = \\prod_{i=0}^{s} G^{\\bullet}_{\\mathfrak{p}_i}, \n \\end{displaymath}\nwhere \n\\begin{equation}\\label{BZGpi1e}\n G^{\\bullet}_{\\mathfrak{p}_i} = \\{g \\in\n \\Aut_{B_{\\mathfrak{p}_i}} V_{\\mathfrak{p}_i} \\; | \\;\n \\psi_{\\mathfrak{p}_i}(g v, g w) =\\psi_{\\mathfrak{p}_i}(\\mu_{\\mathfrak{p}_i}(g) v, w),\n \\; \\mu_{\\mathfrak{p}_i}(g) \\in F^{\\times}_{\\mathfrak{p}_i}\\}. \n\\end{equation}\nThis defines the homomorphism\n$\\mu_{\\mathfrak{p}_i}: G^{\\bullet}_{\\mathfrak{p}_i} \\rightarrow F^{\\times}_{\\mathfrak{p}_i}$. \nAccording to the decomposition\n$V \\otimes_{F} F_{\\mathfrak{p}_i} = V_{\\mathfrak{q}_i} \\oplus V_{\\bar{\\mathfrak{q}}_i}$ \nwe can write $g = (g_{\\mathfrak{q}_i}, g_{\\bar{\\mathfrak{q}}_i})$. Then we obtain\n\\begin{equation}\\label{BZGpi2e} \n G^{\\bullet}_{\\mathfrak{p}_i} = \\{g = (g_{\\mathfrak{q}_i}, g_{\\bar{\\mathfrak{q}}_i}), \\; \n g_{\\mathfrak{q}_i} \\in \\Aut_{B_{\\mathfrak{q}_i}} \\! V_{\\mathfrak{q}_i}, \\, \n g_{\\bar{\\mathfrak{q}}_i} \\in \\Aut_{B_{\\bar{\\mathfrak{q}}_i}} \\! V_{\\bar{\\mathfrak{q}}_i} \\;\n | \\;\n g'_{\\mathfrak{q}_i} g_{\\bar{\\mathfrak{q}}_i} \\in F^{\\times}_{\\mathfrak{p}_i} \\} ,\n\\end{equation}\ncf. (\\ref{BZGQ_p4e}). We note that\n$\\mu_{\\mathfrak{p}_i}(g) = g'_{\\mathfrak{q}_i} g_{\\bar{\\mathfrak{q}}_i}$. By this\nequation we obtain an isomorphism of groups\n\\begin{equation}\\label{BZGdot1e}\n \\begin{array}{cccr} \n G^{\\bullet}_{\\mathfrak{p}_i} & \\cong & G^{\\bullet}_{\\mathfrak{q}_i} \\times\n F^{\\times}_{\\mathfrak{p}_i}, & \\quad \\; G^{\\bullet}_{\\mathfrak{q}_i}\n = \\Aut_{B_{\\mathfrak{q}_i}} \\! V_{\\mathfrak{q}_i} ,\\\\\n g & \\mapsto & g_{\\mathfrak{q}_i} \\times \\mu_{\\mathfrak{p}_i}(g) & \n \\end{array}\n\\end{equation}\ncf. (\\ref{BZGQ_p5e}). Altogether we obtain an isomorphism\n\\begin{equation}\\label{BZGdot2e}\n G^{\\bullet}(\\mathbb{Q}_p) = \\prod_{i=0}^{s} G^{\\bullet}_{\\mathfrak{q}_i} \\times\n \\prod_{i=0}^{s} F^{\\times}_{\\mathfrak{p}_i}. \n\\end{equation}\nWe use the notations $\\Lambda_{\\mathfrak{q}_i}$, $\\Lambda_{\\bar{\\mathfrak{q}}_i}$,\n$\\Lambda_{\\mathfrak{p}_i}$, $O_{B_{\\mathfrak{q}_i}}$, $O_{B_{\\bar{\\mathfrak{q}}_i}}$, and\n$O_{B_{\\mathfrak{p}_i}} = O_{B_{\\mathfrak{q}_i}} \\oplus O_{B_{\\bar{\\mathfrak{q}}_i}}$\nas before \\eqref{BZLambda1e}. \nFor each prime $\\mathfrak{p}_i$, $i= 0, \\ldots, s$, we choose an open subgroup\n$\\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} \\subset O_{F_{\\mathfrak{p}_i}}^{\\times}$.\nWe set \n\\begin{equation}\\label{BZ20e}\n \\mathbf{M}^{\\bullet} = \\prod_{i=0}^{s} \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} \\subset\n \\prod_{i=0}^{s} O_{F_{\\mathfrak{p}_i}}^{\\times} = (O_F \\otimes \\mathbb{Z}_p)^{\\times}. \n\\end{equation}\n\nAs in the previous section (see right after \\eqref{BZLambda1e}), we set \n$\\mathbf{K}_{\\mathfrak{q}_0} = \\Aut_{O_{B_{\\mathfrak{q}_0}}} \\Lambda_{\\mathfrak{q}_0}$, and \n choose for $j > 0$ arbitrarily open compact subgroups \n$\\mathbf{K}_{\\mathfrak{q}_j}\\subset\\Aut_{O_{B_{\\mathfrak{q}_j}}}\\Lambda_{\\mathfrak{q}_j}$.\nWe define, for $i=0,\\ldots,s$, \n\\begin{displaymath}\n \\mathbf{K}_{\\mathfrak{p}_i} = \\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i} \\times\n \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} \\subset G^\\bullet_{\\mathfrak{q}_i} \\times\n F^{\\times}_{\\mathfrak{p}_i} \\cong G_{\\mathfrak{p}_i} ,\n\\end{displaymath}\ncf. (\\ref{BZGdot1e}). We obtain \n\\begin{displaymath}\n \\begin{array}{ll}\n \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i} = & \n \\{g = (g_{\\mathfrak{q}_i}, g_{\\bar{\\mathfrak{q}}_i}), \\; \n g_{\\mathfrak{q}_i} \\in \\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i} ,\\; \n g_{\\bar{\\mathfrak{q}}_i} \\in \\Aut_{O_{B_{\\bar{\\mathfrak{q}}_i}}} \\! \\Lambda_{\\bar{\\mathfrak{q}}_i} \n \\; | \\; \\\\ & \n \\psi_{\\mathfrak{p}_i}(g_{\\mathfrak{q}_i}v_1, g_{\\bar{\\mathfrak{q}}_i} v_2) =\n \\psi_{\\mathfrak{p}_i}(mv_1, v_2), \\; v_1 \\in \\Lambda_{\\mathfrak{q}_i}, \\,\n v_2\\in \\Lambda_{\\bar{\\mathfrak{q}}_i}, \\, m\\in \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} \\}, \n \\end{array}\n\\end{displaymath}\nand in particular \n\\begin{equation*}\n \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_0} = \\{g \\in\n\\Aut_{O_{B_{\\mathfrak{p}_0}}} \\Lambda_{\\mathfrak{p}_0} \\; | \\;\n\\psi_{\\mathfrak{p}_0}(gv_1, gv_2) = \\psi_{\\mathfrak{p}_0}(\\mu(g)v_1, v_2), \\;\n\\mu(g) \\in \\mathbf{M}^\\bullet_{\\mathfrak{p}_0} \\}. \n\\end{equation*}\nFinally we set\n\\begin{equation}\\label{BZKpPkt1e} \n \\mathbf{K}^{\\bullet}_p = \\prod_{i=0}^{s} \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i} =\n \\prod_{i=0}^{s} (\\mathbf{K}_{\\mathfrak{q}_i} \\times\n \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i}) .\n \\end{equation}\nWe choose $\\mathbf{K}^{\\bullet, p}$ arbitrarily and set\n\\begin{equation}\\label{BZ14e}\n\\mathbf{K}^{\\bullet} = \\mathbf{K}^{\\bullet}_p \\mathbf{K}^{\\bullet, p}.\n \\end{equation}\nWith these assumptions on $\\mathbf{K}^{\\bullet}$ we can rewrite the definition\nof the functor $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}$ as follows.\n\\begin{definition}\\label{BZApkt3altd} (alternative of Definition \\ref{BZApkt3d})\nLet $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}. Let $S$ be an \n$E$-scheme. A point of $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}(S)$ consists of the\nfollowing data: \n\\begin{enumerate} \n\\item[(a)] An abelian scheme $A$ over $S$ up to isogeny with an\n action $\\iota: B \\rightarrow \\End^{o} A$. \n\n\\item[(b)] A $F$-homogeneous polarization $\\bar{\\lambda}$ of $A$ which induces\n on $B$ the involution $b \\mapsto b^{\\star}$. \n \n\\item[(c)] A class $\\bar{\\eta}^p$ modulo $\\mathbf{K}^{\\bullet, p}$\nof $B \\otimes \\mathbb{A}^p_f$-module\n isomorphisms\n \\begin{displaymath}\n \\eta^p: V \\otimes \\mathbb{A}^p_f \\isoarrow \\mathrm{V}^p_f(A) ,\n \\end{displaymath}\n such that for each $\\lambda \\in \\bar{\\lambda}$ there is locally for the\n Zariski topology a constant \n $\\xi^{(p)}(\\lambda) \\in (F \\otimes \\mathbb{A}^p_f)(1)$\n with\n \\begin{displaymath}\n\\psi(\\xi^{(p)}(\\lambda) v_1, v_2) = E^{\\lambda}(\\eta^p(v_1), \\eta^p(v_2)). \n \\end{displaymath}\n\\item[(d)]\n For each polarization $\\lambda \\in \\bar{\\lambda}$ and for each prime\n $\\mathfrak{p}|p$ of $O_F$ a section\n $\\xi_{\\mathfrak{p}}(\\lambda) \\in F^{\\times}_{\\mathfrak{p}}(1)\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$\n such that $\\xi_{\\mathfrak{p}}(\\lambda u) = u \\xi_{\\mathfrak{p}}(\\lambda)$ for\n each $u \\in F^{\\times}$. \n\\item[(e)]\n For each $i = 0, 1, \\ldots, s$ a class $\\bar{\\eta}_{\\mathfrak{q}_i}$ modulo $\\mathbf{K}_{\\mathfrak{q}_i}^{\\bullet}$\n of $B_{\\mathfrak{q}_i}$-module isomorphisms \n \\begin{displaymath}\n \\eta_{\\mathfrak{q}_i}: V_{\\mathfrak{q}_i} \\isoarrow \\mathrm{V}_{\\mathfrak{q}_i}(A) .\n \\end{displaymath} \n \\end{enumerate}\nWe require that the following condition ${\\rm (KC)}$ holds,\n \\begin{displaymath}\n {\\rm char}(T, \\iota(b) \\mid \\Lie A) = \\prod_{\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}}\n \\varphi(\\Nm^o_{B\/K} (T -b))^{r_{\\varphi}} . \n \\end{displaymath}\n\n \\end{definition}\nWe write a point of this functor in the form\n\\begin{equation}\\label{BZGdot4e} \n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i,\n (\\xi_{\\mathfrak{p}})_{\\mathfrak{p}}) \n\\end{equation}\nor alternatively\n$(A,\\iota,\\bar{\\lambda},\\bar{\\eta}^p,(\\bar{\\eta}_{\\mathfrak{q}_i})_i,(\\xi_{\\mathfrak{p}_i})_i)$, $i = 0, \\ldots, s$. \n\nTo make the relationship of the last two Definitions \\ref{BZApkt3d} and\n\\ref{BZApkt3altd} explicit, we consider an $S$-valued point\n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta})$ of Definition \\ref{BZApkt3d}.\nWe fix $\\eta \\in \\bar{\\eta}$. Then $\\eta_p$ is an isomorphism\n\\begin{displaymath}\n \\eta_p = \\oplus_{i=0}^{s} \\eta_{\\mathfrak{p}_i}: \\bigoplus_{i=0}^{s} V_{\\mathfrak{p}_i} \n \\isoarrow \\bigoplus_{i=0}^{s} \\mathrm{V}_{\\mathfrak{p}_i} (A). \n\\end{displaymath}\nThe component\n$\\eta_{\\mathfrak{p}_i} = \\eta_{\\mathfrak{q}_i} \\oplus \\eta_{\\bar{\\mathfrak{q}}_i}$\nsatisfies an equation\n\\begin{displaymath}\n \\psi(\\xi_{\\mathfrak{p}_i}(\\lambda) v, w) =\n E^{\\lambda}(\\eta_{\\mathfrak{q}_i}(v), \\eta_{\\bar{\\mathfrak{q}}_i}(w)), \\quad\n v \\in V_{\\mathfrak{q}_i}, \\; w \\in V_{\\bar{\\mathfrak{q}}_i}. \n \\end{displaymath}\nWe see that the data $\\eta_{\\mathfrak{q}_i}$ and $\\eta_{\\bar{\\mathfrak{q}}_i}$\nfor $i =0, \\ldots, s$ determine the data $\\eta_{\\mathfrak{q}_i}$ and\n$\\xi_{\\mathfrak{p}_i}(\\lambda)$ and vice versa. Therefore we obtain the\n$S$-valued point\n$(A,\\iota,\\bar{\\lambda},\\bar{\\eta}^p,(\\bar{\\eta}_{\\mathfrak{q}_i})_i,(\\xi_{\\mathfrak{p}_i})_i)$\nof (\\ref{BZApkt3altd}). \n\n\nLet $g \\in G^{\\bullet}(\\mathbb{Q}_p)$. According to (\\ref{BZGdot2e}) we write\nit in the form $(\\ldots, g_{\\mathfrak{q}_i}, \\ldots, a_{\\mathfrak{p}_i}, \\ldots)$,\nwith $a_{\\mathfrak{p}_i} = \\mu_{\\mathfrak{p}_i}(g)$. Then the Hecke operator\n$g: \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}} \\rightarrow \\mathcal{A}^{\\bullet}_{g^{-1}\\mathbf{K}^{\\bullet}g},$ \n sends (\\ref{BZGdot4e}) to the point\n\\begin{equation}\\label{BZGdot5e} \n (A,\\iota,\\bar{\\lambda},\\bar{\\eta}^p, \n (\\overline{{\\eta}_{\\mathfrak{q}_i} g_{\\mathfrak{q}_i}})_i,\n (a_{\\mathfrak{p}_i}\\xi_{\\mathfrak{p}_i})_i),\n \\end{equation}\ncomp. (\\ref{Heckeoperator1e}).\n\nIt is convenient for us to introduce another action of \n$(F \\otimes \\mathbb{Q}_p)^{\\times}$, \n\\begin{equation}\\label{xi-action1e} \n a_{\\mid \\xi}: \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}} \\rightarrow\n \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}, \\quad\n a \\in (F \\otimes \\mathbb{Q}_p)^{\\times}. \n\\end{equation}\nWe write \n$a = (\\ldots, a_{\\mathfrak{p}}, \\ldots) \\in (F \\otimes \\mathbb{Q}_p)^{\\times} = \\prod_{\\mathfrak{p}|p} F_{\\mathfrak{p}}^{\\times}$.\nThen $a_{\\mid \\xi}$ maps a point (\\ref{BZGdot4e}) to\n\\begin{displaymath}\n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i,\n (a_{\\mathfrak{p}}\\xi_{\\mathfrak{p}})_{\\mathfrak{p}}) .\n \\end{displaymath} \nFor a fixed $\\mathfrak{p}$ let $a_{\\mathfrak{p}} \\in F_{\\mathfrak{p}}^{\\ast}$. The we\ndefine ${a_{\\mathfrak{p}}}_{\\mid \\xi_{\\mathfrak{p}}} := a_{\\mid \\xi}$ where $a$ is the\nelement with component $a_{\\mathfrak{p}}$ at $\\mathfrak{p}$ and with\n$a_{\\mathfrak{p}'} = 1$ for $\\mathfrak{p}' \\neq \\mathfrak{p}$. \n\nThe action of $a_{\\mid \\xi}$ of (\\ref{xi-action1e}) coincides with the\naction of the Hecke operator\n$g = (\\ldots, g_{\\mathfrak{q}_i}, g_{\\bar{\\mathfrak{q}}_i}. \\ldots) \\in G^{\\bullet}(\\mathbb{Q}_p)$,\nwhere $g_{\\mathfrak{q}_i} = 1$ and\n$g_{\\bar{\\mathfrak{q}}_i}=a_{\\mathfrak{p}_i}\\in F^\\times_{\\mathfrak{p}_i}\\cong K^\\times_{\\bar{\\mathfrak{q}}_i}$\nfor $i = 0, \\ldots, s$. \n\nWe will denote by $U_p(F) \\subset F^{\\times}$ the set of all $a \\in F^{\\times}$\n which are units in all fields $F_{\\mathfrak{p}}$ with $\\mathfrak{p} | p$.\n\\begin{definition}\\label{BZApkt4d}\nLet $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}. We define a functor$\\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}$ on the category of\n$E$-schemes $S$. A point of $\\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}(S)$ consists of the\nfollowing data: \n\\begin{enumerate} \n\\item[(a)] An abelian scheme $A$ over $S$ up to isogeny prime to $p$ with an\n action\n $\\iota: O_{B,(p)} \\rightarrow \\End A \\otimes_{\\mathbb{Z}} \\mathbb{Z}_{(p)}$.\n\n\\item[(b)] \n An $U_p(F)$-homogeneous polarization $\\bar{\\lambda}$ of $A$ which is\n principal in $p$ and which induces on $B$ the involution $b\\mapsto b^\\star$. \n\n \n\\item[(c)]\n A class $\\bar{\\eta}^p$ modulo $\\mathbf{K}^{\\bullet,p}$\nof\n $B \\otimes \\mathbb{A}^p_f$-module isomorphisms\n \\begin{displaymath}\n \\eta^p: V \\otimes \\mathbb{A}^p_f \\isoarrow \\mathrm{V}^p_f(A) , \\;\n \\end{displaymath}\n such that for each $\\lambda \\in \\bar{\\lambda}$ there is locally for the\n Zariski topology a constant \n $\\xi^{(p)}(\\lambda) \\in (F \\otimes \\mathbb{A}^p_f)^{\\times}(1)$\n with\n \\begin{displaymath}\n\\psi(\\xi^{(p)}(\\lambda) v_1, v_2) = E^{\\lambda}(\\eta^p(v_1), \\eta^p(v_2)). \n \\end{displaymath}\n\\item[(d)]\n For each polarization $\\lambda \\in \\bar{\\lambda}$ and for each prime\n $\\mathfrak{p}|p$ of $O_F$ a section\n $\\xi_{\\mathfrak{p}}(\\lambda) \\in O^{\\times}_{F_{\\mathfrak{p}}}(1)\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$\n such that\n $\\xi_{\\mathfrak{p}}(\\lambda u) = u \\xi_{\\mathfrak{p}}(\\lambda)$ for\n each $u \\in U_p(F)$. \n\\item[(e)]\n For each $j= 1, \\ldots s$, a class $\\bar{\\eta}_{\\mathfrak{q}_j}$ modulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_j}$\n of $O_{B_{\\mathfrak{q}_j}}$-module isomorphisms \n \\begin{displaymath}\n \\eta_{\\mathfrak{q}_j}: \\Lambda_{\\mathfrak{q}_j} \\isoarrow T_{\\mathfrak{q}_j}(A) .\\;\n \\end{displaymath} \n \\end{enumerate}\n We require that the following condition ${\\rm (KC)}$ holds,\n \\begin{displaymath}\n {\\rm char}(T, \\iota(b) \\mid \\Lie A) = \\prod_{\\varphi: K \\rightarrow \\bar{\\mathbb{Q}}}\n \\varphi(\\Nm^o_{B\/K} (T -b))^{r_{\\varphi}} . \n \\end{displaymath}\n\\end{definition}\n\\begin{variant}\\label{varia}\nWe will also use a modified version of this Definition. We obtain a\nfunctor isomorphic to $\\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}$ if we modify\nthe items $(b)$ and $(d)$ of Definition \\ref{BZApkt4d} as follows.\n\n\\begin{enumerate}\n\\item[($b^\\prime$)] An $F$-homogeneous polarization $\\bar{\\lambda}$ on $A$ which\n induces on $B$ the involution $\\star$ from \\eqref{defstar}. \n\n\\item[($d^\\prime$)] For each polarization $\\lambda \\in \\bar{\\lambda}$ and for\n each prime $\\mathfrak{p}|p$ of $O_F$ a section\n $\\xi_{\\mathfrak{p}}(\\lambda) \\in F^{\\times}_{\\mathfrak{p}}(1)\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$ such that\n $\\xi_{\\mathfrak{p}}(u \\lambda) = u \\xi_{\\mathfrak{p}}(\\lambda)$ for\neach $u \\in F^{\\times}$ and such that $\\lambda$ is principal in $\\mathfrak{p}$\niff $\\xi_{\\mathfrak{p}}(\\lambda) \\in O_{F^{\\times}_{\\mathfrak{p}}}^{\\times}(1)$. \n\\end{enumerate}\n\\end{variant}\n\\begin{proposition}\\label{BZ11p} \nLet $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}. The \nfunctors $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}$ and\n$\\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}$ on the category of\n$E$-schemes are canonically isomorphic. \n\\end{proposition}\n\\begin{proof}\n The proof is an obvious modification of the proof of Proposition \\ref{BZ1p}.\n But for later use we indicate the point of\n $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}(S)$ which corresponds to a point\n$(A_0,\\bar{\\lambda}_0, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j, (\\xi_{\\mathfrak{p_i}})_i)$ \n of $\\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}(S)$\n (recall from Definition \\ref{BZsAK2d} that the index $i$ runs from\n $0,\\ldots, s$ and the index $j$ from $1,\\ldots, s$). Since we work over $E$,\n the Tate module $T_{\\mathfrak{q}_0}(A)$ makes sense. By\n our choice of $\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_0}$ there is a unique class of \n $O_{B_{\\mathfrak{q}_0}}$-module isomorphisms\n $\\eta_{\\mathfrak{q}_0}: \\Lambda_{\\mathfrak{q}_0} \\rightarrow T_{\\mathfrak{q}_0}(A)$\n modulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_0}$. Therefore we may replace $j$ in\n datum $(e)$ by $i = 0, \\ldots s$ in Definition \\ref{BZApkt4d},\n without changing anything.\n \n Let $A$ be the isogeny\n class of $A_0$ and choose $\\lambda_0 \\in \\bar{\\lambda}_0$ and\n $\\eta^{p} \\in \\bar{\\eta}^p$ to construct a point of\n $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}(S)$.\nFor $i = 0, \\ldots, s$, the isomorphisms\n $\\eta_{\\mathfrak{q}_i}: \\Lambda_{\\mathfrak{q}_i} \\rightarrow T_{\\mathfrak{q}_i}(A_0)$\n induce by duality (using $\\psi$ and $E^{\\lambda_0}$) an isomorphism\n $T_{\\bar{\\mathfrak{q}}_i}(A_0) \\rightarrow \\Lambda_{\\bar{\\mathfrak{q}}_i}(1)$.\n If we multiply the inverse of this map by $\\xi_{\\mathfrak{p}_i}(\\lambda_0)$\n we obtain an isomorphism\n \\begin{displaymath}\n \\eta_{\\bar{\\mathfrak{q}}_i}: \n \\Lambda_{\\bar{\\mathfrak{q}}_i} \\isoarrow T_{\\bar{\\mathfrak{q}}_i}(A_0) \n \\end{displaymath}\n which satisfies\n \\begin{equation}\\label{BZ17S1e}\nE^{\\lambda_0}(\\eta_{\\mathfrak{q}_i}(v), \\eta_{\\bar{\\mathfrak{q}}_i}(w)) \n= \\psi(\\xi_{\\mathfrak{p}_i}(\\lambda_0) v, w), \\quad v \\in \\Lambda_{\\mathfrak{q}_i},\n\\; w \\in \\Lambda_{\\bar{\\mathfrak{q}}_i}. \n \\end{equation}\n We set $\\eta_{\\mathfrak{p}_i} = \\eta_{\\mathfrak{q}_i} \\oplus \\eta_{\\bar{\\mathfrak{q}}_i}$\n and $\\eta_p = \\oplus_{i=0}^{s} \\eta_{\\mathfrak{p}_i}$. We denote by $\\bar{\\eta}_p$\n the class modulo $\\mathbf{K}_p$ of $\\eta_p$, and by $\\lambda$ the\n $\\mathbb{Q}$-homogeneous polarization which contains $\\lambda_0$. Then\n $(A, \\iota, \\bar{\\lambda}, \\bar{\\eta})$ is the corresponding point of \n $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}(S)$.\n\\end{proof}\n\nWe reformulate the action of the Hecke operator (\\ref{BZGdot5e}) in terms\nof Variant \\ref{varia} of Definition \\ref{BZApkt4d}. This will not be used\nuntil section \\ref{s:uniform}. We consider an element \n$g \\in G^{\\bullet}(\\mathbb{Q}_p) \\subset G^{\\bullet}(\\mathbb{A}_f)$. We write\n$g = (\\ldots g_{\\mathfrak{q}_i}, \\bar{g}_{\\mathfrak{q}_i}, \\ldots)$ as before, where\n$i = 0, \\ldots, s$. \nWe consider an open compact subgroup\n$\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$\nas in \\eqref{BZKpPkt1e}, such that\n$k g\\Lambda_{\\mathfrak{p}_i} = g\\Lambda_{\\mathfrak{p}_i}$, for\n$k \\in \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$. \n\nLet\n\\begin{equation}\\label{Heckevar1e}\n(A_0, \\iota_0, \\bar{\\lambda}, \\bar{\\eta}^{p},(\\bar{\\eta}_{\\mathfrak{q}_j})_j,\n (\\xi_{\\mathfrak{p_i}})_i) \\in \\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}(S) \n \\end{equation}\nbe a point of Variant \\ref{varia}. We recall that there is a unique isomorphism\nof $O_{B_{\\mathfrak{q}_0}}$-modules \n$\\eta_{\\mathfrak{q}_0}: \\Lambda_{\\mathfrak{q}_0} \\rightarrow T_{\\mathfrak{q}_0}(A_0)$ \nmodulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_0}$. This defines the unique class\n$\\bar{\\eta}_{\\mathfrak{q}_0}$. First we use this class to describe the Hecke\noperator $g$. \n\nA point \n\\begin{equation}\\label{RZ15e} \n (A_1, \\iota_1, \\bar{\\lambda}_1, \\bar{\\theta}^{p},(\\bar{\\theta}_{\\mathfrak{q}_j})_j,\n (\\xi'_{\\mathfrak{p_i}})_i) \\in \\mathcal{A}^{\\bullet bis}_{g^{-1}\\mathbf{K}^{\\bullet}g}(S)\n\\end{equation}\nis the image of (\\ref{Heckevar1e}) by the Hecke operator $g$ if the following\nconditions are fulfilled. There exists a quasi-isogeny\n\\begin{equation}\\label{Heckevar2e} \n\\alpha: (A_1, \\iota_1) \\rightarrow (A_0, \\iota_0) \n\\end{equation}\nsuch that \n\\begin{equation}\\label{RZ16e}\n \\begin{array}{l} \n\\alpha^{*}( \\bar{\\lambda}) = \\bar{\\lambda}_1, \\quad \n\\alpha^{*} (\\bar{\\eta}^p) = \\bar{\\theta}^p,\\\\[2mm] \n \\xi'_{\\mathfrak{p_i}}(\\alpha^{*}(\\lambda)) =\n \\mu_{\\mathfrak{p_i}}(g) \\xi_{\\mathfrak{p_i}}(\\lambda), \\quad \\text{for} \\;\n \\lambda \\in \\bar{\\lambda}.\n \\end{array}\n\\end{equation}\nMoreover we require that the data $\\bar{\\theta}_{\\mathfrak{q}_i}$ and\n$\\bar{\\eta}_{\\mathfrak{q}_i}$ for $i = 0, \\ldots, s$ are connected by the\nfollowing diagrams \n\\begin{equation}\\label{Heckevar3e}\n \\xymatrix{\n T_{\\mathfrak{q}_i}(A_1) \\otimes \\mathbb{Q} \\ar[r]^{\\alpha} &\n T_{\\mathfrak{q}_i}(A_0) \\otimes \\mathbb{Q}\\\\\n \\Lambda_{\\mathfrak{q}_i} \\otimes \\mathbb{Q} \\ar[u]_{\\theta_{\\mathfrak{q}_i}} \n \\ar[r]_{g_{\\mathfrak{q}_i}} & \\Lambda_{\\mathfrak{q}_i} \\otimes \\mathbb{Q} , \n \\ar[u]_{\\eta_{\\mathfrak{q}_i}} \\\\ \n } \n \\end{equation} \nwhere $\\theta_{\\mathfrak{q}_i} \\in \\bar{\\theta}_{\\mathfrak{q}_i}$, \n$\\eta_{\\mathfrak{q}_i} \\in \\bar{\\eta}_{\\mathfrak{q}_i}$. The diagrams are \nrequired to be commutative after replacing $\\eta_{\\mathfrak{q}_i}$ by\n$\\eta_{\\mathfrak{q}_i} k$ for some $k \\in \\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i}$.\n\nWe may reformulate the condition (\\ref{Heckevar3e}) for $i=0$ without mentioning\nthe classes $\\bar{\\eta}_{\\mathfrak{q}_0}$ and $\\bar{\\theta}_{\\mathfrak{q}_0}$.\nRecall that\n$G^{\\bullet}_{\\mathfrak{q}_0}\\cong (B^{\\rm opp}_{\\mathfrak{q}_0})^{\\times}\\cong D^{\\times}_{\\mathfrak{p}_0}$,\ncf. (\\ref{BZGdot1e}), (\\ref{Dsplit-in0}). \nLet $m = \\ord g_{\\mathfrak{q}_0}$ be the valuation in the division algebra\n$D_{\\mathfrak{p}_0}$. Let $\\Pi\\in O_{D_{\\mathfrak{q}_0}}$ be a prime element which \nwe regard also as an element of $O_{D^{opp}_{\\mathfrak{q}_0}}$. We may define $m$ by\n$g_{\\mathfrak{q}_0} (\\Lambda_{\\mathfrak{q}_0}) = \\Pi^{m} \\Lambda_{\\mathfrak{q}_0}$.\nLet $X_0$ be the $p$-divisible group of $A_0$ and $X_1$ the $p$-divisible\ngroup of $A_1$. The condition (\\ref{Heckevar3e}) for $i=0$ is equivalent\nwith the condition that the quasi-isogeny of $p$-divisible groups \n\\begin{equation}\n\\Pi^{-m} \\alpha: (X_1)_{\\mathfrak{q}_0} \\rightarrow (X_0)_{\\mathfrak{q}_0} \n \\end{equation}\nis an isomorphism. \n\nWe note that (\\ref{RZ16e}) means implicitly that a polarization\n$\\lambda_1 \\in \\bar{\\lambda}_1$ is $\\mathfrak{p}_i$-principal if it differs\nfrom $\\mu_{\\mathfrak{p}_i}(g)^{-1} \\alpha^{*} (\\lambda)$ by a unit in\n$O_{F_{\\mathfrak{p_j}}}^{\\times}$, for a polarization\n$\\lambda \\in \\bar{\\lambda}$ of $A_0$ which is principal in $p$. \n\nThat (\\ref{RZ15e}) indeed describes the image by the Hecke operator given by\n$g \\in G^{\\bullet}(\\mathbb{Q}_p)$ follows immediately from (\\ref{BZGdot5e}) \nif we pass from $A_0$ to its isogeny class as in Definition \\ref{BZApkt3altd}.\nOur conditions for $A_1$ only ensure that we obtain a point of Variant \n\\ref{varia}. This description of the Hecke operators will allow us to extend\nthem in section \\ref{s:uniform} to a model of the functor Variant \\ref{varia}\nover $O_{E_{\\nu}}$. \n\n Using Proposition \\ref{BZ11p}, we identify the functors $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}$ and\n $\\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}$\n and use the notation $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}$ for this functor. \nFinally we can define a functor on the category of $O_{E_{\\nu}}$-schemes.\n\\begin{definition}\\label{BZsApkt4d} \nLet $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}. We define a functor $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ on the\ncategory of $O_{E_{\\nu}}$-schemes $S$. \nAn $S$-valued point consists of the data $(a), (b), (c), (e)$ as in Definition\n\\ref{BZApkt4d}. But we replace $(d)$ by the following datum,\n\\begin{enumerate}\n\\item[($d^t$)]\n For each polarization $\\lambda \\in \\bar{\\lambda}$ and for each prime \n $\\mathfrak{p}|p$ of $O_F$ a section\n $\\xi_{\\mathfrak{p}}(\\lambda) \\in O^{\\times}_{F_{\\mathfrak{p}}}\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$\n such that\n $\\xi_{\\mathfrak{p}}(u \\lambda) = u \\xi_{\\mathfrak{p}}(\\lambda)$ for\n each $u \\in U_p(F)$.\n \\end{enumerate}\n\\end{definition}\n\n\n Let $\\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ be the functor on the category of\n $E$-schemes $S$ which is obtained by changing in Definition \\ref{BZApkt3altd}\n the item $(d)$ into\n \\begin{enumerate}\n\\item[($d^t$)]\n For each polarization $\\lambda \\in \\bar{\\lambda}$ and for each prime\n $\\mathfrak{p}|p$ of $O_F$ a section\n $\\xi_{\\mathfrak{p}}(\\lambda)\\in F^{\\times}_{\\mathfrak{p}}\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$\n such that $\\xi_{\\mathfrak{p}}(u \\lambda) = u \\xi_{\\mathfrak{p}}(\\lambda)$ for\n each $u \\in F^{\\times}$.\n \\end{enumerate}\n By changing $(d)$ in Definition \\ref{BZApkt4d} in the same way (i.e., replacing $O^{\\times}_{F_{\\mathfrak{p}}}(1)\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$ by $O^{\\times}_{F_{\\mathfrak{p}}}\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$), we obtain\n another description of $\\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$.\nWe call this the $t$-version of Definition \\ref{BZApkt4d}.\n \n By the proof of Proposition \\ref{BZ11p}, we have a canonical isomorphism \n \\begin{equation}\\label{BZGdot6e} \n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec E} \\Spec E_{\\nu} \\cong\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec E_{\\nu}. \n \\end{equation}\n\\begin{remark}\\label{remHecke}\nFor $g \\in G^{\\bullet}(\\mathbb{A}_f)$ we have the Hecke operators\n$g: \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\rightarrow \\mathcal{A}^{\\bullet t}_{g^{-1}\\mathbf{K}^{\\bullet} g}$ . For $g\\in G^{\\bullet}(\\mathbb{A}^p_f)$ these extend\nobviously to\n$g: \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\rightarrow \\tilde{\\mathcal{A}}^{\\bullet t}_{g^{-1}\\mathbf{K}^{\\bullet} g}$. \nLet $g \\in G^{\\bullet}(\\mathbb{Q}_p)$, we have defined the Hecke operator by\n(\\ref{BZGdot5e}).\n In section \\ref{s:uniform} we will extend this Hecke operator to\n $g: \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\rightarrow \\tilde{\\mathcal{A}}^{\\bullet t}_{g^{-1}\\mathbf{K}^{\\bullet} g}$,\n whenever both $\\mathbf{K}^{\\bullet}_p$ and $g^{-1}\\mathbf{K}^{\\bullet}_pg$ are as specified in \\eqref{BZKpPkt1e}, so that both source and target make sense.\n This will be done by the remark after Proposition \\ref{BZ11p}.\nFor the time being, we only need the extensions of Hecke operators defined by elements in $(K\\otimes{\\ensuremath{\\mathbb {Q}}\\xspace_p})^\\times\\subset G^\\bullet(\\ensuremath{\\mathbb {Q}}\\xspace_p)$. For these Hecke operators we give an ad hoc definition, cf. \\eqref{Hecke2e}.\n \\end{remark}\n \nLet $\\zeta_{p^{\\infty}} \\in \\bar{\\mathbb{Q}}$ be a compatible system of\nprimitive $p^n$-th roots of unity. It defines over $\\bar{\\ensuremath{\\mathbb {Q}}\\xspace}$ an isomorphism\nof \\'etale sheaves for each $\\mathfrak p$,\n \\begin{equation}\\label{BZ24e} \n \\kappa_{\\mathfrak{p}}\\colon O_{F_{\\mathfrak{p}}}^{\\times}\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet} \n \\isoarrow O_{F_{\\mathfrak{p}}}^{\\times}(1)\/\\mathbf{M}_{\\mathfrak{p}}^{\\bullet}, \\quad\\quad\n \\xi_{\\mathfrak{p}} \\mapsto \\zeta_{p^{\\infty}}\\xi_{\\mathfrak{p}}.\n \\end{equation} \nIt is defined over a finite abelian extension $L\/E$ which we choose\nindependently of $\\mathfrak{p}$. The isomorphism (\\ref{BZ24e}) defines an\nisomorphism of functors\n\\begin{equation}\\label{BZ33e}\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}\\times_{\\Spec E} \\Spec L\n \\isoarrow\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}\\times_{\\Spec E} \\Spec L.\n\\end{equation}\nHere, for an $L$-scheme $S$ a point\n \\begin{equation}\\label{BZ23e} \n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j,\n ( \\xi_{\\mathfrak{p_i}})_i) \\in \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(S) \n \\end{equation}\nis mapped by the isomorphism (\\ref{BZ33e}) to \n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j, (\\zeta_{p^{\\infty}} \\xi_{\\mathfrak{p_i}})_i)$.\nThe isomorphism (\\ref{BZ33e}) is compatible with the Hecke operators\n$G^{\\bullet}(\\mathbb{A}_f)$ acting on both sides. \n\n\\begin{proposition}\\label{BZ3c} \n Let $\\tau \\in \\Gal(L\/E)$ be an automorphism. \n The isomorphism (\\ref{BZ33e}) fits into a commutative diagram\n \\begin{displaymath}\n \\xymatrix{\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec E} \\Spec L\n \\ar[d]_{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\n \\ar[r] & \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec E} \\Spec L \n \\ar[d]^{{\\varsigma_{p^{\\infty}}(\\tau^{-1})}_{\\mid \\xi} \\times \\tau_c}\\\\\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec E} \\Spec L \\ar[r] & \n \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec E} \\Spec L .\\\\\n }\n \\end{displaymath}\n Here we take the composite of the cyclotomic character by the inclusion \n \\begin{displaymath} \n \\varsigma_{p^{\\infty}}: \\Gal(\\bar{E}\/E) \\rightarrow \\mathbb{Z}_p^{\\times} \\subset\n (O_F \\otimes \\mathbb{Z}_p)^{\\times}. \n \\end{displaymath}\n {\\rm See (\\ref{xi-action1e}) for the definition of the automorphism $a_{\\mid \\xi}$ of $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}$.\n A similiar definition applies to\n $\\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$. }\n\\end{proposition}\n\\noindent \nThe proof coincides with that of Proposition \\ref{Gtwist1p}. As a consequence\nwe have an analogue of Proposition \\ref{Gtwist3p}, i.e. there is a morphism\nof functors\n\\begin{equation}\\label{compshim}\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}} \n \\Spec E^{nr}_{\\nu} \n\\rightarrow \n{\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G, h \\delta^{-1})_{E_{\\nu}} \\times_{\\Spec E_{\\nu}} \n\\Spec E^{nr}_{\\nu} .\n\\end{equation}\nThe descent data relative to $E^{nr}_{\\nu}\/E_{\\nu}$ on both sides are compatible\nup to the factor $p^{f_{\\nu}}_{\\mid \\xi}$ which can be expressed by a diagram\nsimiliar to that of Proposition \\ref{Gtwist1p}. In contrast to Proposition\n\\ref{Gtwist1p}, the morphism \\eqref{compshim} is no longer an isomorphism\nsince we are dealing with a coarse moduli scheme. \n\nWe will next show that the action of the group \n$(K \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p)^{\\times} \\subset G^{\\bullet}(\\mathbb{Q}_p)$ on\n$\\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ by Hecke operators extends naturally to\nan action on the $O_{E_{\\nu}}$-scheme\n$\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$. We write an element of that\ngroup as \n\\begin{displaymath}\n z = (\\ldots, a_i, b_i, \\ldots) \\in (K \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p)^{\\times}\n \\cong \\prod_{i=0}^{s} (K_{\\mathfrak{q}_i}^{\\times} \\times K_{\\bar{\\mathfrak{q}}_i}^{\\times})\n \\cong \\prod_{i=0}^{s} (F_{\\mathfrak{p}_i}^{\\times} \\times F_{\\mathfrak{p}_i}^{\\times}), \n \\end{displaymath}\nwhere $a_i, b_i \\in F_{\\mathfrak{p}_i}^{\\times}$, for $i = 0, 1, \\ldots, s$. We note\nthat $\\mu_{\\mathfrak{p}_i}(z) = a_i b_i$.\n\nWe consider a point $(A, \\iota, \\bar{\\lambda}, \\bar{\\eta})$ of\n$\\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet}(S)$ as in Definition \\ref{BZApkt3d}.\nWe write $\\eta = \\eta^p \\eta_p$ and\n\\begin{displaymath}\n \\eta_p = \\oplus_{i=0}^s (\\eta_{\\mathfrak{q}_i} \\oplus \\eta_{\\bar{\\mathfrak{q}}_i}),\n \\quad\n \\eta_{\\mathfrak{q}_i}: V_{\\mathfrak{q}_i} \\rightarrow \\mathrm{V}_{\\mathfrak{q}_i}(A),\n \\; \n \\eta_{\\bar{\\mathfrak{q}}_i}: V_{\\bar{\\mathfrak{q}}_i} \\rightarrow\n \\mathrm{V}_{\\bar{\\mathfrak{q}}_i}(A). \n \\end{displaymath}\nThe Hecke operator\n$z: \\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet} \\rightarrow \\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet}$ \non $S$-valued points is given by\n\\begin{equation}\\label{Hecke2e} \n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i,\n (\\bar{\\eta}_{\\bar{\\mathfrak{q}}_i})_i) \\longmapsto (A, \\iota, \\bar{\\lambda},\n \\bar{\\eta}^p, \\; (\\bar{\\eta}_{\\mathfrak{q}_i}\\cdot a_i)_i, \\;\n (\\bar{\\eta}_{\\bar{\\mathfrak{q}}_i} \\cdot b_i)_i). \n\\end{equation}\nLet $\\mathbf{x} \\in K^{\\times}$. We write its image in\n$(K\\otimes_{\\mathbb{Q}_p} \\mathbb{Q}_p)^\\times$ as\n\n\\begin{displaymath}\n (\\ldots, x_i, y_i, \\ldots) \\in (K \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p)^{\\times}\n \\cong \\prod_{i=0}^{s} (K_{\\mathfrak{q}_i}^{\\times} \\times K_{\\bar{\\mathfrak{q}}_i}^{\\times})\n \\cong \\prod_{i=0}^{s} (F_{\\mathfrak{p}_i}^{\\times} \\times F_{\\mathfrak{p}_i}^{\\times}), \n\\end{displaymath}\nwhere $x_i, y_i \\in F_{\\mathfrak{p}_i}^{\\times}$. We note that $x_i y_i$ is the\nimage of $\\Nm_{K\/F} \\mathbf{x}$ in $F_{\\mathfrak{p}_i}^{\\times}$.\n\nWe consider the quasi-isogeny $\\mathbf{x}: A \\rightarrow A$ induced by multiplication by $\\mathbf{x}$.\nThe inverse image of the data\n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, \\; (\\bar{\\eta}_{\\mathfrak{q}_i}\\cdot a_i)_i, \\; (\\bar{\\eta}_{\\bar{\\mathfrak{q}}_i} \\cdot b_i)_i)$ \nby this quasi-isogeny is\n\\begin{equation}\\label{Hecke3e}\n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p \\cdot \\mathbf{x}^{-1}, \\;\n (\\bar{\\eta}_{\\mathfrak{q}_i}\\cdot a_i x_i^{-1})_i,\n \\; (\\bar{\\eta}_{\\bar{\\mathfrak{q}}_i} \\cdot b_i y_i^{-1})_i) .\n\\end{equation}\nTherefore this is just another way to write the image under the Hecke operator\n(\\ref{Hecke2e}). \n\nWe rewrite (\\ref{Hecke2e}) in terms of the alternative\nDefinition \\ref{BZApkt3altd}. \nIn terms of this definition, the left hand side of (\\ref{Hecke2e}) corresponds to \n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i,\n (\\xi_{\\mathfrak{p}_i})_i)$ \nand (\\ref{Hecke3e}) corresponds to\n$$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p \\mathbf{x}^{-1}, (\\eta_{\\mathfrak{q}_i} a_i x_i^{-1})_i, \\; (a_i b_i (\\Nm_{K\/F} \\mathbf{x}^{-1}) \\xi_{\\mathfrak{p}_i})_i) .\n$$ \n\nSummarizing, the Hecke operator\n$z: \\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet} \\rightarrow \\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet}$ \nbecomes in terms of Definition \\ref{BZApkt3altd} the map \n\\begin{equation}\\label{Hecke4e}\n(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_i})_i, \n(\\xi_{\\mathfrak{p}_i})_i) \\longmapsto\n(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p \\mathbf{x}^{-1},(\\bar{\\eta}_{\\mathfrak{q}_i}\na_i x_i^{-1})_i, \\; (a_i b_i (\\Nm_{K\/F} \\mathbf{x}^{-1}) \\xi_{\\mathfrak{p}_i})_i) .\n\\end{equation}\nIn the same way $z$ acts on $\\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$.\nIn fact, we are only interested in the latter functor. \nLet us choose $\\mathbf{x} \\in K^{\\times}$, such that\n$a_i x_i^{-1}$ and $b_i y_i^{-1}$ are units in $O_{F_{\\mathfrak{p}_i}}^{\\times}$ for\n$i = 0, \\ldots, s$. \nAssume we have chosen the left hand side of (\\ref{Hecke4e})\nin the form of the t-version of Definition \\ref{BZApkt4d}. Note that we\nhave added to the data of this definition the unique class of\n$O_{B_{\\mathfrak{q}_0}}$-module isomorphisms\n$\\bar{\\eta}_{\\mathfrak{q}_0}: \\Lambda_{\\mathfrak{q}_0} \\isoarrow T_{\\mathfrak{q}_0}(A)$ \nmodulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_0}$. We then see that the right hand\nside of (\\ref{Hecke4e}) is also a point in the sense of Definition\n\\ref{BZApkt4d}. For $i=0$ we have the isomorphism \n$\\bar{\\eta}_{\\mathfrak{q}_0} a_0 x_0^{-1}: \\Lambda_{\\mathfrak{q}_0} \\isoarrow T_{\\mathfrak{q}_0}(A)$\nas required. Hence we may forget about $i=0$ and obtain a definition of the\nHecke operator in terms of the t-version of Definition \\ref{BZApkt4d}, \n\\begin{displaymath}\n z: \\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet t} \\rightarrow\n \\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet t} .\n \\end{displaymath}\nThis definition of $z$ makes sense for the functor\n$\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$. We define \n\\begin{equation}\\label{BZGdot7e} \n \\tilde{z}: \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\rightarrow\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}\n\\end{equation}\nas follows. Let\n$(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j, ( \\xi_{\\mathfrak{p_i}})_i)$\nbe a point of $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$\n with values in an $O_{E_{\\nu}}$-scheme $S$. We define the image by morphism \n (\\ref{BZGdot7e}) as\n\\begin{displaymath}\n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p \\mathbf{x}^{-1},\n (\\bar{\\eta}_{\\mathfrak{q}_j} a_j x_j^{-1})_j,\n (a_i b_i (\\Nm_{K\/F} \\mathbf{x}^{-1})\\xi_{\\mathfrak{p_i}})_i).\n \\end{displaymath}\nIt is clear that this is an extension of $z$ with respect to the isomorphism\n(\\ref{BZGdot6e}). \n\n\nRecall from \\eqref{defhD} \n \\begin{equation}\\label{h_D1e}\n h_D\\colon \\mathbb{S} \\rightarrow\n (D \\otimes \\mathbb{R})^{\\times } \\cong \n \\mathrm{GL}_2(\\mathbb{R}) \\times\n \\prod_{\\chi \\neq \\chi_0} (D \\otimes_{F, \\chi} \\mathbb{R})^\\times ,\\quad \n z = a + b\\mathbf{i} \\mapsto \n \\left(\n \\begin{array}{rr}\n a & -b\\\\\n b & a\n \\end{array}\n \\right) \\times (1, \\ldots, 1).\n \\end{equation}\n Moreover, we consider the composite \n\\begin{equation}\\label{h_D2e}\n \\begin{array}{rcccl} \n h^{\\bullet}_D: &\\mathbb{S} & \\rightarrow & \\quad (D \\otimes \\mathbb{R})^{\\times}\n \\times (K \\otimes \\mathbb{R})^{\\times} & \\rightarrow G^{\\bullet}_{\\mathbb{R}} ,\\\\ \n& z & \\mapsto & h_D(z)\\; \\times \\; 1 &\n \\end{array}\n \\end{equation}\n cf. Lemma \\ref{BZ1l}.\n \n Recall from \\eqref{def:bulletshim} the Shimura datum $(G^\\bullet, h_D^\\bullet)$. The next proposition relates the Shimura varieties ${\\rm Sh}(G^\\bullet, h)$ and ${\\rm Sh}(G^\\bullet, h_D^\\bullet)$. \n\\begin{proposition}\\label{Sh_D1p} \n Let $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}, where\n $\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. Denote by\n $f_{\\nu}$ the inertia index of $E_{\\nu}\/\\mathbb{Q}_p$. \n Let $\\pi_{\\mathfrak{p}_0}$ be an arbitrary prime element of $F_{\\mathfrak{p}_0}$.\n We consider the element \n \\begin{displaymath}\n \\dot{z} = (\\pi_{\\mathfrak{p}_0}^{-1} p^{f_{\\nu}}, p^{f_{\\nu}}, \\ldots, p^{f_{\\nu}})\n \\in (F \\otimes \\mathbb{Q}_p)^{\\times} = \\prod_{i=0}^{s} F_{\\mathfrak{p}_i}^{\\times}. \n \\end{displaymath}\n Let $\\tau \\in \\Gal(E^{nr}_{\\nu}\/E_{\\nu})$ be the Frobenius automorphism.\n Then there is a morphism of functors\n \\begin{equation}\\label{BZGdot10e}\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{nr}_{\\nu} \\rightarrow\n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{nr}_{\\nu}, \n \\end{equation}\n such that the following diagram is commutative\n \\begin{equation*}\n \\xymatrix{\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet},E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu} \n \\ar[d]_{\\dot{z}_{\\mid \\xi} \\times \\tau_c} \\ar[r]\n & {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu} \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\\\\\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{nr}_{\\nu} \\ar[r] & {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}} \\Spec E^{nr}_{\\nu}, \n }\n \\end{equation*} \n Here, the right hand side of (\\ref{BZGdot10e}) is the coarse moduli scheme of\n the functor on the left hand side. {\\rm See (\\ref{xi-action1e}) for the definition of $\\dot{z}_{\\mid \\xi}$.} \n \\end{proposition}\n We will show in Proposition \\ref{BZ7p} that ${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{nr}_{\\nu}$ \n is in fact the \\'etale sheafification of $ \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{nr}_{\\nu}$. \n\\begin{proof}\n Recall the morphism to the coarse moduli space\n \\begin{equation}\\label{BZGdot11e}\n \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet},E_{\\nu}} \\rightarrow\n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h)_{E_{\\nu}}. \n \\end{equation}\n Let $T^{\\bullet} = (K \\otimes \\mathbb{Q})^{\\times} \\subset G^{\\bullet}$ be the central\n torus. We consider $\\delta: \\mathbb{S} \\rightarrow T^{\\bullet}_{\\mathbb{R}}$\n cf. (\\ref{delta1e}). The local reciprocity law\n $r_{\\nu}(T^{\\bullet},\\delta^{-1}): E_{\\nu}^{\\times}\\rightarrow T^{\\bullet}(\\mathbb{Q}_p)$\n is the composite of the local reciprocity law $r_{\\nu}(T, \\delta^{-1})$ for $T$ (given by \\eqref{rec-delta1e}) with the inclusion\n $T(\\mathbb{Q}_p) \\subset T^{\\bullet}(\\mathbb{Q}_p)$. Let $e \\in E_{\\nu}^{\\times}$\n and let $\\sigma \\in \\Gal(E^{ab}_{\\nu}\/E_{\\nu})$ be the automorphism which\n corresponds to it by local class field theory. \n If we twist the morphism (\\ref{BZGdot11e}) by $r_{\\nu}(T^{\\bullet},\\delta^{-1})$\n we obtain as in the proof of Proposition \\ref{Gtwist3p} a commutative diagram\n \\begin{equation}\\label{Gtwist5e} \n \\begin{aligned}\\xymatrix{\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet},E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu} \n \\ar[d]_{(p^{f_{\\nu} \\ord e})_{\\mid \\xi} \\times \\sigma_c} \\ar[r]\n & {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h \\delta^{-1})_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu} \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\sigma_c}\\\\\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{ab}_{\\nu} \\ar[r] & {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h\\delta^{-1})_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}} \\Spec E^{ab}_{\\nu}. \n }\n \\end{aligned}\n \\end{equation}\n We consider the homomorphism\n $\\gamma: \\mathbb{S} \\rightarrow (K \\otimes_{\\mathbb{Q}} \\mathbb{R})^{\\times}$\n which in terms of the isomorphism (\\ref{CMXi1e}) is defined by\n \\begin{equation}\n \\gamma: \\mathbb{S} \\rightarrow \\mathbb{C}^{\\times} \\times\n \\prod_{\\varphi, r_{\\varphi}=2} \\mathbb{C}^{\\times} , \\quad\n z \\mapsto (z, 1, \\ldots, 1) ,\n \\end{equation}\n i.e., on the right hand side we have $z$ at the factor which corresponds to\n $\\bar{\\varphi}_0$. We find that \n \\begin{equation}\n h^{\\bullet}_{D} = h \\delta^{-1} \\gamma .\n \\end{equation}\n Therefore we must twist the horizontal line of (\\ref{Gtwist5e}) by the \n local reciprocity law of $\\gamma$. The one-parameter\n group $\\mu_{\\gamma}$ associated to the Shimura datum $\\gamma$ is\n \\begin{displaymath}\n \\mu_{\\gamma}: \\mathbb{C}^{\\times} \\rightarrow \n \\prod_{\\Phi} \\mathbb{C}^{\\times}, \\quad z \\mapsto (1,\\ldots, 1,z,1,\\ldots, 1) ,\n \\end{displaymath}\n where $z$ is exactly at the place $\\bar{\\varphi}_0$. Since we are interested\n in the local reciprocity law we replace $\\mathbb{C}$ by $\\bar{\\mathbb{Q}}_p$\n cf. (\\ref{local-rec1e}). The field of definition $E_{\\nu}$ of $\\mu_{\\gamma}$\n is the image of\n $\\bar{\\varphi}_0: K_{\\bar{\\mathfrak{q}}_0} \\rightarrow \\bar{\\mathbb{Q}}_p$.\n There is a canonical isomorphism of $K_{\\bar{\\mathfrak{q}}_0}$-algebras \n \\begin{displaymath}\n K_{\\bar{\\mathfrak{q}}_0} \\otimes_{\\mathbb{Q}_p} E_{\\nu} \\cong\n K_{\\bar{\\mathfrak{q}}_0} \\times C_{\\bar{\\mathfrak{q}}_0}, \n \\end{displaymath} \n where the first factor corresponds to the compositum $K_{\\bar{\\mathfrak{q}}_0}$ of\n the fields $K_{\\bar{\\mathfrak{q}}_0}$ and $E_{\\nu}$, given by $\\ensuremath{\\mathrm{id}}\\xspace_{K_{_{\\bar{\\mathfrak{q}}_0}}}$\n and $\\bar{\\varphi}_0^{-1}$.\n \nWe consider the homomorphism \n\\begin{equation}\\label{Gtwist7e}\n E_{\\nu}^{\\times} \\rightarrow \n (K_{_{\\bar{\\mathfrak{q}}_0}} \\otimes_{\\mathbb{Q}_p} E_{\\nu})^{\\times}\\cong K_{\\bar{\\mathfrak{q}}_0}^{\\times} \\times\n C_{\\bar{\\mathfrak{q}}_0}^{\\times}, \\quad \n e \\mapsto \\bar{\\varphi}_0^{-1} (e) \\times 1.\n \\end{equation}\n The one-parameter group $\\mu_{\\gamma}$ over $E_{\\nu}$ is the homomorphism\n \\begin{displaymath}\n E_{\\nu}^{\\times} \\rightarrow (K \\otimes_{\\mathbb{Q}} E_{\\nu})^{\\times} \\cong \n \\prod_i (K_{_{\\mathfrak{q}_i}} \\otimes_{\\mathbb{Q}_p} E_{\\nu})^{\\times} \\times\n \\prod_{i} (K_{_{\\bar{\\mathfrak{q}}_i}} \\otimes_{\\mathbb{Q}_p} E_{\\nu})^{\\times} , \n \\end{displaymath}\n which is given by (\\ref{Gtwist7e}) on the factor\n $(K_{_{\\bar{\\mathfrak{q}}_0}} \\otimes_{\\mathbb{Q}_p} E_{\\nu})^{\\times}$ and is trivial\n on all other factors. The map\n \\begin{displaymath}\n\\Nm_{E_{\\nu}\/\\mathbb{Q}_p} =\n\\Nm_{K_{_{\\bar{\\mathfrak{q}}_0}}\\otimes_{\\mathbb{Q}_p} E_{\\nu}\/K_{\\bar{\\mathfrak{q}}_0}}: \nK_{\\bar{\\mathfrak{q}}_0} \\otimes_{\\mathbb{Q}_p} E_{\\nu}\\rightarrow K_{_{\\bar{\\mathfrak{q}}_0}} \n \\end{displaymath}\n becomes in terms of (\\ref{Gtwist7e})\n \\begin{displaymath}\n (a,c) \\in K_{\\bar{\\mathfrak{q}}_0} \\times C_{\\bar{\\mathfrak{q}}_0} \\mapsto\n a \\Nm_{C_{\\bar{\\mathfrak{q}}_0}\/K_{\\bar{\\mathfrak{q}}_0}}. \n \\end{displaymath}\n We conclude that the local reciprocity law associated to $\\gamma$ \n \\begin{equation}\n r(T^{\\bullet}, \\gamma): E_{\\nu}^{\\times} \\rightarrow \\prod_i K_{\\mathfrak{q}_i}^{\\times}\n \\times \\prod_i K_{\\bar{\\mathfrak{q}}_i}^{\\times}\n \\end{equation}\n maps $e$ to the element with component\n $\\bar{\\varphi}_{\\bar{\\mathfrak{q}}_0}^{-1}(e^{-1})$ at the\n factor $K_{\\bar{\\mathfrak{q}}_0}$ and with trivial component at all other\n factors.\n\n By Corollary \\ref{zentralerTwist1c} and our remarks about the Hecke operators\n (\\ref{BZGdot5e}) we obtain from (\\ref{Gtwist5e}) a commutative diagram\n \\begin{displaymath}\n \\xymatrix{\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet},E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu} \n \\ar[d]_{(\\varphi^{-1}_{\\mathfrak{p}_0}(e^{-1})_{\\mid \\xi_{\\mathfrak{p}_0}} (p^{f_{\\nu} \\ord e})_{\\mid \\xi} \\times \\sigma_c} \\ar[r]\n & {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h \\delta^{-1} \\gamma)_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu} \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\sigma_c}\\\\\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{ab}_{\\nu} \\ar[r] &\n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h\\delta^{-1} \\gamma)_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}} \\Spec E^{ab}_{\\nu}. \n }\n \\end{displaymath}\n The $\\xi_{\\mathfrak{p}_0}$-part of the datum ($d^t$) in\n Definition \\ref{BZsApkt4d} is a function with values in\n $F_{\\mathfrak{p}_0}^{\\times}\/O_{F_{\\mathfrak{p}_0}}^{^{\\times}}$. \n Therefore $(\\varphi^{-1}_{\\mathfrak{p}_0}(e^{-1})_{\\mid \\xi_{\\mathfrak{p}_0}}$ acts on\n this datum exactly like $(\\pi_{\\mathfrak{p}_0}^{- \\ord e})_{\\mid \\xi_{\\mathfrak{p}_0}}$.\n This shows that for $e \\in O_{E_{\\nu}}^{\\times}$ the vertical arrow on the left \n hand side in the above diagram is equal to $\\ensuremath{\\mathrm{id}}\\xspace \\times \\sigma_c$. Therefore\n the horizontal arrow in this diagram is defined over $E_{\\nu}^{nr}$. \n The proposition follows.\n \\end{proof}\nWe use Proposition \\ref{Sh_D1p} to define a model of\n${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}}$ over $O_{E_{\\nu}}$.\nThe group \n$T^{\\bullet}(\\mathbb{A}_f)\/(\\mathbf{K}^{\\bullet}\\cap T^{\\bullet}(\\mathbb{A}_f))$ acts through a\nfinite quotient. Therefore the Hecke operator associated to $\\dot{z}$ has finite order.\nIt follows that the field $E^{nr}_{\\nu}$ in Proposition \\ref{Sh_D1p} can be\nreplaced by a finite unramified extension $L\/E_{\\nu}$. We have extended\n$\\dot{z}$ to an automorphism of the functor \n$\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ over $O_{E_{\\nu}}$. \n\\begin{definition}\\label{tildeSh_D1d} \n Let $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}, where\n $\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. We define \n $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ to be the\n $O_{E_{\\nu}}$-scheme given by the descent datum $\\dot{z} \\times \\tau_c$\n on $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}} \\Spec O_L$, where $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ is the coarse moduli scheme of $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$.\n \\end{definition}\nThe diagram of Proposition \\ref{Sh_D1p} becomes\n\\begin{equation}\\label{tildeSh_D1e} \n \\begin{aligned}\n \\xymatrix{\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \n \\ar[d]_{\\dot{z}_{\\mid \\xi} \\times \\tau_c} \\ar[r]\n & \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \n \\times_{\\Spec O_{E_{\\nu}}} \\Spec O_{E^{nr}_{\\nu}} \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\\\\ \n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \\ar[r] &\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \n \\times_{\\Spec O_{E_{\\nu}}} \\Spec O_{E^{nr}_{\\nu}} .\\\\ \n } \n \\end{aligned}\n \\end{equation}\n\n\\begin{remark} Let us drop the assumption that\n$\\mathbf{M}_{\\mathfrak{p}_0} = O^\\times_{F_{\\mathfrak{p}_0}}$. We can write the diagram \nat the end of the proof of Proposition \\ref{Sh_D1p} in the form \n \\begin{displaymath}\n \\xymatrix{\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet},E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu} \n \\ar[d]_{(\\pi_{\\mathfrak{p}_0}^{-\\ord e})_{\\mid \\xi_{\\mathfrak{p}_0}} (p^{f_{\\nu} \\ord e})_{\\mid \\xi}}^{\\times \\sigma_c} \\ar[r]\n & {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu}\n \\ar[d]_{(\\pi_{\\mathfrak{p}_0}^{-\\ord e})_{\\mid \\xi_{\\mathfrak{p}_0}} (\\varphi_{\\mathfrak{p}_0}^{-1}(e))_{\\mid \\xi_{\\mathfrak{p}_0}}}^{\\times \\sigma_c}\\\\\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{ab}_{\\nu} \\ar[r] & \n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}} \\Spec E^{ab}_{\\nu}. \n }\n \\end{displaymath}\nAs before $e$ corresponds to $\\sigma$ by local class field theory. \n We define the Galois twist \n ${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}(\\pi_{\\mathfrak{p}_0})$ of\n ${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}$ by the commutative\n diagram\n \\begin{displaymath}\n \\xymatrix{\n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}\n \\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu}\n \\ar[d]^{(\\pi_{\\mathfrak{p}_0}^{-\\ord e})_{\\mid \\xi_{\\mathfrak{p}_0}} (\\varphi_{\\mathfrak{p}_0}^{-1}(e))_{\\mid \\xi_{\\mathfrak{p}_0}} \\times \\sigma_c} \\ar[r] & \n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}(\\pi_{\\mathfrak{p}_0})\n \\times_{\\Spec E_{\\nu}}\\Spec E^{ab}_{\\nu}\n \\ar[d]^{id \\times \\sigma_c}\\\\\n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{ab}_{\\nu} \\ar[r] & \n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}(\\pi_{\\mathfrak{p}_0})\n \\times_{\\Spec E_{\\nu}} \\Spec E^{ab}_{\\nu}. \n }\n \\end{displaymath}\n Then we obtain \n a commutative diagram \n\\begin{displaymath}\n \\xymatrix{\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet},E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu} \n \\ar[d]^{(\\pi_{\\mathfrak{p}_0}^{-\\ord e})_{\\mid \\xi_{\\mathfrak{p}_0}} (p^{f_{\\nu} \\ord e})_{\\mid \\xi} \\times \\sigma_c} \\ar[r]\n & {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}(\\pi_{\\mathfrak{p}_0})\n \\times_{\\Spec E_{\\nu}}\\Spec E^{nr}_{\\nu}\n \\ar[d]^{(\\ensuremath{\\mathrm{id}}\\xspace \\times \\sigma_c}\\\\\n \\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}, E_{\\nu}} \\times_{\\Spec E_{\\nu}}\n \\Spec E^{nr}_{\\nu} \\ar[r] & \n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}(\\pi_{\\mathfrak{p}_0})\n \\times_{\\Spec E_{\\nu}} \\Spec E^{nr}_{\\nu}. \n }\n \\end{displaymath}\nIn the same way as in Definition \\ref{BZKpPkt1e} we obtain a model\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)(\\pi_{\\mathfrak{p}_0})$ over\n$O_{E_{\\nu}}$. The diagram (\\ref{tildeSh_D1e}) continues to hold for arbitrary\n$\\mathbf{K}^{\\bullet}$ if we substitute\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)(\\pi_{\\mathfrak{p}_0})$ for\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)$. \nWe note that the last two schemes are canonically identified if\n$\\mathbf{K}^{\\bullet}$ is of the type\n$\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. \n\nOne could regard\n${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}(\\pi_{\\mathfrak{p}_0})$ as the\ntwist of ${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_D)_{E_{\\nu}}$ by the\ncharacter of $\\Gal(E_{\\nu}^{ab}\/E_{\\nu})$ associated to the Lubin-Tate group\ndefined by $\\pi_{\\mathfrak{p}_0}$. \n\n\\end{remark} \n\nOur next aim is to compare the functors ${\\mathcal A}_{\\mathbf K}$ and ${\\mathcal A}_{\\mathbf K^\\bullet}$. For this we need the following variant of a theorem of Chevalley \\cite{Che}. \n\\begin{proposition}\\label{Chevalley1p}\n Let $F$ be a totally real number field. We set $[F: \\mathbb{Q}] = d = 2^h d'$\n such that $d'$ is odd. Let $M \\geq 2$ be a natural number and let $\\ell$ be a prime number such\n that\n \\begin{displaymath}\n \\ell \\equiv 2 \\mod d', \\qquad \\ell \\equiv 3 \\mod 4.\n \\end{displaymath}\nFor a natural number $N$, let $U_{N\\ell}$ be the principal congruence subgroup of $(O_F \\otimes \\hat{\\mathbb{Z}})^{\\times}$,\n $$U_{N\\ell} =\\{u \\equiv 1 \\mod N \\ell (O_F \\otimes \\hat{\\mathbb{Z}}) \\}.\n $$\nFor each natural number $m$ there is a power $N$ of $M$ with the following\n property: for each element $f \\in F^{\\times}$ which is totally positive and such\n that $f \\in U_{N\\ell} \\cdot \\mathbb{A}_f^{\\times}$, there is a unit $g \\in O_F^{\\times}$\n such that\n \\begin{displaymath}\nf = g^m q, \\quad \\text{for some} \\; q \\in \\mathbb{Q}^{\\times}, \\; q >0. \n \\end{displaymath}\n \\end{proposition}\n\\begin{proof} Set $U=U_{N\\ell}$, where $N$ will be determined in the proof. \n We write $f = u \\alpha$ with $u \\in U$, $\\alpha \\in \\mathbb{A}_f^{\\times}$.\n We find $q$ such that $\\alpha = q \\beta$ and\n $\\beta \\in \\hat{\\mathbb{Z}}^{\\times}$. Therefore we may assume\n that $f \\in O_F^{\\times}$ and hence $q = 1$. We obtain\n \\begin{displaymath}\nf = u \\alpha, \\quad u \\in U, \\; \\alpha \\in \\hat{\\mathbb{Z}}^{\\times}.\n \\end{displaymath}\n We note that $\\Nm_{F\/\\mathbb{Q}} u \\in U$. We find\n \\begin{displaymath}\n f^d (\\Nm_{F\/\\mathbb{Q}} f)^{-1} = u^d (\\Nm_{F\/\\mathbb{Q}} u)^{-1}\n \\alpha^d (\\Nm_{F\/\\mathbb{Q}} \\alpha)^{-1} = u^d (\\Nm_{F\/\\mathbb{Q}} u)^{-1}\n \\in U. \n \\end{displaymath}\n Since $f$ is a totally positive unit $\\Nm_{F\/\\mathbb{Q}} f = 1$ and therefore\n $f^d \\in U$. By Chevalley \\cite{Che} there exists for a suitable $N$ a unit\n $g \\in O_F^{\\times}$ such that $f^d = g^{md}$. Replacing $m$ by a multiple, we may \n assume that $m$ is even and that\n \\begin{displaymath}\ng^m \\equiv 1 \\; \\mod \\ell (O_F \\otimes \\hat{\\mathbb{Z}}). \n \\end{displaymath}\n\n We consider the $d$-th root of unity \n \\begin{displaymath}\nf\/g^m = \\zeta. \n \\end{displaymath}\n Since $f \\equiv \\alpha \\mod \\ell (O_F \\otimes \\hat{\\mathbb{Z}})$\n we obtain\n \\begin{displaymath}\n\\zeta \\equiv \\alpha \\mod \\ell (O_F \\otimes \\hat{\\mathbb{Z}}). \n \\end{displaymath}\nThe right hand side is in $\\mathbb{Z}\/ \\ell \\mathbb{Z} \\subset O_F\/ \\ell O_F$.\nThis shows $\\zeta^{\\ell -1} \\equiv 1$. Here and below, this is meant\n$\\!\\!\\mod \\ell (O_F \\otimes \\hat{\\mathbb{Z}})$. On the other hand,\n we have\n \\begin{displaymath}\n\\zeta^{2^h d'} \\equiv 1.\n \\end{displaymath}\n Since $\\ell -1 \\equiv 1 \\mod d'$, we obtain that\n $\\zeta^{2^h} \\equiv 1$. If $h = 0$ we conclude from Serre's lemma that\n $\\zeta = 1$. Let $h > 0$. By our assumption $(\\ell -1)\/2$ is odd. Therefore\n the greatest common divisor of $\\ell - 1$ and $2^{h}$ is $2$. We conclude\n that $\\zeta^2 \\equiv 1$ and by Serre's \n lemma that $\\zeta^2 = 1$. We obtain\n \\begin{displaymath}\nf\/g^m = \\pm 1.\n \\end{displaymath}\n Since $m$ is even, the left hand side is totally positive by assumption.\n This gives finally $f = g^m$. \n\\end{proof}\n\nLet $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ be an open and compact subgroup.\n We set\n$\\mathbf{K} = \\mathbf{K}^{\\bullet} \\cap G(\\mathbb{A}_f)$. For an open compact subgroup\n$U \\subset (F \\otimes \\mathbb{A}_f)^{\\times}$, \nwe define\n\\begin{displaymath}\n \\mathbf{K}^{\\bullet}_U = \\{ g \\in \\mathbf{K}^{\\bullet} \\; | \\;\n \\mu(g) \\in U \\hat{\\mathbb{Z}}^{\\times} \\}. \n\\end{displaymath}\nThen \n\\begin{equation}\\label{BZ17e}\n \\mathbf{K}= \\mathbf{K}_U^{\\bullet} \\cap G(\\mathbb{A}_f). \n\\end{equation}\n\n\\begin{proposition}\\label{BZ4p}\n We fix $M$ and $\\ell$ as in Proposition \\ref{Chevalley1p}. \n Let $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ be an open compact\n subgroup. Then there exists a power $N$ of $M$ such that for the principal congruence \n subgroup $U =U_{N\\ell}\\subset (F \\otimes \\mathbb{A}_f)^{\\times}$ of Proposition\n \\ref{Chevalley1p}, the natural map of functors\n \\begin{equation}\\label{BZ8e}\n\\mathcal{A}_{\\mathbf{K}} \\rightarrow \\mathcal{A}^{\\bullet}_{\\mathbf{K}_U^{\\bullet}}\n \\end{equation}\n is a monomorphism. \n\\end{proposition}\nTo show this, it is enough to check injectivity for points with values in\n$S=\\Spec R$, where $R$ is a noetherian $E_{\\nu}${\\rm -Alg}ebra and $\\Spec R$ is\nconnected. We begin with two lemmas. Since the meaning of the class in the notation $(A, \\bar{\\lambda}, \\bar{\\eta})$ depends on whether this is an object of $\\mathcal{A}_{\\mathbf{K}}$ or of $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}}$, we use the notation $(A, \\tilde{\\lambda}, \\tilde{\\eta})$ in the latter case. \n\\begin{lemma}\\label{BZ4l}\nLet $(A, \\bar{\\lambda}, \\bar{\\eta}) \\in \\mathcal{A}_{\\mathbf{K}}(R)$\nwith image \n$(A', \\tilde{\\lambda}', \\tilde{\\eta}') \\in \\mathcal{A}^{\\bullet}_{\\mathbf{K}_U^{\\bullet}}(R)$.\nThen there is a polarization $\\lambda' \\in \\tilde{\\lambda}'$ and level structure\n$\\eta' \\in \\tilde{\\eta}'$ such that the point\n$(A, \\bar{\\lambda}, \\bar{\\eta})$ may be represented in the form\n$(A', \\bar{\\lambda}', \\bar{\\eta}')$.\n\\end{lemma}\n\\begin{proof}\n We start with arbitrary polarizations $\\lambda' \\in \\tilde{\\lambda'}$ and\n $\\lambda \\in \\bar{\\lambda}$ and arbitrary level structures $\\eta' \\in \\tilde{\\eta'}$,\n $\\eta \\in \\bar{\\eta}$. Since we have the same point in\n $\\mathcal{A}^{\\bullet}_{\\mathbf{K}_U^{\\bullet}}(R)$, there is an isogeny\n $\\alpha: A' \\rightarrow A$ such that\n $\\alpha^{*} (\\lambda) = f \\lambda'$. Since we have chosen polarizations,\n $f \\in F^{\\times}$ must be totally positive. Moreover, $\\alpha$ must respect\n $\\eta$ and $\\eta'$ up to a factor in $\\mathbf{K}_U^{\\bullet}$,\n \\begin{displaymath}\n\\alpha \\circ \\eta' \\dot{c} = \\eta, \\quad \\dot{c} \\in \\mathbf{K}_U^{\\bullet}. \n \\end{displaymath}\n We claim that $(A', \\overline{f \\lambda'}, \\overline{\\eta' \\dot{c}})$ is\n a point of $\\mathcal{A}_{\\mathbf{K}}(R)$. We have to check that the isomorphism \n \\begin{displaymath}\n\\eta_1 \\dot{c}: V \\otimes \\mathbb{A}_f \\isoarrow \\hat{V} (A') \n \\end{displaymath}\n respects the form $\\psi$ and the Riemann form $E^{f \\lambda'}$, up to a factor\n $a \\in \\mathbb{A}^{\\times}_f(1)$. But for\n $x, y \\in V \\otimes \\mathbb{A}_f$ we have\n \\begin{equation*}\n \\begin{aligned}\n E^{f \\lambda'}(\\eta' (\\dot{c} x), \\eta' (\\dot{c} y)) &=\n E^{\\alpha^{*} (\\lambda)}(\\alpha^{-1}\\circ \\eta (x), \\alpha^{-1}\\circ \\eta (y)) =\n E^{\\lambda}(\\eta (x), \\eta (y))\\\\ \n &= a \\psi(x, y) \n \\end{aligned}\n \\end{equation*}\n for some $a \\in \\mathbb{A}^{\\times}_f(1)$, because\n $(A, \\bar{\\lambda}, \\bar{\\eta})$ is a point of\n $\\mathcal{A}_{\\mathbf{K}}(R)$.\n\n It is obvious that\n \\begin{displaymath}\n \\alpha: (A', \\overline{f \\lambda'}, \\overline{\\eta' \\dot{c}})\n \\rightarrow (A, \\bar{\\lambda}, \\bar{\\eta}) \n \\end{displaymath}\n is an isomorphism and therefore both sides give the same point of\n $\\mathcal{A}_{\\mathbf{K}}(R)$. \n\\end{proof}\n\\begin{lemma}\\label{BZ42}\n Let $(A_1, \\bar{\\lambda}_1, \\bar{\\eta}_1)$ and\n $(A_2, \\bar{\\lambda}_2, \\bar{\\eta}_2)$ be two points of\n $\\mathcal{A}_{\\mathbf{K}}(R)$ whose images in\n $\\mathcal{A}^{\\bullet}_{\\mathbf{K}_U^{\\bullet}}(R)$ by (\\ref{BZ8e}) are the\n same. Then there exists a totally positive $f \\in F^{\\times}$ and an element\n $\\dot{c} \\in \\mathbf{K}_U^{\\bullet}$, such that\n \\begin{displaymath}\nf (\\dot{c}' \\dot{c}) \\in \\mathbb{A}_f^{\\times}, \n \\end{displaymath}\n and such that $(A_2, \\bar{\\lambda}_2, \\bar{\\eta}_2)$ is isomorphic\n to $(A_1, f \\bar{\\lambda}_1, \\bar{\\eta}_1 \\dot{c})$. \n \\end{lemma}\n\\begin{proof}\n We choose arbitrary polarizations $\\lambda_1 \\in \\bar{\\lambda}_1$ and\n $\\lambda_2 \\in \\bar{\\lambda}_2$ and arbitrary level structures $\\eta_1 \\in \\bar{\\eta}_1$ and\n $\\eta_2 \\in \\bar{\\eta}_2$. We remark that for each\n $\\dot{c} \\in \\mathbf{K}_U^{\\bullet}$ the class $\\eta_1 \\dot{c} \\mathbf{K}$\n is invariant under the action of $\\pi_1(\\bar{s}, S)$ because\n $\\mathbf{K} \\subset \\mathbf{K}_U^{\\bullet}$ is a normal subgroup. \n\n By the Lemma \\ref{BZ4l} we may assume that\n $(A_2, \\bar{\\lambda}_2, \\bar{\\eta}_2) = (A_1, \\overline{f \\lambda_1}, \\overline{\\eta_1 \\dot{c}})$.\n We have factors $a_1, a_2 \\in \\mathbb{A}_f(1)$ such that for all\n $x, y \\in V \\otimes \\mathbb{A}_f$ \n \\begin{equation}\\label{BZ18e} \n \\begin{aligned}\n a_2 \\psi(x,y) &= E^{f \\lambda_1}(\\eta_1(\\dot{c} x), \\eta_1(\\dot{c} y)) =\n E^{\\lambda_1}(\\eta_1(\\dot{c} x), \\eta_1(\\dot{c} fy))= \\\\\n &=a_1 \\psi(\\dot{c} x, \\dot{c} fy) = a_1 \\psi (x, \\dot{c}' \\dot{c} fy). \n \\end{aligned}\n \\end{equation}\n The assertion follows. \n \\end{proof}\n\\begin{proof}[Proof of Proposition \\ref{BZ4p}]\n We may assume that $S=\\Spec R$ is connected. We consider a point\n $(A, \\bar{\\lambda}, \\bar{\\eta}) \\in \\mathcal{A}_{\\mathbf{K}}(R)$. Any\n other point with the same image by (\\ref{BZ8e}) is of the form \n \\begin{equation}\\label{BZ9e}\n (A, f \\bar{\\lambda}, \\bar{\\eta} \\dot{c}), \\quad \\text{such that} \\;\n f \\in F^{\\times}, \\; \\dot{c} \\in \\mathbf{K}^{\\bullet}_U , \\;\n f \\dot{c}' \\dot{c} = a \\in \\mathbb{A}^{\\times}_f. \n \\end{equation}\n Moreover $f$ is totally positive. Replacing $f$ by $f q$ for some\n $q \\in \\mathbb{Q}^{\\times}$, $q >0$, does not change the point (\\ref{BZ9e}).\n Therefore we may assume that $a \\in \\hat{\\mathbb{Z}}^{\\times}$ and that\n $f$ is a unit.\n\n By Proposition \\ref{Chevalley1p}, for each natural number $m$ we find\n $U=U_{N\\ell}$ in such a way that $f = g_m^{2m}$ for some $g_m \\in O_F^{\\times}$.\n Since $\\mathbf{K}^{\\bullet} \\cap (F \\otimes \\mathbb{A}_f)^{\\times}$ is open\n in $(F \\otimes \\mathbb{A}_f)^{\\times}$ \n we may choose $m$ such that $g_m^m \\in \\mathbf{K}^{\\bullet}$.\n We set $g = g_m^m$. Since $f = g^2$, the multiplication isomorphism\n by $g$ is an isomorphism \n \\begin{displaymath}\n g: (A, f \\bar{\\lambda}, \\bar{\\eta} \\dot{c}) \\isoarrow\n (A, \\bar{\\lambda}, \\bar{\\eta} g \\dot{c}).\n \\end{displaymath}\n We obtain\n \\begin{displaymath}\n (g \\dot{c})' \\cdot (g \\dot{c}) = g^2 \\dot{c}' \\dot{c} =\n f \\dot{c}' \\dot{c} = a \\in \\mathbb{A}_f^{\\times}, \n \\end{displaymath}\n and therefore\n $g \\dot{c} \\in G(\\mathbb{A}_f) \\cap \\mathbf{K}^{\\bullet} = \\mathbf{K}$.\n We see that $(A, f \\bar{\\lambda}, \\bar{\\eta} \\dot{c})$ and\n $(A, \\bar{\\lambda}, \\bar{\\eta})$ define the same point of\n $\\mathcal{A}_{\\mathbf{K}}$. \n\\end{proof}\n\n\nWe know that for $\\mathbf{K} \\subset G(\\mathbb{A}_f)$ small enough the\nfunctor $\\mathcal{A}_{\\mathbf{K}, E_{\\nu}}$ is representable by the scheme\n${\\mathrm{Sh}}(G,h)_{\\mathbf{K}, E_{\\nu}}$. In general the latter is a coarse moduli scheme.\n\\begin{proposition}\\label{BZ7p} \n Let $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ be an open and\n compact subgroup. We set\n $\\mathbf{K} = G(\\mathbb{A}_f) \\cap \\mathbf{K}^{\\bullet}$. We\n assume that there is an $O_K$-lattice $\\Gamma \\subset V$ and \n an integer $m \\geq 3$ such that for each $u \\in \\mathbf{K}^{\\bullet}$ we have\n $u \\Gamma\\otimes \\hat{\\mathbb{Z}}\\subset \\Gamma \\otimes \\hat{\\mathbb{Z}}$\n and such that $u$ acts trivially on\n $\\Gamma\/m \\Gamma$, so that $\\mathcal{A}_{\\mathbf{K}, E_{\\nu}}$ is representable (this is the analogue of condition \\eqref{BZneat1e} for $\\mathbf{K}^{\\bullet}$ instaed of $\\mathbf{K}$).\n\n Let $U$ be as in Proposition \\ref{BZ4p}. Then the \\'etale sheafification of\n the presheaf $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}_{U}, E_{\\nu}}$ on the big\n \\'etale site is represented by\n ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_{U}, E_{\\nu}}$. \n \\end{proposition}\n\\begin{proof}\n We begin with some general remarks on \\cite[Prop. 1.15]{D-TS} in our case.\n The morphism of schemes (not of finite type) \n ${\\mathrm{Sh}}(G, h) \\rightarrow {\\mathrm{Sh}} (G^{\\bullet}, h)$ is an open and closed immersion.\n More precisely, for any open compact subgroup\n $\\mathbf{K}_1 \\subset G(\\mathbb{A}_f)$, there is an open compact subgroup \n $\\mathbf{K}^{\\bullet}_1 \\subset G^{\\bullet}(\\mathbb{A}_f)$ such that\n $\\mathbf{K}^{\\bullet}_1 \\cap G(\\mathbb{A}_f) = \\mathbf{K}_1$ and such that\n ${\\mathrm{Sh}}(G, h)_{\\mathbf{K}_1} \\subset {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_1}$ is\n an open and closed immersion. Indeed, it is a closed immersion by\n \\cite[Prop. 1.15]{D-TS} and\n it is open because the local rings of these varieties are normal and have\n both the same constant dimension. If $Z$ is a connected component of\n ${\\mathrm{Sh}}(G, h)_{\\mathbb{C}}$, then its \n image $Z^{\\bullet}$ in $ {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbb{C}}$ is a connected component.\n For arbitrary open compact subgroups $\\mathbf{K} \\subset G(\\mathbb{A}_f)$\n resp. $\\mathbf{K}^{\\bullet} \\subset G(\\mathbb{A}_f)$ the image\n $Z_{\\mathbf{K}}$ of $Z$ in $ {\\mathrm{Sh}}(G, h)_{\\mathbf{K}, \\mathbb{C}}$, resp. the image\n $Z^{\\bullet}_{\\mathbf{K}^{\\bullet}}$ of $Z^{\\bullet}$ in \n ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}, \\mathbb{C}}$, is a connected component.\n For $\\mathbf{K}_1$ and $\\mathbf{K}_1^{\\bullet}$ as above, the map\n$Z_{\\mathbf{K}_1} \\rightarrow Z^{\\bullet}_{\\mathbf{K}^{\\bullet}_1}$ is an isomorphism. For\n$g \\in G^{\\bullet}(\\mathbb{A}_f)$, the multiplication by $g$ induces a map\n \\begin{displaymath}\n g: {\\mathrm{Sh}}(G^{\\bullet}, h)_{g\\mathbf{K}^{\\bullet}_1g^{-1}} \\rightarrow\n {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_1}. \n \\end{displaymath}\nNow $G^{\\bullet}(\\mathbb{A}_f)$ acts transitively on the connected components of \n ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbb{C}}$, cf. \\cite[Prop. 2.2]{D-TS}. Therefore the sets\n $g Z^{\\bullet}_{g\\mathbf{K}^{\\bullet}_1 g^{-1}, \\mathbb{C}}$ cover \n ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_1, \\mathbb{C}}$, as $g$ runs through all elements\n of $G^{\\bullet}(\\mathbb{A}_f)$. We note that\n$g\\mathbf{K}^{\\bullet}_1g^{-1} \\cap G(\\mathbb{A}_f) = g\\mathbf{K}_1g^{-1}$ because\n$G(\\mathbb{A}_f) \\subset G^{\\bullet}(\\mathbb{A}_f)$ is a normal subgroup. \nWe conclude that the images of the following composite maps cover ${\\mathrm{Sh}}(G^{\\bullet},h)_{\\mathbf{K}^{\\bullet}_1}$, as $g$ varies in $G^{\\bullet}(\\mathbb{A}_f)$ ,\n\\begin{equation}\\label{BZ30e}\n \\varkappa_g: {\\mathrm{Sh}}(G,h)_{g\\mathbf{K}_1g^{-1}} \\rightarrow\n {\\mathrm{Sh}}(G^{\\bullet}, h)_{g\\mathbf{K}^{\\bullet}_1g^{-1}} \\overset{g}{\\rightarrow} \n {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_1}.\n \\end{equation}\n\nNow we turn to the proof of the proposition. We show that the\nmorphism\n \\begin{equation}\\label{BZ10e} \n {\\mathrm{Sh}}(G,h)_{\\mathbf{K}} \\rightarrow \n {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_{U}} \n \\end{equation}\n is an open and closed immersion. Indeed by Proposition \\ref{BZ4p} we know\n that (\\ref{BZ10e}) induces an injection on the $\\mathbb{C}$-valued points.\n By the remarks above there exists a open and compact subgroup\n $\\mathbf{K}_1^{\\bullet} \\subset \\mathbf{K}^{\\bullet}_{U}$ such that\n $\\mathbf{K}_1^{\\bullet} \\cap G(\\mathbb{A}_f) = \\mathbf{K}$ and such that\n $ {\\mathrm{Sh}}(G,h)_{\\mathbf{K}} \\rightarrow {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}_1^{\\bullet}}$\n is an open and closed immersion. We consider the commutative diagram\n \\begin{displaymath}\n \\xymatrix{\n {\\mathrm{Sh}}(G,h)_{\\mathbf{K}} \\ar[r] \\ar[rd] & {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_1}\n \\ar[d]\\\\\n & {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_{U}} .\n } \n \\end{displaymath}\n By our assumption on $\\mathbf{K}^{\\bullet}$, the vertical arrow is a finite\n \\'etale morphism. Hence the same is true for the oblique arrow. Since by Proposition \\ref{BZ4p} its \n geometric fibres contain at most one element, the claim follows.\n \n Let $Y$ be the \\'etale sheafification of\n $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}}$. We consider the preimage\n $Y^{o}$ of ${\\mathrm{Sh}}(G,h)_{\\mathbf{K}, E_{\\nu}}$ by the natural morphism\n \\begin{displaymath}\n Y \\rightarrow\n {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_{U}, E_{\\nu}}. \n \\end{displaymath}\n Then $Y^{o} \\subset Y$ is an open and closed subfunctor. We consider the\n natural morphism\n \\begin{displaymath}\n \\mathcal{A}_{\\mathbf{K}, E_{\\nu}} \\rightarrow\n \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}} \\rightarrow Y \\rightarrow\n {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_{U}, E_{\\nu}}. \n \\end{displaymath}\n Since $Y^{o}$ is a fibre product we obtain a factorization\n \\begin{equation}\\label{et-sheaf1e} \n \\mathcal{A}_{\\mathbf{K}, E_{\\nu}} \\rightarrow Y^{o} \\rightarrow\n {\\mathrm{Sh}}(G,h)_{\\mathbf{K}, E_{\\nu}}. \n \\end{equation} \n We claim that both arrows are isomorphisms. Since their composite is an\n isomorphism the first arrow is a monomorphism. Therefore it suffices to\n show that the first arrow is a surjection of \\'etale sheaves. \n Since both functors $\\mathcal{A}_{\\mathbf{K}, E_{\\nu}}$ and \n $\\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}}$ commute with inductive limits,\n the stalks at a geometric point $\\xi$ of $\\Spec R$ of the sheafifications are\n the points of these functors with values in the strict henselization\n $R^{sh}_{\\xi}$. Therefore it is enough to show that\n \\begin{equation}\\label{BZ11e}\n\\mathcal{A}_{\\mathbf{K}, E_{\\nu}}(R) \\rightarrow Y^o(R) \n \\end{equation}\n is surjective for a\n strictly henselian local ring $R$. For an algebraically closed field\n $R$ both sides have the same coarse moduli space. Therefore the map\n is bijective in this case. In general the residue field $\\kappa_R$ of $R$ is\n algebraically closed, since we are in characteristic $0$. \n We consider a point\n $(A, \\tilde{\\lambda}, \\tilde{\\eta}) \\in \\mathcal{A}^{\\bullet}_{\\mathbf{K}_U^{\\bullet}}(R) = Y(R)$ \n which is in $Y^{o}(R)$. Over $\\kappa_R$ this point is in the image of\n (\\ref{BZ11e}). By Lemma \\ref{BZ4l}, the preimage by (\\ref{BZ11e}) has the form\n $(A_{\\kappa_R}, \\bar{\\lambda}, \\bar{\\eta})$ for some \n $\\lambda \\in \\tilde{\\lambda}$ and $\\eta \\in \\tilde{\\eta}$. This is justified\n because the reduction to $\\kappa_R$ defines a bijection between the class\n $\\tilde{\\lambda}$ on $A$ and its reduction on $A_{\\kappa_R}$. The same applies\n to $\\tilde{\\eta}$. We must verify that\n $(A, \\lambda, \\eta)$ defines a point of $\\mathcal{A}_{\\mathbf{K}, E_{\\nu}}(R)$. \n Since there is no difference of a rigidification $\\eta$ over $\\kappa_R$ \n or over $R$, this is already decided over $\\kappa_R$. This proves that\n (\\ref{BZ11e}) is bijective. Consequently the arrows of (\\ref{et-sheaf1e})\n are isomorphism and therefore the functor $Y^o$ is representable. \n\n\n Now we deduce the representability of $Y$. Let\n $g \\in G^{\\bullet}(\\mathbb{A}_f)$. We already noted\nthat $g\\mathbf{K}^{\\bullet}g^{-1} \\cap G(\\mathbb{A}_f) = g\\mathbf{K}g^{-1}$. The\n multiplication by $g$ induces an isomorphism\n \\begin{equation}\\label{BZ12e}\n \\mathcal{A}^{\\bullet}_{g\\mathbf{K}^{\\bullet}_U g^{-1}, E_{\\nu}}\n \\overset{\\sim}{\\longrightarrow}\n \\mathcal{A}^{\\bullet}_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}} \\rightarrow Y . \n \\end{equation}\n We have shown that ${\\mathrm{Sh}}(G,h)_{g\\mathbf{K}g^{-1}, E_{\\nu}}$ is an open and closed\n subfunctor of the sheafification of the left hand side of (\\ref{BZ12e}).\n (We note that the same $U=U_{N\\ell}$ suffices for each $g\\in G^{\\bullet}(\\mathbb{A}_f)$.) \n Taking the composite with (\\ref{BZ12e}), we obtain an open and closed\n immersion\n \\begin{equation}\\label{BZ13e}\n{\\mathrm{Sh}}(G,h)_{g\\mathbf{K}g^{-1}, E_{\\nu}} \\rightarrow Y.\n \\end{equation}\n Its image is equal to the pullback of\n ${\\mathrm{Sh}}(G,h)_{g\\mathbf{K}g^{-1}, E_{\\nu}} \\overset{g}{\\rightarrow} {\\mathrm{Sh}}(G^{\\bullet},h)_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}}$\n by the natural morphism\n $Y \\rightarrow {\\mathrm{Sh}}(G^{\\bullet},h)_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}}$.\n Therefore (\\ref{BZ13e}) gives, for varying $g \\in G^{\\bullet}(\\mathbb{A}_f)$, an open\n covering of $Y$ by representable subfunctors. \n\\end{proof} \n\nFor later use we formulate a variant of the last argument.\n\n\\begin{lemma}\\label{BZCover1l} \n Let $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ be an open compact\n subgroup and let $\\mathbf{K} = G(\\mathbb{A}_f) \\cap \\mathbf{K}^{\\bullet}$. \n Assume that $U \\subset (F \\otimes \\mathbb{A}_f)^{\\times}$ is a principal congruence subgroup as constructed in the\n proof of Proposition \\ref{BZ4p}. Then for all $g \\in G^{\\bullet}(\\mathbb{A}_f)$ the canonical map\n \\begin{equation}\\label{BZ28e}\n{\\mathrm{Sh}}(G, h)_{g\\mathbf{K}g^{-1}} \\rightarrow {\\mathrm{Sh}}(G^{\\bullet}, h)_{g\\mathbf{K}^{\\bullet}_U g^{-1}} \n \\end{equation}\n is an open and closed immersion. The composite of this map with\n$g\\colon{\\mathrm{Sh}}(G^{\\bullet},h)_{g\\mathbf{K}^{\\bullet}_U g^{-1}}\\rightarrow{\\mathrm{Sh}}(G^{\\bullet},h)_{\\mathbf{K}^{\\bullet}_U}$\n gives an open and closed immersion,\n \\begin{displaymath}\n \\varkappa_g: {\\mathrm{Sh}}(G, h)_{g\\mathbf{K}g^{-1}} \\rightarrow\n {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_U} .\n \\end{displaymath}\nThe maps $\\varkappa_g$ for varying $g \\in G^{\\bullet}(\\mathbb{A}^p_f)$ are an open\ncovering of ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_U}$. \n\nIf the group $\\mathbf{K}^{\\bullet}$ satisfies the assumptions of Proposition \\ref{BZ7p}, then \nthe set of maps \n \\begin{displaymath}\n \\{\\varkappa_g: {\\mathrm{Sh}}(G, h)_{g\\mathbf{K}g^{-1}} \\rightarrow\n {\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}} \\}_{g \\in G^{\\bullet}(\\mathbb{A}_f^p)} \n \\end{displaymath}\n is an \\'etale covering by finite \\'etale maps. \n\\end{lemma}\n\\begin{proof}\n Only the last assertion remains to be proved. \n Let $Z$ be a connected component of ${\\mathrm{Sh}}(G,h)_{\\mathbb{C}}$ and let \n $Z^{\\bullet} \\in {\\mathrm{Sh}}(G^{\\bullet}, h)$ be its image as in the proof of Proposition\n \\ref{BZ7p}. \n As in that proof, it is enough to show that the sets\n $g Z^{\\bullet}_{g \\mathbf{K}^{\\bullet}_U g^{-1}}$ cover ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_U}$, as \n $g$ runs through all elements of $G^\\bullet(\\mathbb{A}_f^p)$. \n\n We consider \n $\\tilde{G}^{\\bullet} = \\{b \\in B^{{\\rm opp}} \\; | \\; b'b \\in F^{\\times} \\}$ as algebraic\n group over $F$. Then $\\Res_{F\/\\mathbb{Q}} \\tilde{G}^{\\bullet} = G^{\\bullet}$, cf.\n (\\ref{Gpunkt2e}). We consider the homomorphisms\n \\begin{equation}\\label{Gpunkt3e} \n \\begin{array}{rcc}\n \\mu: \\tilde{G}^{\\bullet} & \\rightarrow & F^{\\times},\\\\\n b\\; & \\mapsto & b'b\n \\end{array}\n \\qquad \n \\begin{array}{rcc}\n \\det: \\tilde{G}^{\\bullet} & \\rightarrow & K^{\\times}\\\\\n b\\; & \\mapsto & \\Nm^o_{B\/K}. \n \\end{array}\n \\end{equation}\n Let $\\tilde{T}^{\\bullet}$ be the algebraic torus over $F$ given by\n \\begin{displaymath}\n \\tilde{T}^{\\bullet}(F) = \\{(f, k) \\in F^{\\times} \\times K^{\\times} \\; | \\;\n f^2 = k \\bar{k} \\}. \n \\end{displaymath}\n By (\\ref{Gpunkt3e}) we obtain a homomorphism\n $\\nu:\\tilde{G}^{\\bullet}\\rightarrow\\tilde{T}^{\\bullet}$, $b \\mapsto (\\mu (b),\\det b)$.\n Let $\\tilde{H}^{\\bullet}$ be the kernel of this map. One can check that\n$\\tilde{H}^{\\bullet} \\times_{\\Spec F}\\Spec\\mathbb{C}\\cong\\mathrm{SL}_2(\\mathbb{C})$. \n Therefore we obtain an exact sequence\n \\begin{displaymath}\n 0 \\rightarrow \\tilde{G}^{\\bullet}_{\\rm der} \\rightarrow \\tilde{G}^{\\bullet}\n \\overset{\\nu}{\\longrightarrow} \\tilde{T}^{\\bullet} \\rightarrow 0 ,\n \\end{displaymath}\n where the derived group is simply connected. By \\cite[Thm. 2.4]{D-TS}, we obtain a\n bijection\n \\begin{equation}\\label{BZ29e}\n \\pi_0({\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_U} \\overset{\\sim}{\\longrightarrow}\n \\nu(\\mathbf{K}^{\\bullet}_{\\infty} \\times \\mathbf{K}^{\\bullet}_U)\\backslash\n \\tilde{T}^{\\bullet}(\\mathbb{A}_{F})\/\\tilde{T}^{\\bullet}(F). \n \\end{equation}\n The right hand side may also be written as \n $\\nu(\\mathbf{K}^{\\bullet}_{\\infty} \\times \\mathbf{K}^{\\bullet}_U)\\backslash\n T^{\\bullet}(\\mathbb{A})\/T^{\\bullet}(\\mathbb{Q})$. \n \nBecause the cyclic extension $K\/F$ splits the torus $\\tilde{T}^{\\bullet}$,\n weak approximation holds for $\\tilde{T}^{\\bullet}$, cf. \\cite[Thm. 6.36]{V}.\nIn particular $\\tilde{T}^{\\bullet}(F)$ is dense in\n $\\tilde{T}^{\\bullet}(F \\otimes_{\\mathbb{Q}} \\ensuremath{\\mathbb {R}}\\xspace) \\tilde{T}^{\\bullet}(F \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p)$. \n This implies that\n \\begin{displaymath}\n\\tilde{T}^{\\bullet}(F) \\nu(\\mathbf{K}^{\\bullet}_{\\infty} \\times \\mathbf{K}^{\\bullet}_U)\n \\tilde{T}^{\\bullet}(\\mathbb{A}^p_{F,f}) = \\tilde{T}^{\\bullet}(\\mathbb{A}_{F}).\n \\end{displaymath}\n Hence\n $\\tilde{T}^{\\bullet}(\\mathbb{A}^p_{F,f}) = T^{\\bullet}(\\mathbb{A}^p_f)$\n acts transitively on the right hand side of (\\ref{BZ29e}). Since\n $G^{\\bullet}(\\mathbb{A}^p_f) \\rightarrow T^{\\bullet}(\\mathbb{A}^p_f)$ is surjective,\n the sets $g Z^{\\bullet}_{g \\mathbf{K}^{\\bullet}_U g^{-1}}$ for\n $g \\in G^{\\bullet}(\\mathbb{A}^p_f)$ cover ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_U}$.\n Therefore the last assertion of the proposition follows as in the proof of\n Proposition \\ref{BZ7p}. \n\\end{proof}\n\nLet $\\mathbf{K}^{\\bullet}_p \\subset G^{\\bullet}(\\mathbb{Q}_p)$ be the subgroup\nassociated to a choice of $\\Lambda_p$, $\\mathbf{M}^{\\bullet}$ and\n$\\mathbf{K}_{\\mathfrak{q}_i}$, for $i = 1, \\ldots, s$, cf. (\\ref{BZKpPkt1e}). \n We set\n$\\mathbf{M} = \\mathbb{Z}_p^{\\times} \\cap \\mathbf{M}^{\\bullet}$. We denote by\n$\\mathbf{K}_p \\subset G(\\mathbb{Q}_p)$ the subgroup associated to \nthe choice of $\\Lambda_p$, $\\mathbf{M}$ and $\\mathbf{K}_{\\mathfrak{q}_i}$,\ncf. (\\ref{BZKp1e}). We see easily that\n\\begin{equation}\\label{BZ34e}\n\\mathbf{K}_p = \\mathbf{K}^{\\bullet}_p \\cap G(\\mathbb{Q}_p). \n\\end{equation}\nUnder these hypotheses, we have an integral version of Proposition \\ref{BZ4p}. It concerns $ \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}}$ instead of $\\ensuremath{\\mathcal {A}}\\xspace_{\\mathbf K}$ and $ \\widetilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}}$ instead of $\\ensuremath{\\mathcal {A}}\\xspace^\\bullet_{\\mathbf K}$. \n\\begin{proposition}\\label{BZ6p}\n We fix $M$ and $\\ell$ as in Proposition \\ref{Chevalley1p}, but we assume that\n both are prime to $p$. Let $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}, where\n $\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. Let\n $\\mathbf{K} = \\mathbf{K}^{\\bullet} \\cap G^{\\bullet}(\\mathbb{A}_f)$. \n \n Then there exists a power $N$ of $M$ such that for the open compact\n subgroup $U \\subset (F \\otimes \\mathbb{A}_f)^{\\times}$ of Proposition\n \\ref{Chevalley1p}, the natural map\n \\begin{equation}\\label{BZ15e}\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}} \\rightarrow\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}_U^{\\bullet}}\n \\end{equation}\n is a monomorphism of functors. \n\\end{proposition}\nBy assumption we have \n$\\mathbf{K}^{\\bullet} = \\mathbf{K}^{\\bullet}_p \\mathbf{K}^{\\bullet,p}$ and\n$\\mathbf{K}_U^{\\bullet} = \\mathbf{K}^{\\bullet}_p \\mathbf{K}^{\\bullet, p}_U$. The group\n$\\mathbf{K} = \\mathbf{K}^{\\bullet} \\cap G(\\mathbb{A}_f) = \\mathbf{K}^{\\bullet}_U \\cap G(\\mathbb{A}_f)$\nhas a similiar decomposition and therefore the functor\n$\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ makes sense. Similarly to the proof of Proposition \\ref{BZ4p}, we need two lemmas which are analogous to Lemmas \\ref{BZ4l} and \\ref{BZ42}. \n\\begin{lemma}\\label{BZ7l}\nLet $(A, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j, \\xi_{ p})$ be a point\nof $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}(R)$, with image \n$(A', \\tilde{\\lambda}', \\tilde{\\eta'}^p, (\\bar{\\eta'}_{\\mathfrak{q}_j})_j, (\\xi'_{ \\mathfrak{p}_i})_i)$\n in $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}_U^{\\bullet}}(R)$. Then there is a polarization $\\lambda' \\in \\tilde{\\lambda}'$ and a level structure \n$\\eta'^p \\in \\tilde{\\eta}'^p$ and an element\n$\\xi'_{p}(\\lambda') \\in \\mathbb{Z}_p^{\\times}$ such that for $i = 0, \\ldots, s$ \n\\begin{displaymath}\n \\xi'_{p}(\\lambda') \\equiv \\xi'_{ \\mathfrak{p}_i} \\; \n \\mod \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} \n \\end{displaymath}\nand such that the point\n$(A, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j, \\xi_{ p})$\nis isomorphic to\n$(A', \\bar{\\lambda}', \\bar{\\eta'}^p, (\\bar{\\eta'}_{\\mathfrak{q}_j})_j, \\xi'_{p}(\\lambda'))$.\nThe function $\\xi'_{p}$ on $\\bar{\\lambda}'$ is given by\n$\\xi'_{p}(u \\lambda') = u \\xi'_{p}(\\lambda')$ for $u \\in U_p(\\mathbb{Q})$. \n\\end{lemma}\n\\begin{proof}\n The proof is similar to that of Lemma \\ref{BZ4l}. We may assume that\n $S = \\Spec R$ is connected. Then we can argue over a geometric point $\\bar{s}$ of\n $S$, as explained after Definition \\ref{BZAK1d}. We choose arbitrarily polarizations \n $\\lambda' \\in \\tilde{\\lambda}'$ and $\\lambda \\in \\bar{\\lambda}$, and prime-to-$p$ level structures\n $\\eta'^{p} \\in \\tilde{\\eta'}^p$ and $\\eta^p \\in \\bar{\\eta}^p$, and $p$-level structures \n $\\eta'_{ \\mathfrak{q}_j} \\in \\bar{\\eta'}_{ \\mathfrak{q}_j}$ and \n $\\eta_{ \\mathfrak{q}_j} \\in \\bar{\\eta}_{ \\mathfrak{q}_j}$. By assumption there exists an isogeny $\\alpha: A' \\rightarrow A$ of order\n prime to $p$ such that\n \\begin{equation*}\n \\begin{aligned} \n \\alpha^{*} (\\lambda) = f \\lambda', \\quad \\alpha \\circ \\eta^p \\dot{c}^p = \\eta'^p, \\quad\n \\alpha \\circ \\eta'_{ \\mathfrak{q}_j} c_{\\mathfrak{q}_j} = \\eta_{ \\mathfrak{q}_j}, \\quad\n \\xi'_{\\mathfrak{p}}(f \\lambda') \\varepsilon_{\\mathfrak{p}} =\n \\xi_{p}(\\lambda), \n \\end{aligned}\n \\end{equation*}\n where $f \\in U_p(F)$ is totally positive,\n $\\dot{c}^p \\in G^{\\bullet}(\\mathbb{A}_f^p)$ and\n $c_{\\mathfrak{q}_j} \\in \\mathbf{K}_{\\mathfrak{q}_j}$ and\n $\\varepsilon_{\\mathfrak{p}} \\in \\mathbf{M}_{\\mathfrak{p}}^{\\bullet}$.\n From this the assertion follows easily. \n \\end{proof}\n\n\n\\begin{lemma}\\label{Leminj}\n Let $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ and $U$ as in\n Proposition \\ref{BZ6p}. Let\n \\begin{displaymath}\n (A_1, \\bar{\\lambda}_1, \\bar{\\eta}_1^{p},(\\bar{\\eta}_{1,\\mathfrak{q}_j})_j,\\xi_{1,p}),\n \\; \n (A_2, \\bar{\\lambda}_2, \\bar{\\eta}_2^{p}, (\\bar{\\eta}_{2,\\mathfrak{q}_j})_j,\\xi_{2,p})\n \\end{displaymath}\n be two points of $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}(R)$ which have the same\n image in $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(R)$. \nThen there exists a totally positive $f \\in O_{F}^{\\times}$, an element\n $\\theta \\in (O_F \\otimes \\hat{\\mathbb{Z}})^{\\times}$ such that $f \\theta = a \\in \\hat{\\mathbb{Z}}^{\\times}$ and \n $\\theta_p \\in \\prod_{i=0}^s \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i}$, and an element $\\dot{c} \\in \\mathbf{K}_U^{\\bullet,p}$ with\n $\\theta^p = \\dot{c}' \\dot{c}$, such that the point\n$(A_2, \\bar{\\lambda}_2, \\bar{\\eta}_2^{p}, (\\bar{\\eta}_{2,\\mathfrak{q}_j})_j,\\xi_{2,p})$ \n is isomorphic to\n \\begin{displaymath}\n (A_1, \\bar{\\lambda}_1 f, \\bar{\\eta}_1^{p} \\dot{c}, \n (\\bar{\\eta}_{1,\\mathfrak{q}_j})_j,\\xi'_{1,p}). \n \\end{displaymath}\n Here the function $\\xi'_{1,p}$ on $\\bar{\\lambda}_1 f$ is defined by\n \\begin{displaymath}\n \\xi'_{1,p}(\\lambda_1 f) = a \\xi_{1,p}(\\lambda_1). \n \\end{displaymath}\n\\end{lemma}\n\\begin{proof}\n We fix a polarization $\\lambda_1 \\in \\bar{\\lambda}_1$ and\n $\\eta_1 \\in \\bar{\\eta}_1$. By Lemma \\ref{BZ7l}, the point\n$(A_2, \\bar{\\lambda}_2, \\bar{\\eta}_2^{p}, (\\bar{\\eta}_{2,\\mathfrak{q}_j})_j,\\xi_{2,p})$\n is isomorphic to a point of the form\n \\begin{displaymath}\n (A_1, \\bar{\\lambda}_1 f, \\bar{\\eta_1^{p} \\dot{c}}, \n (\\bar{\\eta}_{1,\\mathfrak{q}_j})_j, \\xi'_{1,p}). \n \\end{displaymath}\n The value $\\xi'_{1,p}(f\\lambda_1) \\in \\mathbb{Z}_p$ satisfies the following\n congruence in $O^{\\times}_{F_{\\mathfrak{p}_i}}$ for each $i = 0, \\ldots s$, \n \\begin{displaymath}\n \\xi'_{1,p}(f \\lambda_1) \\equiv f \\xi_{1,p}(\\lambda_1) \n \\mod \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} . \n \\end{displaymath}\n This implies that there is an element\n $\\theta_p \\in \\prod_{i=1}^s \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i}$ such that\n $f \\theta_p = a_p \\in \\mathbb{Z}_p^{\\times}$. Then we obtain\n \\begin{displaymath}\n \\xi'_{1,p}(f \\lambda_1) \\equiv a_p \\xi_{1,p}(\\lambda_1) \n \\mod \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} . \n \\end{displaymath}\n Moreover an $\\mathbb{A}_f^{p}$-version of (\\ref{BZ18e}) shows that there\n is an element $a^p \\in (\\ensuremath{\\mathbb {A}}\\xspace_f^p)^{\\times}$ such that\n $f \\dot{c}' \\dot{c} = a^p$. We have the right to multiply $f$ by an\n element of $U_p(\\mathbb{Q})$. Therefore we may assume that\n $a^p \\in \\hat{\\mathbb{Z}}^{p}$. The result follows by setting\n $a = a_p a^p$. \n\\end{proof}\n\\begin{proof}[Proof of Proposition \\ref{BZ6p}]\n As in the proof of Proposition \\ref{BZ4p}, it is enough to show that\n \\begin{displaymath}\n\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}(R) \\rightarrow\n\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}_U^{\\bullet}}(R) \n \\end{displaymath}\n is injective if $\\Spec R$ is connected.\n Assume that we are given two points as in Lemma \\ref{Leminj} which are mapped\n to the same point of $\\tilde{\\mathcal{A}}^{\\bullet}_{\\mathbf{K}_U^{\\bullet}}(R)$. \n For suitable $U$ we conclude as in the proof of Proposition \\ref{BZ4p} that\n $f = g^2$, for some $g \\in O_{F}^{\\times} \\cap \\mathbf{K}^{\\bullet}$.\n If in the argument of that proof we choose $m$ big enough we may assume that\n $g \\in \\mathbf{M}^{\\bullet}$. We obtain\n$a_p = f \\theta_p\\in\\mathbf{M}^{\\bullet} \\cap \\mathbb{Z}_p^{\\times} = \\mathbf{M}$\n (see before (\\ref{BZ34e})). The multiplication by $g$ induces an isomorphism\n \\begin{equation}\\label{BZ19e} \ng: (A_1, \\bar{\\lambda}_1 f, \\bar{\\eta}_1^{p} \\dot{c}, (\\bar{\\eta}_{1,\\mathfrak{q}_j})_j, \\xi'_{1,p}) \\rightarrow\n (A_1, \\bar{\\lambda}_1, \\bar{\\eta}_1^{p},(\\bar{\\eta}_{1,\\mathfrak{q}_j})_j,\\xi_{1,p}).\n \\end{equation}\n Indeed, we have\n $g^{*}(\\lambda_1) = \\lambda_1 g^2$ and the morphism (\\ref{BZ19e}) respects the data\n $\\bar{\\eta}_1^{p} \\dot{c}$ and $\\bar{\\eta}_1^{p}$, comp. the proof of Proposition\n \\ref{BZ4p}. Furthermore, for $\\lambda_1 \\in \\bar{\\lambda}_1$ we obtain\n \\begin{displaymath}\n g^{*}( \\xi_{1,p}(\\lambda_1 f)) = g^{*} (\\xi_{1,p})(g^{*} (\\lambda_1)) :=\n \\xi_{1,p}(\\lambda_1) = a_p \\xi_{1,p}(\\lambda_1) = \\xi'_{1,p}(\\lambda_1 f). \n \\end{displaymath}\n The second to last equation holds because $a_p \\in \\mathbf{M}$. \n \\end{proof}\n\nWe recall that we assume that $D_{\\mathfrak{p}_0}$ is a quaternion division\nalgebra, cf. (\\ref{Dsplit-in0}). In this case, we have the following integral version of Proposition \\ref{BZ7p}. \n\n\\begin{proposition}\\label{BZ8p} \n Let $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}, where\n $\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. We set\n $\\mathbf{K} = G(\\mathbb{A}_f) \\cap \\mathbf{K}^{\\bullet}$. We\n assume that there is an $O_K$-lattice $\\Gamma\\subset V$ and an integer\n $m \\geq 3$ prime to $p$ such that for each $g \\in \\mathbf{K}^{\\bullet}$ we have\n $g \\Gamma \\subset \\Gamma$ and such that $g$ acts trivially on\n $\\Gamma\/m \\Gamma$. (In this case $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ is\n representable.) Let $U$ be as in Proposition \\ref{BZ4p}. \n\n Let $\\tilde{\\mathsf{A}}^t_{\\mathbf{K}} $ be the $\\Spec O_{E_{\\nu}}$-scheme which represents the functor $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ and let\n $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$\n be the coarse moduli scheme of\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$. It is a normal scheme\n which is proper over $\\Spec O_{E_{\\nu}}$. \n\n The canonical map \n $\\tilde{\\mathsf{A}}^t_{\\mathbf{K}} \\rightarrow \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ \n is an open and closed immersion. The arrow \n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}} \\rightarrow \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ \n is the \\'etale sheafification of the presheaf \n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ on the big \\'etale site. \n\\end{proposition}\n\\begin{proof} \n The scheme $\\tilde{\\mathsf{A}}^t_{\\mathbf{K}}$ is regular and the\n morphism $\\tilde{\\mathsf{A}}^t_{\\mathbf{K}} \\rightarrow \\Spec O_{E_{\\nu}}$ is\n generically smooth and proper.\n Its special fibre is a divisor with normal crossings.\n This follows from deformation theory because the $p$-divisible group\n $X_{\\mathfrak{q}_0}$ is a special formal $O_{B_{\\mathfrak{q}_0}}$-module in the sense\n of Drinfeld. The properness follows from a standard argument using that\n $B$ is a division algebra, cf. \\cite[Prop. 4.1]{Dr}.\n\nFor the proof that \na coarse moduli scheme $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$\nexists we refer to \\cite[1.7 Satz]{Z-sR}. Because this moduli scheme is\nobtained as a quotient of a normal scheme by a finite group, the coarse\nmoduli scheme is normal. Since\n $\\tilde{\\mathsf{A}}^t_{\\mathbf{K},E_{\\nu}}\\subset\\tilde{\\mathsf{A}}^t_{\\mathbf{K}}$\n is an open dense subset of a scheme which is locally integral we obtain\n a bijection between connected components\n \\begin{equation*}\n \\pi_0 (\\tilde{\\mathsf{A}}^t_{\\mathbf{K}, E_{\\nu}}) \\rightarrow \n \\pi_0 (\\tilde{\\mathsf{A}}^t_{\\mathbf{K}}), \\quad\\quad\n Z \\mapsto \\bar{Z} .\n \\end{equation*}\n The same is true for the connected components of\n $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U},E_{\\nu}}$ and\n $\\tilde{\\mathsf{A}}^{\\bullet}_{\\mathbf{K}^{\\bullet t}_{U}}$. \n We claim that \n \\begin{equation}\\label{BZ22e}\n \\tilde{\\mathsf{A}}^t_{\\mathbf{K}} \\rightarrow\n \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}} \n \\end{equation}\n is an open and closed immersion. Indeed, the morphism (\\ref{BZ22e}) is proper\n because $\\tilde{\\mathsf{A}}_{\\mathbf{K}}^t$ is proper over $\\Spec O_{E_{\\nu}}$. The\n general fiber over $E_{\\nu}$ of this morphism coincides up to a Galois twist\n with (\\ref{BZ10e}) and is therefore an open and closed immersion. Let\n $Z \\subset \\tilde{\\mathsf{A}}^t_{\\mathbf{K},E_{\\nu}}$\n be a connected component which we also regard as a connected component of\n $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}, E_{\\nu}}$. We consider the\n closures $\\bar{Z} \\subset \\tilde{\\mathsf{A}}^t_{\\mathbf{K}}$ and\n $\\bar{Z}^{\\bullet} \\subset \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ of $Z$.\n These are connected components and the morphism (\\ref{BZ22e}) induces\n an birational proper morphism $\\bar{Z} \\rightarrow \\bar{Z}^{\\bullet}$ of normal\n schemes. If we take the values in some algebraically closed field the last\n morphism becomes injective. This follows from Proposition \\ref{BZ6p} and\n the definition of a coarse moduli problem. Therefore\n $\\bar{Z} \\rightarrow \\bar{Z}^{\\bullet}$ is an isomorphism, and the claim is proved. \n \n \n Let $Y$ be the \\'etale sheafification of\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ on the big \\'etale site.\n We consider the following commutative diagram\n \\begin{equation}\\label{BZ27e}\n \\begin{aligned}\n \\xymatrix{\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}} \\ar[r] \\ar[d] &\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}} \\ar[d]\\\\\n Y^{o} \\ar[r] \\ar[d] & Y \\ar[d]\\\\\n \\tilde{\\mathsf{A}}^t_{\\mathbf{K}} \\ar[r] &\n \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}} .\\\\\n }\n \\end{aligned}\n \\end{equation}\n Here $Y^{o}$ is defined to be the fiber product in the lower square. We know\n that the horizontal arrows are monomorphisms and the two lower ones are\n open and closed immersions.\n We will show that $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}} \\rightarrow Y^{o}$ is an\n isomorphism of sheaves. \n \n Let $T$ be a noetherian scheme over $\\Spec O_{E_{\\nu}}$ and\n $\\mathcal{O}_{T,t}^{sh}$ the strict henselization at a geometric point $t$\n of $T$. We have to show that the induced map of stalks\n \\begin{displaymath}\n(\\tilde{\\mathcal{A}}^t_{\\mathbf{K}})_{T,t} \\rightarrow (Y^o)_{T,t} \n \\end{displaymath}\n is bijective. (The stalks are the same as the stalks of the restriction\n of both sheaves to the small \\'etale site $T_{et}$.) Since\n $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ commutes with direct limits, we obtain\n $(\\tilde{\\mathcal{A}}^t_{\\mathbf{K}})_{T,t} = \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}}(\\mathcal{O}_{T,t}^{sh})$.\n The same is true for the presheaf\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$. Therefore we obtain \n$(Y^o)_{T,t}\\subset Y_{T,t} =\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}(\\mathcal{O}_{T,t}^{sh})$.\n This subset consists of the points on the right hand side which are mapped to\n $\\tilde{\\mathsf{A}}^t_{\\mathbf{K}}(\\mathcal{O}_{T,t}^{sh})$.\n \n Let $L$ the residue class field of $\\mathcal{O}_{T,t}^{sh}$ which is separably\n closed. We firstly show that\n \\begin{equation}\\label{BZ25e}\n\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}(L) \\rightarrow Y^o(L) \n \\end{equation}\n is bijective. Equivalently we may show that\n $Y^{o}(L) \\rightarrow \\tilde{\\mathsf{A}}^t_{\\mathbf{K}}(L)$ is bijective.\n Clearly this map is surjective because\n $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}} \\cong \\tilde{\\mathsf{A}}^t_{\\mathbf{K}}$.\n Let\n $\\theta_1,\\theta_2\\in Y^o(L)\\subset\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}(L)$\n be two elements with the same image in\n $\\tilde{\\mathsf{A}}^t_{\\mathbf{K}}(L)$. By the properties of a coarse moduli\n scheme, the map \n$\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}(\\bar{L}) \\rightarrow \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}(\\bar{L})$\n is bijective. We conclude that $\\theta_{1,\\bar{L}} = \\theta_{2,\\bar{L}}$ holds for the\n base change. Therefore we find a finite totally inseparable extension $N$\n of $L$ such that $\\theta_{1,N} = \\theta_{2,N}$. If the last two points\n are represented by the data \n$(A_1,\\tilde{\\lambda}_1,\\tilde{\\eta}^p_1,(\\bar{\\eta}_{1,\\mathfrak{q}_j})_j, (\\xi_{1, \\mathfrak{p}_i})_i)$\n and\n$(A_2,\\tilde{\\lambda}_2,\\tilde{\\eta}^p_2,(\\bar{\\eta}_{2,\\mathfrak{q}_j})_j, (\\xi_{2, \\mathfrak{p}_i})_i)$, \n we conclude by the rigidity of abelian varieties, applied to the nilimmersion\n $N \\otimes_{L} N \\rightarrow N$, that $\\theta_1 = \\theta_2$.\n\n Now we consider the map\n \\begin{displaymath}\n \\widetilde{\\mathcal{A}}^t_{\\mathbf{K}}(\\mathcal{O}_{T,t}^{sh}) \\rightarrow\n Y^o(\\mathcal{O}_{T,t}^{sh}). \n \\end{displaymath}\n This map is clearly injective. We show that it is surjective. We consider\n a point\n \\begin{equation}\\label{BZ26e} \n (A_1, \\tilde{\\lambda}_1, \\tilde{\\eta}^p_1, (\\bar{\\eta}_{1,\\mathfrak{q}_j})_j, (\\xi_{1, \\mathfrak{p}_i})_i) \\in Y^o(\\mathcal{O}_{T,t}^{sh}) \\subset\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}(\\mathcal{O}_{T,t}^{sh}).\n \\end{equation}\n Over $L$ this point is in the image of (\\ref{BZ25e}). By Lemma \\ref{BZ7l},\n the preimage has the form\n $(A_{1,L}, \\bar{\\lambda}_1, \\bar{\\eta}^p_1, (\\bar{\\eta}_{1,\\mathfrak{q}_j})_j, (\\xi_{2, \\mathfrak{p}_i})_i)$. \n Since $\\bar{\\lambda}_1 \\subset \\tilde{\\lambda}_1$, the polarizations in\n $\\bar{\\lambda}_1$ lift to polarizations of $A_1$ which are principal in $p$. \n Since there is no difference between a rigidification over\n $\\mathcal{O}_{T,t}^{sh}$ and over the residue class field $L$, we see that \n $(A_{1}, \\bar{\\lambda}_1, \\bar{\\eta}^p_1, (\\bar{\\eta}_{1,\\mathfrak{q}_j})_j, (\\xi_{2, \\mathfrak{p}_i})_i)$\n is a point of $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}(\\mathcal{O}_{T,t}^{sh})$\n which is mapped to the point (\\ref{BZ26e}). We have proved that the two\n vertical arrows on the left hand side of diagram (\\ref{BZ27e}) are\n isomorphisms.\n\n To show that\n$\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}\\rightarrow\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ \n is the \\'etale sheafification, we can argue as in the proof of Proposition\n\\ref{BZ7p} if we substitute Lemma \\ref{BZCover1l} by Lemma \\ref{Lemcover} below. \n\\end{proof}\n\nThe group $G^{\\bullet}(\\mathbb{A}_f^p)$ acts on the projective system of the\nfunctors $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ for varying\n$\\mathbf{K}^{\\bullet,p} \\subset G^{\\bullet}(\\mathbb{A}_f^p)$ via the datum\n$\\eta^p$ of Definition \\ref{BZApkt4d}. More explicitly, each\n$g \\in G^{\\bullet}(\\mathbb{A}_f^p)$ induces by multiplication an isomorphism\n\\begin{displaymath}\n g: \\tilde{\\mathcal{A}}^{\\bullet t}_{g\\mathbf{K}^{\\bullet}g^{-1}} \\rightarrow\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}, \n\\end{displaymath}\nwhich induces an isomorphism of the coarse moduli spaces.\n\\begin{lemma}\\label{Lemcover} \n Let $\\mathbf{K}^{\\bullet}$ be as in Proposition \\ref{BZ8p}. Then there is an\n open subgroup $U \\subset (F \\otimes \\mathbb{A}_f)^{\\times}$ such that for each\n $g \\in G^{\\bullet}(\\mathbb{A}_f^p)$ the natural morphism\n \\begin{equation}\\label{BZ31e}\n \\tilde{\\mathsf{A}}^t_{g\\mathbf{K}g^{-1}} \\rightarrow \n \\tilde{\\mathsf{A}}^{\\bullet t}_{g\\mathbf{K}^{\\bullet}_{U}g^{-1}} \n \\end{equation}\n is an open and closed immersion. If we compose the immersion with the\n morphisms\n $g:\\tilde{\\mathsf{A}}^{\\bullet t}_{g\\mathbf{K}^{\\bullet}_{U}g^{-1}} \\rightarrow \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$,\n we obtain open and closed immersions\n \\begin{equation}\\label{BZ32e}\n \\varkappa_g: \\tilde{\\mathsf{A}}^t_{g\\mathbf{K}g^{-1}} \\rightarrow\n \\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}} .\n \\end{equation}\n For varying $g \\in G^{\\bullet}(\\mathbb{A}_f^p)$ the morphisms $\\varkappa_g$\n are an open covering of $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$. \n\\end{lemma}\n\\begin{proof}\n We have already seen that (\\ref{BZ31e}) is an open and closed\n immersion, cf. (\\ref{BZ22e}). Therefore the same is true of $\\varkappa_g$. The general fibre\n of $\\tilde{\\mathsf{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ is up to a Galois twist \n ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}}$. By Lemma \\ref{BZCover1l}\n each connected component $Z$ of ${\\mathrm{Sh}}(G^{\\bullet}, h)_{\\mathbf{K}^{\\bullet}_U, E_{\\nu}}$\n is in the image of $\\varkappa_g$ for some $g \\in G^{\\bullet}(\\mathbb{A}_f^p)$.\n Since $\\tilde{\\mathsf{A}}^{\\bullet}_{\\mathbf{K}^{\\bullet}_{U}}$ is locally an integral\n scheme which is flat over $O_{E_{\\nu}}$, each connected component is of the form $\\bar{Z}$. It is therefore in the image of the open\n and closed immersion $\\varkappa_g$. Hence (\\ref{BZ32e}) is indeed a\n covering. \n \\end{proof}\nIn the sequel, only the Shimura varieties ${\\rm Sh}(G^\\bullet, h)$ and ${\\rm Sh}(G^\\bullet, h^\\bullet_D)$ attached to the group $G^\\bullet$ will play a role. These are \\emph{ramified} abelian Galois twists of each other, cf. Proposition \\ref{BZ3c}. The following table summarizes the Shimura varieties and their relations to moduli functors. \n\n\n$$\n\\vcenter{\n\\begin{Small}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nShimura variety&Moduli problem&First occurrence&Relation \\\\\n\\hline\n${\\rm Sh}(G^\\bullet, h)$ &$\\ensuremath{\\mathcal {A}}\\xspace^\\bullet$, resp. $\\ensuremath{\\mathcal {A}}\\xspace^{\\bullet, bis}$ &Def. \\ref{BZApkt3d}\/\\ref{BZApkt3altd}, resp. Def. \\ref{BZApkt4d} & coarse moduli scheme \\\\\n\n\\hline\n${\\rm Sh}(G^\\bullet, h^\\bullet_D)$ &$\\ensuremath{\\mathcal {A}}\\xspace^{\\bullet, t}$ & Def. \\ref{BZsApkt4d} &coarse moduli sch. of unram. twist \\\\\n\\hline\n\\end{tabular}\n\\end{Small}\n}\n$$\n\n\n\n\n\\medskip\n\nThe integral model $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ of ${{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ over $O_{E_\\nu}$ (defined if $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}, where\n $\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$) is defined by twisting back the integral extension $\\tilde{\\ensuremath{\\mathcal {A}}\\xspace}^{\\bullet, t}$ of $\\ensuremath{\\mathcal {A}}\\xspace^{\\bullet, t}$. \n\n\n \n \n\\section{The $\\mathrm{RZ}$-spaces}\\label{s:RZ}\n\nIn this section, we discuss the $\\mathrm{RZ}$-spaces \nneeded for the $p$-adic uniformization of the Shimura varieties of the last\nsection. \n\nWe first discuss the banal case of a prime ideal $\\mathfrak{p}_i$, for $i \\neq 0$. \n\\begin{definition}\n Let $S$ be an $O_{E_{\\nu}}$-scheme. The category $\\mathcal{P}_{\\mathfrak{p}_i}(S)$\n is the category of all triples $(Y, \\iota, \\bar{\\lambda})$, where\n $Y$ is a $p$-divisible group of height $8[F_{\\mathfrak{p}_i} : \\mathbb{Q}_p]$\n over $S$, where $\\iota: O_{B_{\\mathfrak{p}_i}} \\rightarrow \\End Y $\n is a $\\mathbb{Z}_p${\\rm -Alg}ebra homomorphism, and where $\\bar{\\lambda}$ is a\n $O_{F_{\\mathfrak{p}_i}}^{\\times}$-homogeneous polarization of $Y$ such that each\n $\\lambda \\in \\bar{\\lambda}$ is principal. We demand that the Rosati involution associated to\n $\\lambda \\in \\bar{\\lambda}$ is compatible with the involution\n $b\\mapsto b^\\star$ on $B_{\\mathfrak{p}_i}$ with respect to $\\iota$. \n The decomposition\n $O_{B_{\\mathfrak{p}_i}} = O_{B_{\\mathfrak{q}_i}} \\times O_{B_{\\bar{\\mathfrak{q}}_i}}$\ninduces a composition $Y = Y_{\\mathfrak{q}_i} \\times Y_{\\bar{\\mathfrak{q}}_i}$.\n We demand moreover that the $p$-divisible group $Y_{\\mathfrak{q}_i}$\n is \\'etale. \n\\end{definition}\n\nThe definition implies that\n$\\lambda = \\lambda_{\\mathfrak{q}_i} \\oplus \\lambda_{\\bar{\\mathfrak{q}}_i}$ where\n$\\lambda_{\\mathfrak{q}_i}:Y_{\\mathfrak{q}_i}\\rightarrow(Y_{\\bar{\\mathfrak{q}}_i})^{\\wedge}$ and \n$\\lambda_{\\bar{\\mathfrak{q}}_i}: Y_{\\bar{\\mathfrak{q}}_i} \\rightarrow (Y_{\\mathfrak{q}_i})^{\\wedge}$ \nare isomorphisms to the dual $p$-divisible groups such that\n$\\lambda_{\\bar{\\mathfrak{q}}_i} = - (\\lambda_{\\mathfrak{q}_i})^{\\wedge}$ and such that\nfor each $b_1 \\in O_{B_{\\mathfrak{q}_i}}$ and $b_2 \\in O_{B_{\\bar{\\mathfrak{q}}_i}}$ the\nfollowing diagrams are commutative,\n\\begin{equation}\\label{RZ4e}\n \\begin{aligned}\n \\xymatrix{\n Y_{\\bar{\\mathfrak{q}}_i} \\ar[r]^{\\iota(b_1^{\\star})} \\ar[d]_{\\lambda_{\\bar{\\mathfrak{q}}_i}}\n & Y_{\\bar{\\mathfrak{q}}_i} \\ar[d]^{\\lambda_{\\bar{\\mathfrak{q}}_i}}\\\\\n (Y_{\\mathfrak{q}_i})^{\\wedge} \\ar[r]_{\\iota(b_1)^{\\wedge}} &\n (Y_{\\mathfrak{q}_i})^{\\wedge}, \\\\ \n } \\hspace{2cm} \n \\xymatrix{\n Y_{\\mathfrak{q}_i} \\ar[r]^{\\iota(b_2^{\\star})} \\ar[d]_{\\lambda_{\\mathfrak{q}_i}}\n & Y_{\\mathfrak{q}_i} \\ar[d]^{\\lambda_{\\mathfrak{q}_i}}\\\\\n (Y_{\\bar{\\mathfrak{q}}_i})^{\\wedge} \\ar[r]_{\\iota(b_2)^{\\wedge}} &\n (Y_{\\bar{\\mathfrak{q}}_i})^{\\wedge} .\\\\ \n } \n \\end{aligned}\n\\end{equation}\nSince one of these diagrams is the dual of the other it is enough to require the\ncommutativity of one of these diagrams.\n\nWe construct an object of $\\mathcal{P}_{\\mathfrak{p}_i}(\\Spec\\bar{\\kappa}_{E_{\\nu}})$\nas follows. Let us denote the action of the Frobenius endomorphism on\n$W(\\bar{\\kappa}_{E_{\\nu}})$ by $\\sigma$. Recall from (\\ref{BZLambda1e}) the lattices $\\Lambda_{\\mathfrak{q}_i} $ and $\\Lambda_{\\bar{\\mathfrak{q}}_i}$. We endow\n$\\Lambda_{\\mathfrak{q}_i} \\otimes_{\\mathbb{Z}_p} W(\\bar{\\kappa}_{E_{\\nu}})$ with\nthe structure of a Dieudonn\\'e module by defining the action of the Frobenius\n$F$ on this module by\n\\begin{displaymath}\n F(u_1 \\otimes \\xi_1) = p u_1 \\otimes \\sigma(\\xi_1), \\quad u_1 \\in\n \\Lambda_{\\mathfrak{q}_i}, \\; \\xi_1 \\in W(\\bar{\\kappa}_{E_{\\nu}}), \n\\end{displaymath}\nand we endow\n$\\Lambda_{\\bar{\\mathfrak{q}}_i} \\otimes_{\\mathbb{Z}_p} W(\\bar{\\kappa}_{E_{\\nu}})$\nwith a structure of a Dieudonn\\'e module by defining the action of the\nFrobenius on this module by \n\\begin{displaymath}\n F(u_2 \\otimes \\xi_2) = u_2 \\otimes \\sigma(\\xi_2), \\quad u_2 \\in\n \\Lambda_{\\bar{\\mathfrak{q}}_i}, \\; \\xi_2 \\in W(\\bar{\\kappa}_{E_{\\nu}}). \n\\end{displaymath}\nThe direct sum of these Dieudonn\\'e modules defines a Dieudonn\\'e module\nstructure on\n$\\Lambda_{\\mathfrak{p}_i}\\otimes_{\\mathbb{Z}_p} W(\\bar{\\kappa}_{E_{\\nu}})$.\nWe consider the perfect alternating $W(\\bar{\\kappa}_{E_{\\nu}})$-bilinear form\n\\begin{equation}\\label{RZ21e} \n\\psi_W: \\Lambda_{\\mathfrak{p}_i}\\otimes_{\\mathbb{Z}_p} W(\\bar{\\kappa}_{E_{\\nu}}) \n\\times \\Lambda_{\\mathfrak{p}_i}\\otimes_{\\mathbb{Z}_p} W(\\bar{\\kappa}_{E_{\\nu}})\n\\rightarrow W(\\bar{\\kappa}_{E_{\\nu}}), \n\\end{equation} \ncf. (\\ref{BZLambda1e}). \nOne checks easily that this is a bilinear form of Dieudonn\\'e modules.\nBy covariant Dieudonn\\'e theory,\n$(\\Lambda_{\\mathfrak{p}_i}\\otimes_{\\mathbb{Z}_p} W(\\bar{\\kappa}_{E_{\\nu}}), \\psi_W)$\ncorresponds to a principally polarized $p$-divisible group\n$(\\Lambda_{\\mathfrak{p}_i}^{pd}, \\lambda_{\\psi})$. We have the decomposition\n$\\Lambda_{\\mathfrak{p}_i}^{pd} = \\Lambda_{\\mathfrak{q}_i}^{et} \\oplus \\Lambda_{\\bar{\\mathfrak{q}}_i}^{mult}$,\nwhere the first factor is an \\'etale $p$-divisible group and the second factor\nis multiplicative. The action of $O_{B_{\\mathfrak{p}_i}}$ \non $\\Lambda_{\\mathfrak{p}_i}$ defines an action of $O_{B_{\\mathfrak{p}_i}}$ on \n$\\Lambda_{\\mathfrak{p}_i}^{pd}$. We see that\n$(\\Lambda_{\\mathfrak{p}_i}^{pd}, \\bar{\\lambda}_{\\psi})$ is an object of the category\n$\\mathcal{P}_{\\mathfrak{p}_i}(\\bar{\\kappa}_{E_{\\nu}})$ and that each other object in\nthis category is isomorphic to it.\n\nRecall the group $G^\\bullet_{\\mathfrak{p}_i}$, cf. (\\ref{BZGpi1e}). An element $g \\in G^\\bullet_{\\mathfrak{p}_i}$ induces a\nquasi-isogeny of the $p$-divisible $O_{B_{\\mathfrak{p}_i}}$-module \n$\\Lambda_{\\mathfrak{p}_i}^{pd}$ which respects the polarization $\\lambda_{\\psi}$\nup to a factor in $F_{\\mathfrak{p}_i}^{\\times}$. In particular we conclude that \n\\begin{displaymath}\n \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i} \\subset\n \\Aut_{\\mathcal{P}_{\\mathfrak{p}_i}} (\\Lambda_{\\mathfrak{p}_i}^{pd}, \\bar{\\lambda}_{\\psi}). \n \\end{displaymath}\n\n\nWe consider schemes $S$ over $\\Spf O_{E_{\\nu}}$ or equivalently\n$O_{E_{\\nu}}$-schemes $S$ where $p$ is locally nilpotent. We set\n$\\bar{S} = S \\times_{\\Spec O_{E_{\\nu}}} \\Spec \\kappa_{E_{\\nu}}$. \n\\begin{definition}\\label{RZ1d}\n Let $S$ be a scheme over $\\Spf O_{\\breve{E}_{\\nu}}$ so that $\\bar{S}$\n is a scheme over $\\bar{\\kappa}_{E_{\\nu}}$. We denote by\n $\\Lambda_{\\mathfrak{q}_i, \\bar{S}}^{et} = \\Lambda_{\\mathfrak{q}_i}^{et}\n \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S}$\n the base change. The unique lift to an \\'etale $p$-divisible group over\n $S$ is denoted by $\\Lambda_{\\mathfrak{q}_i, S}^{et}$. This is a constant\n \\'etale $p$-divisible group. \n \n A rigidification of an object\n $(Y, \\iota, \\bar{\\lambda}) \\in \\mathcal{P}_{\\mathfrak{p}_i}(S)$ modulo \n $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$ consists of a class\n $\\bar{\\eta}_{\\mathfrak{q}_i}$ of isomorphisms of $p$-divisible\n $O_{B_{\\mathfrak{q}_i}}$-modules \n \\begin{displaymath}\n \\eta_{\\mathfrak{q}_i}: \\Lambda_{\\mathfrak{q}_i, S}^{et} \\isoarrow Y_{\\mathfrak{q}_i}\n \\quad \\mod \\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i} \n \\end{displaymath}\n and a class $\\bar{\\xi}_{\\mathfrak{p}_i}$ of maps \n $\\xi_{\\mathfrak{p}_i}: \\bar{\\lambda} \\rightarrow O_{F_{\\mathfrak{p}_i}}^{\\times}\/\\mathbf{M}_{\\mathfrak{p}_i}$ such that\n $\\xi_{\\mathfrak{p}_i}(\\lambda u) = \\xi_{\\mathfrak{p}_i}(\\lambda) u$, for \n $u \\in O_{F_{\\mathfrak{p}_i}}^{\\times}$, $\\lambda \\in \\bar{\\lambda}$. \n\\end{definition}\nEquivalently we could replace $\\bar{\\eta}_{\\mathfrak{q}_i}$ by a class of\n$O_{B_{\\mathfrak{q}_i}}$-module homomorphisms of $p$-adic \\'etale sheaves\n$\\eta_{\\mathfrak{q}_i}:\\Lambda_{\\mathfrak{q}_i, S}^{et}\\rightarrow T_p(Y_{\\mathfrak{q}_i})$ \nmodulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i}$. We will use this definition\nonly in the case where $Y_{\\mathfrak{q}_i}$ is a constant \\'etale $p$-divisible\ngroup. We denote the category of objects of $\\mathcal{P}_{\\mathfrak{p}_i}(S)$\nwith an rigidification by\n$\\mathcal{P}_{\\mathfrak{p}_i}(S)_{\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}$. \n\nWe reformulate the definition of an rigidified object\n$(Y, \\iota, \\bar{\\lambda}, \\bar{\\eta}_{\\mathfrak{q}_i}, \\bar{\\xi}_{\\mathfrak{p}_i})$. \nTo each $\\lambda \\in \\bar{\\lambda}$ we associate an\n$O_{B_{\\bar{\\mathfrak{q}}_i}}$-module isomorphism of $p$-divisible groups\n$\\eta_{\\bar{\\mathfrak{q}}_i}$ by the following commutative\ndiagram \n\\begin{equation}\\label{diagreta}\n\\begin{aligned}\n\\xymatrix{\n \\Lambda_{\\mathfrak{q}_i,S}^{et} \\ar[r]^{\\eta_{\\mathfrak{q}_i}} \n \\ar[d]_{\\xi_{\\mathfrak{p}_i}(\\lambda) \\lambda_{\\psi}} \n & Y_{\\mathfrak{q}_i} \\ar[d]^{\\lambda_{\\bar{\\mathfrak{q}}_i}}\\\\\n (\\Lambda_{\\bar{\\mathfrak{q}}_i}^{mult})_{\\; S}^{\\wedge} & \n (Y_{\\bar{\\mathfrak{q}}_i})^{\\wedge} \\ar[l]_{\\eta_{\\bar{\\mathfrak{q}}_i}^{\\wedge}} . \\\\ \n }\n \\end{aligned}\n \\end{equation} \nThen a rigidification modulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$ of\n$(Y, \\iota, \\bar{\\lambda})$ is equivalently described by a class\n$\\bar{\\eta}_{\\mathfrak{p}_i}$ \nof isomorphisms in the category $\\mathcal{P}_{\\mathfrak{p}_i}(S)$: \n\\begin{equation}\\label{RZ2e} \n \\eta_{\\mathfrak{p}_i}: (\\Lambda_{\\mathfrak{p}_i}^{pd},\\bar{\\lambda}_{\\psi})_S\n \\isoarrow\n (Y,\\iota,\\bar{\\lambda}) \\quad \\mod \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}. \n\\end{equation}\nWe see that, for $S$ connected, $\\eta_{\\mathfrak{p}_i}$ is given by its value at a geometric \npoint $\\omega$ of $S$, where $(\\eta_{\\mathfrak{p}_i})_{\\omega}$ is invariant\nmodulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$. This makes sense because, by the\ndiagram \\eqref{diagreta} above, everything comes down to a morphism between $p$-adic \\'etale\nsheaves. \n\nWe indicate how this allows to extend a Hecke operator\n$g \\in G^{\\bullet}_{\\mathfrak{p}_i} \\subset G^{\\bullet}(\\mathbb{A}_f)$ from the\ngeneric fiber $\\tilde{\\mathcal{A}}^{\\bullet t}_{E_{\\nu}}$ to the\nwhole functor $\\tilde{\\mathcal{A}}^{\\bullet t}$. We consider a congruence\nsubgroup $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ such that\n$k g \\Lambda_{\\mathfrak{p}_i} = g \\Lambda_{\\mathfrak{p}_i}$ for\n$k \\in \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$. We consider a point \n\\begin{displaymath}\n(A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j,\n(\\bar{\\xi}_{\\mathfrak{p}_j})_j) \\in \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(S). \n \\end{displaymath}\n Note here that, by $\\mathbf{M}^{\\bullet}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$, the choice of $\\xi_{\\mathfrak{p}_0}$ is redundant, so we drop it from the notation. Let $Y_{\\mathfrak{p}_i}$ be the $\\mathfrak{p}_i$-part of the $p$-divisible\ngroup of $A$. It inherits the structure of an rigidified object\n$(Y_{\\mathfrak{p}_i},\\iota,\\bar{\\lambda},\\bar{\\eta}_{\\mathfrak{q}_i},\\bar{\\xi}_{\\mathfrak{p}_i})$ \nof $\\mathcal{P}_{\\mathfrak{p}_i}(S)$. We choose \n$\\eta_{\\mathfrak{q}_i} \\in \\bar{\\eta}_{\\mathfrak{q}_i}$ and write a commutative\ndiagram\n\\begin{displaymath}\n\\xymatrix{\n (\\Lambda_{\\mathfrak{p}_i,S}^{pd}, \\bar{\\lambda}_{\\psi})_S\n \\ar[r]^{\\eta_{\\mathfrak{p_i}}} \n & (Y_{\\mathfrak{p}_i}, \\iota, \\bar{\\lambda})\\\\\n (\\Lambda_{\\mathfrak{p}_i,S}^{pd}, \\bar{\\lambda}_{\\psi})_S \\ar[r]^{\\eta'_{\\mathfrak{p_i}}}\n \\ar[u]_g & (Y'_{\\mathfrak{p}_i}, \\iota', \\bar{\\lambda}') \\ar[u]_{a} .\n }\n\\end{displaymath}\nHere the maps $a$ and $g$ are quasi-isogenies and $\\eta_{\\mathfrak{p}_i}$ and\n$\\eta'_{\\mathfrak{p}_i}$ are understood as explained after (\\ref{RZ2e}).\nFor $\\lambda \\in \\bar{\\lambda}$ we define\n$\\lambda' = \\mu_{\\mathfrak{p}_i}^{-1}(g) a^{*} (\\lambda)$. This is a principal\npolarization because\n$g^{*} (\\lambda_{\\psi}) = \\mu_{\\mathfrak{p}_i}(g) \\lambda_{\\psi}$. We define\n$\\xi^{'}(a^{*} (\\lambda)) = \\mu_{\\mathfrak{p}_i}(g) \\xi(\\lambda)$. This gives a\nrigidified object\n\\begin{equation}\\label{RZ20e}\n (Y'_{\\mathfrak{p}_i},\\iota', \\bar{\\lambda}', \\bar{\\eta}'_{\\mathfrak{q}_i},\n \\bar{\\xi}'_{\\mathfrak{p}_i}) .\n \\end{equation}\n\nWe find a quasi-isogeny of abelian varieties\n$\\alpha: (A',\\iota') \\rightarrow (A, \\iota)$ which induces on the\n$\\mathfrak{p}_j$-parts of the $p$-divisible groups an isomorphism\nfor $j \\neq i$ and the map $a$ on the $\\mathfrak{p}_i$-parts. Then\n$(A',\\iota')$ inherits the data $(\\bar{\\eta}^p)'$, $\\bar{\\eta}'_{\\mathfrak{q}_j}$, \n$\\bar{\\xi}'_{\\mathfrak{p}_j}$ for $j \\neq i$ by pull back via $\\alpha$. \nThe data $\\bar{\\eta}'_{\\mathfrak{q}_i}$, $\\bar{\\xi}'_{\\mathfrak{p}_i}$ are inherited\nfrom (\\ref{RZ20e}). The $U_p(F)$-homogeneous polarization of $A'$ consists\nof all $\\lambda' \\in \\alpha^{*} (\\bar{\\lambda})$ which are principal in $p$.\nWe define the image by the Hecke operator $g$ as\n\\begin{displaymath}\n(A', \\iota', \\bar{\\lambda}', (\\bar{\\eta}^p)', (\\bar{\\eta}'_{\\mathfrak{q}_j})_j,\n (\\bar{\\xi}'_{\\mathfrak{p}_j})_j) \\in\n \\tilde{\\mathcal{A}}^{\\bullet t}_{g^{-1}\\mathbf{K}^{\\bullet}g}(S). \n \\end{displaymath}\nIt follows from the discussion after the proof of Proposition \\ref{BZ11p} that\nthis defines an extension of the Hecke operators over the generic fiber \n$\\tilde{\\mathcal{A}}^{\\bullet t}_{E_{\\nu}}$. \n\nWe fix an object $(\\mathbb{X}, \\iota_{\\mathbb{X}}, \\bar{\\lambda}_{\\mathbb{X}})$ \nof the category $\\mathcal{P}_{\\mathfrak{p}_i}(\\bar{\\kappa}_{E_{\\nu}})$, e.g.\n$(\\Lambda_{\\mathfrak{p}_i}^{pd},\\bar{\\lambda}_{\\psi})$. We choose\n$\\lambda_{\\mathbb{X}} \\in \\bar{\\lambda}_{\\mathbb{X}}$ and call\n$(\\mathbb{X}, \\iota_{\\mathbb{X}}, \\lambda_{\\mathbb{X}})$ the framing object.\nWe define\n$(\\mathbb{X}, \\iota_{\\mathbb{X}}, \\bar{\\lambda}_{\\mathbb{X}})_S$ in the same way as\nin Definition \\ref{RZ1d}. The $RZ$-space is defined as follows:\n\\begin{definition}\\label{RZ2d} \n We denote by ${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}$ the functor on the\n category of schemes $S$ over $\\Spf O_{\\breve{E}_{\\nu}}$, where a point of\n ${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}(S)$ is given by the\n following data up to isomorphism:\n \\begin{enumerate}\n \\item[({1})] an object $(Y, \\iota, \\bar{\\lambda}) \\in\n \\mathcal{P}_{\\mathfrak{p}_i}(S)$, \n \\item[({2})] a rigidification $(\\bar{\\eta}_{\\mathfrak{q}_i} \\text{ modulo }\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i},\n \\bar{\\xi}_{\\mathfrak{p}_i} \\text{ modulo }\\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i}$ ) of \n $(Y, \\iota, \\bar{\\lambda})$, \n \\item[({3})] a quasi-isogeny of $p$-divisible $O_{B_{\\mathfrak{p}_i}}$-modules \n $\\rho: (Y, \\iota) \\rightarrow (\\mathbb{X}, \\iota_{\\mathbb{X}})_S$ which\n respects\n the polarizations on both sides up to a factor in $F_{\\mathfrak{p}_i}^{\\times}$. \n \\end{enumerate}\n\\end{definition}\nIt follows from (\\ref{RZ2e}) that we can represent a point of \n${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}(S)$ by a class \n$\\bar{\\rho}$ of quasi-isogenies \n\\begin{equation}\\label{RZ3e}\n \\rho: (\\Lambda_{\\mathfrak{p}_i}^{pd},\\bar{\\lambda}_{\\psi})_S \\rightarrow\n (\\mathbb{X}, \\iota_{\\mathbb{X}}, \\bar{\\lambda}_{\\mathbb{X}}) \\text{ modulo $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$ },\n \\end{equation}\n which respects the polarizations\nof both sides up to a factor in $F_{\\mathfrak{p}_i}^{\\times}$. \n\nWe choose an isomorphism\n\\begin{equation}\\label{RZ1e} \n (\\mathbb{X},\\iota_{\\mathbb{X}},\\lambda_{\\mathbb{X}}) \\isoarrow\n (\\Lambda_{\\mathfrak{p}_i}^{pd}, \\lambda_{\\psi})\n\\end{equation}\nwhich respects the polarizations.\nThen we see from (\\ref{RZ3e}) that a point of\n${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}(S)$ is (locally)\nrepresented by an element $g \\in G^{\\bullet}_{\\mathfrak{p}_i}$. \nTherefore we obtain \n\\begin{proposition}\\label{RZ6p}\n The choice of an isomorphism (\\ref{RZ1e}) defines an isomorphism\n \\begin{displaymath}\n {\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}\n \\overset{\\sim}{\\longrightarrow}\n G^{\\bullet}_{\\mathfrak{p}_i}\/\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}, \n \\end{displaymath}\n where the right hand side denotes the constant sheaf. \n \\end{proposition}\nLet $g \\in G^{\\bullet}_{\\mathfrak{p}_i}$. If we represent a point of \n${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}(S)$ in the form\n(\\ref{RZ3e}), the assignement $\\rho \\mapsto \\rho g$ defines a functor\nmorphism \n\\begin{equation}\\label{Hecke11e}\ng: {\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}} \\rightarrow\n {\\rm RZ}_{\\mathfrak{p}_i, g^{-1}\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}g}. \n\\end{equation}\nWe call this a Hecke operator. Note that (\\ref{Hecke11e}) is only defined\nif $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$ is sufficiently small with respect to\n$g$, i.e. if\n$g^{-1}\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}g \\Lambda_{\\mathfrak{p}_i} \\subset\n\\Lambda_{\\mathfrak{p}_i}$. \nWe could define ${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}$ for\nan arbitrary open compact subgroup\n$\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i} \\subset G_{\\mathfrak{p}_0}$ by making the\nHecke operators part of the definition. \n\n\\smallskip\n\nNow we discuss the case of the prime ideal $\\mathfrak{p}_0$. \n\nWe first define the category $\\mathcal{P}_{\\mathfrak{p}_0}(S)$ for a scheme $S$\nover $\\Spf O_{E_{\\nu}}$. Because of the isomorphism\n$\\varphi_0: O_{F_{\\mathfrak{p}_0}} \\rightarrow O_{E_{\\nu}}$ it makes sense to speak\nof a special formal $O_{B_{\\mathfrak{q}_0}}$-module in the sense of Drinfeld over $S$, cf. \\cite{Dr} or \\cite[\\S 5.1]{KRZ}. \nWe consider $p$-divisible $O_{B_{\\mathfrak{p}_0}}$-modules $(Y,\\iota)$. The\ndecomposition $O_{B_{\\mathfrak{p}_0}} = O_{B_{\\mathfrak{q}_0}} \\times O_{B_{\\bar{\\mathfrak{q}}_0}}$\ninduces a decomposition\n\\begin{displaymath}\nY = Y_{\\mathfrak{q}_0} \\times Y_{\\bar{\\mathfrak{q}}_0}. \n\\end{displaymath}\nWe consider principal polarizations $\\lambda$ on $Y$ which induce on\n$O_{B_{\\mathfrak{p}_0}}$ the given involution $\\star$. They are given by two\nisomorphisms\n$\\lambda_{\\mathfrak{q}_0}:Y_{\\mathfrak{q}_0}\\rightarrow(Y_{\\bar{\\mathfrak{q}}_0})^{\\wedge}$ \nand\n$\\lambda_{\\bar{\\mathfrak{q}}_0}:Y_{\\bar{\\mathfrak{q}}_0}\\rightarrow(Y_{\\mathfrak{q}_0})^{\\wedge}$ \nsuch that $\\lambda_{\\bar{\\mathfrak{q}}_0} = - \\lambda_{\\mathfrak{q}_0}^{\\; \\wedge}$. \nMoreover we have commutative diagrams like (\\ref{RZ4e}) for $i=0$. \n\\begin{definition}\\label{RZ3d}\n An object $(Y, \\iota, \\bar{\\lambda})$ of the category\n $\\mathcal{P}_{\\mathfrak{p}_0}(S)$ consists of the following data:\n \\begin{enumerate}\n \\item[(1)] A $p$-divisible $O_{B_{\\mathfrak{p}_0}}$-module $(Y, \\iota)$ over $S$\n such that $Y_{\\mathfrak{q}_0}$ is a special formal $O_{B_{\\mathfrak{q}_0}}$-module.\n \\item[(2)] An $O_{F_{\\mathfrak{p}_0}}^{\\times}$-homogeneous polarization\n $\\bar{\\lambda}$ of $Y$, such that each $\\lambda \\in \\bar{\\lambda}$ is\n principal and such that the Rosati involution of $\\lambda$ induces on\n $O_{B_{\\mathfrak{p}_0}}$ the involution $\\star$. \n \\end{enumerate}\n \\end{definition}\nWe note that the functor $(Y,\\iota, \\bar{\\lambda})$ from \n$\\mathcal{P}_{\\mathfrak{p}_0}(S)$ to the category of special formal\n$O_{B_{\\mathfrak{q}_0}}$-modules over $S$ it not faithful. But it would be\nan equivalence of categories is we replace $\\mathcal{P}_{\\mathfrak{p}_0}(S)$\nby the category of triples $(Y, \\iota, \\lambda)$ with a given polarization\n$\\lambda$ as in the Definition above. We fix an object\n$(\\mathbb{X}, \\iota_{\\mathbb{X}}, \\lambda_{\\mathbb{X}})$ over\n$\\Spec \\bar{\\kappa}_{E_{\\nu}}$. We call this the framing object. \nWe keep the notation of Definition \\ref{RZ1d}. \n\\begin{definition}\\label{RZ4d}\n We denote by ${\\rm RZ}_{\\mathfrak{p}_0}$ the functor on the\n category of schemes $S$ over $\\Spf O_{\\breve{E}_{\\nu}}$ such that a\n point of ${\\rm RZ}_{\\mathfrak{p}_0}(S)$ consists of the following data\n up to isomorphism: \n \\begin{enumerate}\n \\item[(1)] an object \n $(Y,\\iota, \\bar{\\lambda}) \\in \\mathcal{P}_{\\mathfrak{p}_0}(S)$,\n \\item[(2)] a quasi-isogeny of $p$-divisible $O_{B_{\\mathfrak{p}_0}}$-modules \n \\begin{displaymath}\n \\rho: (Y,\\iota)_{\\bar{S}} \\rightarrow (\\mathbb{X}, \\iota_{\\mathbb{X}})\n \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S}\n \\end{displaymath}\n which respects the polarizations on both sides up to a factor in\n $F_{\\mathfrak{p}_0}^{\\times}$. \n \\end{enumerate}\n\\end{definition}\nLet $(G,\\iota)$ be a $p$-divisible\n$O_{F_{\\mathfrak{p}_0}}$-module over an $O_{F_{\\mathfrak{p}_0}}${\\rm -Alg}ebra $R$. Let\n$F_{\\mathfrak{p}_0}^t \\subset F_{\\mathfrak{p}_0}$ be the maximal subfield\nwhich is unramified over $\\mathbb{Q}_p$ and let \n$\\sigma \\in \\Gal(F_{\\mathfrak{p}_0}^t\/\\mathbb{Q}_p)$ be the Frobenius. Let\n$\\varepsilon: O_{F_{\\mathfrak{p}_0}} \\rightarrow R$ be the structure morphism.\nThere is the natural decomposition\n\\begin{displaymath}\n O_{F_{\\mathfrak{p}_0}} \\otimes_{\\mathbb{Z}_p} R \\cong \\prod_{i=0}^{f-1}\n O_{F_{\\mathfrak{p}_0}} \\otimes_{O_{F^t_{\\mathfrak{p}_0}}, \\sigma^i\\varepsilon} R, \n\\end{displaymath}\nwhere $f = [F^t_{\\mathfrak{p}_0}: \\mathbb{Q}_p]$ is the index of inertia of\n$F_{\\mathfrak{p}_0}$. This decomposition induces a corresponding decomposition of the\n$R$-module given by the Lie algebra of $G$. We set \n\\begin{displaymath}\n \\height_{F_{\\mathfrak{p}_0}} G := \\height_{F_{\\mathfrak{p}_0}} (\\pi \\mid G) =\n [F_{\\mathfrak{p}_0} : \\mathbb{Q}_p]^{-1} \\height G, \n\\end{displaymath}\nwhere $\\pi$ is a prime element of $F_{\\mathfrak{p}_0}$. \n\nLet $\\alpha: G_1 \\rightarrow G_2$\nbe an isogeny which is an $O_{F_{\\mathfrak{p}_0}}$-module homomorphism such that \n$\\rank_R \\Lie_i G_1 = \\rank_R \\Lie_i G_2$ for $i = 0, \\ldots, f-1$. \nThen $\\height \\alpha$ is divisible by $[F^t:\\mathbb{Q}_p]$. In this case\nwe define\n\\begin{displaymath}\n \\height_{F_{\\mathfrak{p}_0}} \\alpha = [F_{\\mathfrak{p}_0}^t:\\mathbb{Q}_p]^{-1}\n \\height \\alpha, \n\\end{displaymath}\n\n\nIf $\\alpha: X \\rightarrow X'$ is a quasi-isogeny of special formal\n$O_{B_{\\mathfrak{p}_0}}$-modules, the integer $\\height_{F_{\\mathfrak{p}_0}} \\alpha$\nis divisible by $2$, because there is a quadratic unramified extension\nof $F_{\\mathfrak{p}_0}$ which is contained in $B_{\\mathfrak{p}_0}$. \n\nWe consider a point of ${\\rm RZ}_{\\mathfrak{p}_0}(S)$ as in Definition\n\\ref{RZ4d}. The quasi-isogeny $\\rho$ is the direct sum of two quasi-isogenies\n\\begin{equation}\n \\rho_{\\mathfrak{q}_0} : Y_{\\mathfrak{q}_0, \\bar{S}} \\rightarrow\n \\mathbb{X}_{\\mathfrak{q}_0} \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S}, \\quad \n \\rho_{\\bar{\\mathfrak{q}}_0} : Y_{\\bar{\\mathfrak{q}}_0, \\bar{S}} \\rightarrow\n \\mathbb{X}_{\\bar{\\mathfrak{q}}_0} \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S}. \n \\end{equation}\n\nFor each pair of integers $(a,b)$ such that $a + b \\equiv 0 \\mod 2$, we\ndefine an open and closed subfunctor of\n${\\rm RZ}_{\\mathfrak{p}_0}$\n\\begin{equation}\n {\\rm RZ}_{\\mathfrak{p}_0}(a,b)(S) = \\{(Y,\\iota, \\bar{\\lambda}, \\rho) \\; | \\;\n \\height_{F_{\\mathfrak{p}_0}} \\rho_{\\mathfrak{q}_0} = 2a, \\;\n \\height_{F_{\\mathfrak{p}_0}} \\rho_{\\bar{\\mathfrak{q}}_0} = 2b \\}\n\\end{equation}\nWe remark that for a point\n$(Y,\\iota, \\bar{\\lambda}, \\rho) \\in {\\rm RZ}_{\\mathfrak{p}_0}(S)$ the sum\n$(\\height_{F_{\\mathfrak{p}_0}} \\rho_{\\mathfrak{q}_0} + \\height_{F_{\\mathfrak{p}_0}} \\rho_{\\bar{\\mathfrak{q}}_0})$\nis always divisible by $4$. Indeed, by definition there is an\n$f \\in F_{\\mathfrak{p}_0}^{\\times}$ such that the following diagram of\nquasi-isogenies is commutative,\n\\begin{equation}\\label{RZ14e}\n\\xymatrix{\n Y_{\\mathfrak{q}_0, \\bar{S}} \\ar[rr]^{\\rho_{\\mathfrak{q}_0}} \\ar[d]_{\\lambda f} & \n & \\mathbb{X}_{\\mathfrak{q}_0} \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S}\n \\ar[d]^{\\lambda_{\\mathbb{X}}}\\\\\n (Y_{\\bar{\\mathfrak{q}}_0, \\bar{S}})^{\\wedge} \n & & \\ar[ll]_{\\rho_{\\bar{\\mathfrak{q}}_0}^{\\;\\wedge}}\n (\\mathbb{X}_{\\bar{\\mathfrak{q}}_0})^{\\wedge}\n \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S} .\\\\ \n }\n \\end{equation}\nSince $\\lambda$ and $\\lambda_{\\mathbb{X}}$ are isomorphisms, we obtain that\n\\begin{displaymath}\n \\height_{F_{\\mathfrak{p}_0}} \\rho_{\\mathfrak{q}_0} +\n \\height_{F_{\\mathfrak{p}_0}} \\rho_{\\bar{\\mathfrak{q}}_0} = \\height_{F_{\\mathfrak{p}_0}}(f\n \\mid Y_{\\mathfrak{q}_0, \\bar{S}}). \n\\end{displaymath} \nThe right hand side is divisible by $4$ because\n$\\height_{F_{\\mathfrak{p}_0}} Y_{\\mathfrak{q}_0, \\bar{S}} = 4$. \n\nWe conclude that\n\\begin{equation}\\label{RZ12e}\n {\\rm RZ}_{\\mathfrak{p}_0} = \\coprod_{a+b\\equiv 0\\,{\\rm mod}\\, 2}\n {\\rm RZ}_{\\mathfrak{p}_0}(a,b) .\n \\end{equation}\nWe introduce Hecke operators acting on ${\\rm RZ}_{\\mathfrak{p}_0}$.\nLet ($Y_{\\mathfrak{q}_0}, \\iota_{\\mathfrak{q}_0})$ be a $p$-divisible\n$O_{B_{\\mathfrak{q}_0}}$-module. For $u_1 \\in B_{\\mathfrak{q}_0}^{\\times}$ we\ndefine\n\\begin{displaymath}\n \\iota_{\\mathfrak{q}_0}^{u_1}: O_{B_{\\mathfrak{q}_0}} \\rightarrow \\End Y_{\\mathfrak{q}_0},\n \\quad \\iota_{\\mathfrak{q}_0}^{u_1}(b) = \\iota_{\\mathfrak{q}_0}(u_1^{-1} b u_1). \n\\end{displaymath}\nWe set $Y_{\\mathfrak{q}_0}^{u_1} = Y_{\\mathfrak{q}_0}$ and write\n$(Y_{\\mathfrak{q}_0}^{u_1}, \\iota_{\\mathfrak{q}_0}^{u_1})$. \nWe use the same definition for a $p$-divisible $O_{B_{\\bar{\\mathfrak{q}}_0}}$-module.\n\nLet\n$u =(u_1,u_2) \\in B_{\\mathfrak{p}_0}^{\\times} = B_{\\mathfrak{q}_0}^{\\times} \\times B_{\\bar{\\mathfrak{q}}_0}^{\\times}$.\n For a $p$-divisible $O_{B_{\\mathfrak{p}_0}}$-module $(Y,\\iota)$ we set\n$\\iota^{u}(b) = \\iota(u^{-1}b u)$, $b \\in O_{B_{\\mathfrak{p}_0}}$.\n\n\\begin{lemma}\\label{RZ4l} \n Let $(Y, \\iota, \\bar{\\lambda}) \\in \\mathcal{P}_{\\mathfrak{p}_0}(S)$. \n Let $u = (u_1, u_2) \\in B_{\\mathfrak{p}_0}^{\\times}$ such that\n $u_2^{\\star} u_1 \\in F_{\\mathfrak{p}_0}^{\\times}$.\nThen for each $\\lambda \\in \\bar{\\lambda}$ the Rosati involution of $\\lambda$ \n on $\\End Y^u$ induces via\n \\begin{displaymath}\n\\iota^u: O_{B_{\\mathfrak{p}_0}} \\rightarrow \\End Y^u \n \\end{displaymath}\n the involution $\\star$ on $O_{B_{\\mathfrak{p}_0}}$. In particular $(Y^u, \\iota^u, \\bar{\\lambda}) \\in \\mathcal{P}_{\\mathfrak{p}_0}(S)$. \n \\end{lemma}\n\\begin{proof}\n We must verify the commutativity of the following diagram,\n \\begin{displaymath}\n \\xymatrix{\n Y_{\\mathfrak{q}_0}^{u_1} \\ar[r]^{\\iota^{u_1}(b_2^{\\star})} \\ar[d]_{\\lambda}\n & Y_{\\bar{\\mathfrak{q}}_0}^{u_1} \\ar[d]^{\\lambda}\\\\\n (Y_{\\bar{\\mathfrak{q}}_0}^{u_2})^{\\wedge} \\ar[r]_{\\iota^{u_2}(b_2)^{\\wedge}} &\n (Y_{\\bar{\\mathfrak{q}}_0}^{u_2})^{\\wedge} .\\\\ \n }\n \\end{displaymath}\n Indeed,\n \\begin{displaymath} \n \\lambda^{-1} \\iota^{u_2}(b_2)^{\\wedge} \\lambda =\n \\lambda^{-1} \\iota(u_2^{-1}b_2u_2)^{\\wedge} \\lambda = \n \\iota(u_2^{\\star} b_2^{\\star} (u_2^{\\star})^{-1}) \n = \\iota(u_1^{-1} b_2^{\\star} u_1) = \\iota^{u_1}(b_2^{\\star}). \n \\end{displaymath}\n The second equation holds because the Rosati involution of $\\lambda$ induces\n via $\\iota$ the involution $\\star$ on $O_{B_{\\mathfrak{p}_0}}$. The third equation\n holds because $u_2^{\\star} u_1 \\in F_{\\mathfrak{p}_0}$ implies that\n \\begin{displaymath}\n u_1^{-1} b_2^{\\star} u_1 = u_2^{\\star} u_1 u_1^{-1} b_2^{\\star} u_1\n (u_2^{\\star} u_1)^{-1} = u_2^{\\star} b_2^{\\star} (u_2^{\\star})^{-1}. \n \\end{displaymath}\n\\end{proof}\n\nLet $(Y, \\iota, \\bar{\\lambda}) \\in \\mathcal{P}_{\\mathfrak{p}_0}(S)$. Then\n\\begin{equation}\\label{RZ5e} \n\\iota(u): (Y^u, \\iota^u) \\rightarrow (Y, \\iota) \n\\end{equation}\nis a quasi-isogeny of $p$-divisible $O_{B_{\\mathfrak{p}_0}}$-modules. \n\\begin{lemma}\\label{RZ5l}\n Let $\\lambda \\in \\bar{\\lambda}$. We assume that for $u = (u_1, u_2)$\n we have $u_2^{\\star}u_1 \\in F_{\\mathfrak{p}_0}^{\\times}$.\nThen the quasi-isogeny (\\ref{RZ5e}) respects the polarization $\\lambda$ on\n both sides of (\\ref{RZ5e}) up to a factor in $F_{\\mathfrak{p}_0}^{\\times}$.\n \\end{lemma}\n\\begin{proof}\n We must show that there exists $f \\in O_{F_{\\mathfrak{p}_0}}^{\\times}$ such that the\n following diagram in commutative.\n \\begin{displaymath}\n \\xymatrix{\n Y_{\\mathfrak{q}_0} \\ar[r]^{\\lambda \\iota(f)\\;} \n & (Y_{\\bar{\\mathfrak{q}}_0})^{\\wedge} \\ar[d]^{\\iota(u_2)^{\\wedge}}\\\\\n Y_{\\mathfrak{q}_0}^{u_1} \\ar[r]_{\\lambda \\quad} \\ar[u]^{\\iota(u_1)} &\n (Y_{\\bar{\\mathfrak{q}}_0}^{u_2})^{\\wedge} .\\\\ \n }\n \\end{displaymath}\n The commutativity is equivalent to the first of the following equations, \n \\begin{displaymath}\n \\lambda = \\iota(u_2)^{\\wedge} \\lambda \\iota(f) \\iota(u_1) = \n \\lambda \\lambda^{-1} \\iota(u_2)^{\\wedge} \\lambda \\iota(f) \\iota(u_1) =\n \\lambda \\iota(u_2^{\\star}) \\iota(f) \\iota(u_1) =\n \\lambda \\iota(f u_2^{\\star}u_1) .\n \\end{displaymath}\n Therefore we obtain a commutative diagram if we choose $f u_2^{\\star}u_1 =1$. \n\\end{proof}\n\nWe define the group\n\\begin{displaymath}\n \\mathcal{H}_{\\mathfrak{p}_0} = \\{u \\in B_{\\mathfrak{p}_0} \\; |\n u^{\\star} u \\in F^{\\times}_{\\mathfrak{p}_0} \\; \\} \\subset B^{\\times}_{\\mathfrak{p}_0}\n\\end{displaymath}\nNote that for $u = (u_1, u_2)$ as above, the conditions\n$u^{\\star} u \\in F^{\\times}_{\\mathfrak{p}_0}$, resp.\n$u_2^{\\star} u_1 \\in F^{\\times}_{\\mathfrak{p}_0}$, resp. \n$u_1 u_2^{\\star} \\in F^{\\times}_{\\mathfrak{p}_0}$ are equivalent.\n\nThe group $\\mathcal{H}_{\\mathfrak{p}_0}$ acts from the left on the functor\n${\\rm RZ}_{\\mathfrak{p}_0}$. Let\n$(Y,\\iota,\\bar{\\lambda}, \\rho) \\in {\\rm RZ}_{\\mathfrak{p}_0}(S)$.\nFor $u \\in \\mathcal{H}_{\\mathfrak{p}_0}$ we define the Hecke correspondence \n\\begin{equation}\\label{HOat0}\n \\mathfrak{h}(u): {\\rm RZ}_{\\mathfrak{p}_0} \\rightarrow\n {\\rm RZ}_{\\mathfrak{p}_0}, \\quad \n \\mathfrak{h}(u)((Y,\\iota,\\bar{\\lambda}, \\rho) )=\n (Y^u,\\iota^{u},\\bar{\\lambda}, \\rho \\iota(u)). \n\\end{equation} \nThis definition makes sense because of Lemmas \\ref{RZ4l} and\n\\ref{RZ5l}. We note that for $v \\in \\mathcal{H}_{\\mathfrak{p}_0}$ we obtain\n$(Y^{u})^{v} = Y^{vu}$, $\\iota(u) \\iota^{u}(v) = \\iota(vu)$.\nThe map $\\mathfrak{h}(u)$ is the identity on ${\\rm RZ}_{\\mathfrak{p}_0}$\nif $u \\in O_{B_{\\mathfrak{p}_0}}^{\\times}$ because \n$\\iota(u): (Y^u,\\iota^u,\\bar{\\lambda}) \\rightarrow (Y,\\iota,\\bar{\\lambda})$\nis then an isomorphism.\n\nIf $u = (u_1, u_2) \\in \\mathcal{H}_{\\mathfrak{p}_0}$, then the Hecke operator\ninduces maps\n\\begin{equation}\\label{RZ13e}\n \\mathfrak{h}((u_1,u_2)): {\\rm RZ}_{\\mathfrak{p}_0}(a, b) \\rightarrow\n {\\rm RZ}_{\\mathfrak{p}_0}(a + \\ord_{B_{\\mathfrak{q}_0}} u_1, \\; \n b + \\ord_{B_{\\bar{\\mathfrak{q}}_0}} u_2). \n \\end{equation}\nWe conclude that $\\mathcal{H}_{\\mathfrak{p}_0}$ acts transitively on the\nset of subspaces $\\{ {\\rm RZ}_{\\mathfrak{p}_0}(a,b) \\}$ in the decomposition\n(\\ref{RZ12e}). Indeed, if $c + d$ is an even sum of integers we can find\n$u_1 \\in B_{\\mathfrak{q}_0}^{\\times}$ and $f \\in F_{\\mathfrak{p}_0}^{\\times}$ such that\n$\\ord_{B_{\\mathfrak{q}_0}} u_1 = c$ and $\\ord_{B_{\\mathfrak{q}_0}} f = c + d$. We define\n$u_2$ by the equation $u_2^{\\star} u_1 = f$. Then the right hand side of\n(\\ref{RZ13e}) becomes ${\\rm RZ}_{\\mathfrak{p}_0}(a +c, b + d)$. \n\nIf we use the right action of $B_{\\mathfrak{p}_0}$ on $V_{\\mathfrak{p}_0}$, \nwe can write \n\\begin{displaymath}\n G^\\bullet_{\\mathfrak{p}_0} = \\{g \\in B_{\\mathfrak{p}_0}^{{\\rm opp}} \\mid\n g g' \\in F^{\\times}_{\\mathfrak{p}_0}\\; \\} .\n \\end{displaymath}\nThe anti-isomorphism\n\\begin{equation}\\label{Hecke12e}\n \\begin{array}{ccc} \n B_{\\mathfrak{q}_0} \\times B_{\\bar{\\mathfrak{q}}_0} & \\rightarrow & \n B^{{\\rm opp}}_{\\mathfrak{q}_0} \\times B^{{\\rm opp}}_{\\bar{\\mathfrak{q}}_0}\\\\\n (b_1,b_2) & \\mapsto & (b_1, (b_2^{\\star})') \n \\end{array}\n\\end{equation}\ndefines an anti-isomorphism\n$\\mathcal{H}_{\\mathfrak{p}_0} \\rightarrow G^{\\bullet}_{\\mathfrak{p}_0}$.\nTherefore $G^{\\bullet}_{\\mathfrak{p}_0}$ acts from the right on\n${\\rm RZ}_{\\mathfrak{p}_0}$. We write this action\n\\begin{equation}\\label{Heckep0e} \n (Y,\\iota,\\bar{\\lambda}, \\rho) \\mapsto (Y,\\iota,\\bar{\\lambda}, \\rho)_{\\mid g},\n \\quad g \\in G^{\\bullet}_{\\mathfrak{p}_0}. \n \\end{equation}\nFrom the properties of the action of $\\mathcal{H}_{\\mathfrak{p}_0}$ above,\nwe conclude that\n$\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_0} \\subset G^{\\bullet}_{\\mathfrak{p}_0}$ acts trivially\non ${\\rm RZ}_{\\mathfrak{p}_0}$. Therefore the group \n$G^{\\bullet}_{\\mathfrak{p}_0} \/ \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_0}$ acts\non ${\\rm RZ}_{\\mathfrak{p}_0}$. This group isomorphic to $\\mathbb{Z}^2$ and\nacts simply transitively on the set of subspaces\n$\\{ {\\rm RZ}_{\\mathfrak{p}_0}(a,b) \\}$ in the decomposition (\\ref{RZ12e}). \n\n\nWe denote by $\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2$ the Drinfeld upper half plane\nover $\\Spf O_{F_{\\mathfrak{p}_0}}$. This is a regular formal scheme of dimension $2$,\ncomp. \\cite{Dr}, \\cite{RZ}, \\cite[\\S 5.1]{KRZ}. The formal scheme\n$\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\\Spf O_{\\breve{E}_\\nu}$ \nrepresents the Drinfeld functor $\\mathcal{M}_{\\mathrm{Dr}}(0)$ whose points with values in\na scheme $S$ over $\\Spf O_{\\breve{E}_\\nu}$ are given by pairs\n$(Y_{\\mathfrak{q}_0},\\rho_{\\mathfrak{q}_0})$, where $Y_{\\mathfrak{q}_0}$ is a special\nformal $O_{B_{\\mathfrak{q}_0}}$-module over $S$ and $\\rho$ a quasi-isogeny of\nspecial formal $O_{B_{\\mathfrak{q}_0}}$-modules of height zero,\n\\begin{displaymath}\n\\rho_{\\mathfrak{q}_0} : Y_{\\mathfrak{q}_0, \\bar{S}} \\rightarrow\n \\mathbb{X}_{\\mathfrak{q}_0} \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S}. \n\\end{displaymath}\nWe denote by $\\tilde{\\mathcal{M}}_{\\mathrm{Dr}}$ the functor whose points are\ngiven by pairs $(Y_{\\mathfrak{q}_0},\\rho_{\\mathfrak{q}_0})$ where $\\rho_{\\mathfrak{q}_0}$\nis allowed to have arbitrary height. Note that\n$\\mathrm{height}_{O_{F_{\\mathfrak{p}_0}}} \\rho_{\\mathfrak{q}_0} = 2a$ is automatically\neven. We obtain the decomposition\n\\begin{displaymath}\n \\tilde{\\mathcal{M}}_{\\mathrm{Dr}} = \\coprod_{a \\in \\mathbb{Z}}\n \\mathcal{M}_{\\mathrm{Dr}}(a), \n\\end{displaymath}\ncf. \\cite{KRZ} \\S 5.1. Let $\\mathbf{J}_{\\mathfrak{q}_0}$ be the group of all\nquasi-isogenies \n$\\delta \\in \\End^{o}_{B_{\\mathfrak{q}_0}} \\mathbb{X}_{\\mathfrak{q}_0}$. Then\n$\\mathbf{J}_{\\mathfrak{q}_0} \\cong \\mathrm{GL}_{2}(F_{\\mathfrak{p}_0})$, cf. \\cite{Dr}. \nThis group acts from the left on $\\tilde{\\mathcal{M}}_{\\mathrm{Dr}}$ by\nchanging $\\rho_{\\mathfrak{q}_0}$ to $\\delta \\rho_{\\mathfrak{q}_0}$.\n\nLet $\\Pi \\in O_{B_{\\mathfrak{q}_0}}$ be a prime element. The Hecke operator\n $\\mathfrak{h}(\\Pi)$ in the sense of \\cite{KRZ} (5.1.16) \nacts on $\\tilde{\\mathcal{M}}_{\\mathrm{Dr}}$ as \n\\begin{equation}\\label{Hecke14e}\n \\mathfrak{h}(\\Pi): \\mathcal{M}_{\\mathrm{Dr}}(a) \\overset{\\sim}{\\longrightarrow}\n \\mathcal{M}_{\\mathrm{Dr}}(a+1), \\quad \n (Y_{\\mathfrak{q}_0},\\rho_{\\mathfrak{q}_0}) \\mapsto \n (Y_{\\mathfrak{q}_0}^{\\Pi}, \\iota_{\\mathbb{X}_{\\mathfrak{q}_0}}(\\Pi) \\circ\n \\rho^{\\Pi}_{\\mathfrak{q}_0}). \n\\end{equation}\nWe define an action of $\\mathbf{J}_{\\mathfrak{q}_0}$ on\n$\\mathcal{M}_{\\mathrm{Dr}}(0)$. For $\\delta \\in \\mathbf{J}_{\\mathfrak{q}_0}$, we set\n\n\\begin{equation}\\label{PGL-O1e}\n \\mathrm{pr}(\\delta) (Y_{\\mathfrak{q}_0},\\rho_{\\mathfrak{q}_0}) =\n \\mathfrak{h}(\\Pi)^{-\\ord_{\\mathfrak{p}_0} \\det \\delta} \\circ \n \\delta(Y_{\\mathfrak{q}_0},\\rho_{\\mathfrak{q}_0}). \n\\end{equation}\nBecause the Hecke operators commute with the action of \n$\\mathbf{J}_{\\mathfrak{q}_0}$ this is an action of the group\n$\\mathbf{J}_{\\mathfrak{q}_0}$. One can easily see that this action of\n$\\mathcal{M}_{\\mathrm{Dr}}(0)$ factors through\n$\\mathbf{J}_{\\mathfrak{q}_0} = \\mathrm{GL}_{2}(F_{\\mathfrak{p}_0}) \\rightarrow\n{\\mathrm{PGL}}_{2}(F_{\\mathfrak{p}_0})$.\nUsing (\\ref{Hecke14e}) as identifications we obtain an isomorphism\n\\begin{equation}\\label{PGL-O2e}\n \\tilde{\\mathcal{M}}_{\\mathrm{Dr}} \\cong \\mathcal{M}_{\\mathrm{Dr}}(0) \\times\n \\mathbb{Z} \\cong (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu}) \\times \\mathbb{Z}. \n\\end{equation}\n\\begin{proposition}\n The isomorphism (\\ref{PGL-O2e}) does not depend on the choice of the prime\n element $\\Pi$. \n The action of $\\mathbf{J}_{\\mathfrak{q}_0}$ on the left hand side induces on the \n right hand side\n \\begin{displaymath}\n \\delta (\\omega, m) = (\\mathrm{pr}(\\delta) \\omega, \\; \n \\ord_{\\mathfrak{p}_0} \\det \\delta+m), \\quad \\delta\\in \\mathbf{J}_{\\mathfrak{q}_0},\n \\; \\omega \\in \\mathcal{M}_{\\mathrm{Dr}}(0), \\; m \\in \\mathbb{Z}, \n \\end{displaymath}\n cf. (\\ref{PGL-O1e}).\n\n In the next section we will write\n $\\delta \\omega := \\mathrm{pr}(\\delta) \\omega$. \n \\end{proposition}\n\\begin{proof}\nThis is clear. \n \\end{proof}\n\n\\begin{lemma}\\label{RZ7l} \n There is a canonical isomorphism of functors over $\\Spf O_{\\breve{E}_\\nu}$ \n \\begin{displaymath}\n {\\rm RZ}_{\\mathfrak{p}_0}(0,0) \\isoarrow \\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\\Spf O_{\\breve{E}_\\nu}. \n \\end{displaymath}\n\\end{lemma}\n\\begin{proof}\n We begin with a general remark which is useful later on. We consider the\n isomorphism of rings \n \\begin{equation}\\label{RZ9e} \n \\begin{array}{ccc} \n B_{\\mathfrak{p}_0} = B_{\\mathfrak{q}_0} \\times B_{\\bar{\\mathfrak{q}}_0} & \n \\overset{\\sim}{\\longrightarrow} & B_{\\mathfrak{q}_0} \\times\n B_{\\mathfrak{q}_0}^{\\mathrm{opp}}. \\\\\n \\quad (b,c) & \\mapsto & (b, c^{\\star}) \n \\end{array}\n \\end{equation}\n The involution $\\star$ on $B_{\\mathfrak{p}_0}$ induces on the right hand side\n the involution $(b_1, b_2) \\mapsto (b_2, b_1)$. The maximal orders defined on each\n side (cf. (\\ref{Dsplit-in0})) are mapped isomorphically to each other.\n Consider a point $(Y, \\iota, \\bar{\\lambda}) \\in \\mathcal{P}_{\\mathfrak{p}_0}(S)$. We\n choose $\\lambda \\in \\bar{\\lambda}$. It defines an isomorphism\n $\\lambda: Y_{\\mathfrak{q}_0} \\rightarrow (Y_{\\bar{\\mathfrak{q}}_0})^{\\wedge}$. \n It becomes an isomorphism of $O_{B_{\\mathfrak{q}_0}}$-modules if we consider\n $(Y_{\\bar{\\mathfrak{q}}_0})^{\\wedge}$ as an $O_{B_{\\mathfrak{q}_0}}$-module via the\n isomorphism (\\ref{RZ9e}), cf. (\\ref{RZ4e}).\n By the choice of $\\lambda$ we may identify\n $Y_{\\bar{\\mathfrak{q}}_0}$ with the $p$-divisible\n $O_{B_{\\mathfrak{q}_0}}^{\\mathrm{opp}}$-module $(Y_{\\mathfrak{q}_0})^{\\wedge}$. The\n $O_{F_{\\mathfrak{p}_0}}^{\\times}$-homogeneous polarization on\n $Y \\cong Y_{\\mathfrak{q}_0} \\oplus (Y_{\\mathfrak{q}_0})^{\\wedge}$ becomes the\n polarization induced by\n $\\ensuremath{\\mathrm{id}}\\xspace: Y_{\\mathfrak{q}_0} \\rightarrow ((Y_{\\mathfrak{q}_0})^{\\wedge})^{\\wedge}$.\n We call this the canonical polarization.\n Let us denote by $B^{o}_{\\mathfrak{p}_0}$ the right hand side of (\\ref{RZ9e})\n with its involution. Then we may describe an object of\n $\\mathcal{P}_{\\mathfrak{p}_0}(S)$ as a triple \n $(Y = Y_{\\mathfrak{q}_0} \\oplus Y_{\\mathfrak{q}_0}, \\iota, \\bar{\\lambda})$,\n where $Y_{\\mathfrak{q}_0}$ is a special formal $O_{B_{\\mathfrak{q}_0}}$-module, where \n $\\iota: O_{B^o_{\\mathfrak{q}_0}} \\rightarrow \\End Y$ is the natural action\n and where $\\bar{\\lambda}$ is the $O_{F_{\\mathfrak{p}_0}}^{\\times}$-homogeneous class\n of the canonical polarization. \n\n Now we remark that for a point $(Y, \\iota, \\bar{\\lambda}, \\rho)$ of \n ${\\rm RZ}_{\\mathfrak{p}_0}(0,0) (S)$ there is a unique\n $\\lambda \\in \\bar{\\lambda}$ which makes the diagram (\\ref{RZ14e}) commute\n with $f = 1$. Indeed, one notes that in this diagram it follows that \n $f \\in O_{F_{\\mathfrak{p}_0}}^{\\times}$ if $\\rho_{\\mathfrak{q}_0}$ and\n $\\rho_{\\bar{\\mathfrak{q}}_0}$ are quasi-isogenies of height zero.\n\n Therefore the point is uniquely determined by\n $(Y_{\\mathfrak{q}_0}, \\iota_{\\mathfrak{q}_0}, \\rho_{\\mathfrak{q}_0})$. This proves\n the Lemma. \n \\end{proof} \n\n \nWe introduce the group $\\mathbf{I}_{\\mathfrak{p}_0}$ of all quasi-isogenies \n$\\gamma\\colon (\\mathbb{X},\\iota_{\\mathbb{X}})\\rightarrow (\\mathbb{X},\\iota_{\\mathbb{X}})$\nwhich respect the polarization\n$\\lambda_{\\mathbb{X}}$ up to a factor in $F_{\\mathfrak{p}_0}^{\\times}$. If we denote\nby $\\gamma \\mapsto \\gamma'$ the involution on $\\End^{o} \\mathbb{X}$ induced by\n$\\lambda_{\\mathbb{X}}$, we obtain\n\\begin{equation}\\label{RZ17e}\n \\mathbf{I}_{\\mathfrak{p}_0} = \\{\\gamma \\in \\End^{o}_{B_{\\mathfrak{p}_0}} \n\\mathbb{X} \\; | \\; \\gamma' \\gamma \\in F_{\\mathfrak{p}_0}^{\\times} \\} .\n\\end{equation}\nThe group $\\mathbf{I}_{\\mathfrak{p}_0}$ acts from the left on the functor\n${\\rm RZ}_{\\mathfrak{p}_0}$,\n\\begin{displaymath}\n (Y,\\iota, \\bar{\\lambda},\\rho) \\mapsto (Y,\\iota, \\bar{\\lambda}, \\gamma\\rho),\n \\qquad \\gamma \\in \\mathbf{I}_{\\mathfrak{p}_0}. \n \\end{displaymath}\nThis action commutes with the action of $G^{\\bullet}_{\\mathfrak{p}_0}$. \n\n\n\nWe make this action more explicit by using the description of the category\n$\\mathcal{P}_{\\mathfrak{p}_0}(S)$ given in the proof of Lemma \\ref{RZ7l}. \nRecall the group $\\mathbf{J}_{\\mathfrak{q}_0}$ of all quasi-isogenies\n$\\delta \\in \\End^{o}_{B_{\\mathfrak{q}_0}} \\mathbb{X}_{\\mathfrak{q}_0}$. Then an \nelement $\\delta_2 \\in \\mathbf{J}^{\\mathrm{opp}}_{\\mathfrak{q}_0}$ acts on\n$\\mathbb{X}^{\\wedge}_{\\mathfrak{q}_0}$ by\n$\\iota_{\\mathbb{X}_{\\mathfrak{q}_0}}(\\delta_2)^{\\wedge}$. \nWe conclude that \n\\begin{displaymath}\n \\mathbf{I}_{\\mathfrak{p}_0} = \\{(\\delta_1, \\delta_2) \\in\n \\mathbf{J}_{\\mathfrak{q}_0} \\times \\mathbf{J}_{\\mathfrak{q}_0}^{{\\rm opp}} \\; | \\;\n \\delta_1 \\delta_2 \\in F_{\\mathfrak{p}_0}^{\\times} \\} .\n \\end{displaymath}\n\nIf we replace in (\\ref{RZ13e}) $(u_1, u_2)$ by any other\n$(v_1,v_2) \\in \\mathcal{H}_{\\mathfrak{p}_0}$ such that\n$\\ord_{B_{\\mathfrak{q}_0}} u_i = \\ord_{B_{\\mathfrak{q}_0}} v_i$ for $i=1,2$, we obtain the\nsame morphism. Using this morphism as an identification of both sides of\n(\\ref{RZ13e}), we obtain an isomorphism\n\\begin{equation}\\label{RZ18e}\n \\mathrm{RZ}_{\\mathfrak{p}_0} = \\mathrm{RZ}_{\\mathfrak{p}_0}(0,0) \\times \\Lambda, \n \\end{equation}\nwhere $\\Lambda = \\{(a,b) \\in \\mathbb{Z}^2 \\; | a+b \\equiv 0 \\mod 2 \\; \\}$.\nCombining this with Lemma \\ref{RZ7l}, we obtain\n\n\\begin{proposition}\\label{RZ7p} \n There is an isomorphism of functors\n \\begin{displaymath}\n (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\n \\Spf O_{\\breve{E}_\\nu}) \\times\n G^{\\bullet}_{\\mathfrak{p}_0} \/ \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_0} \n \\overset{\\sim}{\\longrightarrow}\n{\\rm RZ}_{\\mathfrak{p}_0} \n \\end{displaymath}\n which is equivariant with respect to the actions of\n $G^{\\bullet}_{\\mathfrak{p}_0} \/ \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_0}$\n on both sides. \n\n The right hand side of (\\ref{RZ18e}) can be written as \n $\\mathcal{M}_{\\mathrm{Dr}}(0) \\times \\Lambda$. An element\n $(\\delta_1, \\delta_2) \\in \\mathbf{I}_{\\mathfrak{p}_0}$ then sends \n a point $(\\omega, (m_1,m_2))$ to \n$(\\mathrm{pr}(\\delta_1) \\omega,\\; (m_1 +\\ord_{\\mathfrak{p}_0} \\det \\delta_1, \\; m_2 + \\ord_{\\mathfrak{p}_0} \\det \\delta_2 ))$. \n\n As noted above we will write in the next section\n $\\delta \\omega := \\mathrm{pr}(\\delta) \\omega$. \n\\end{proposition}\n\\begin{proof}\nOnly the last assertion needs a proof. \nWe consider a point $(x, (m_1,m_2))$ from the right hand\nside of (\\ref{RZ18e}), where $x$ corresponds to\n$\\omega = (Y_{\\mathfrak{q}_0} \\iota_{\\mathfrak{q}_0}, \\rho_{\\mathfrak{q}_0}) \\in \\mathcal{M}_{\\mathrm{Dr}}(0)$.\nThe image of a point $(x, (m_1,m_2))$ under the action of\n$(\\delta_1,\\delta_2)$ is computed by looking at $Y_{\\mathfrak{q}_0}$ only.\nBy the description of $\\mathrm{pr}(\\delta_1)$, this shows the result.\n\\end{proof}\n \n\n\\section{The $p$-adic uniformization of Shimura curves}\\label{s:uniform}\nIn this section, we prove the $p$-adic uniformization of the integral model $\\tilde{\\rm Sh}_{\\mathbf{K}^\\bullet}(G^\\bullet, h^\\bullet_D)$, cf. Definition \\ref{tildeSh_D1d}. Here $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}$, with \n$\\mathbf{K}^{\\bullet}_p \\subset G^{\\bullet}(\\mathbb{Q}_p)$ defined by \n(\\ref{BZKpPkt1e}), where\n$\\mathbf{M}^{\\bullet}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. From this, Cherednik uniformization, i.e., Theorem \\ref{MainIntro} will follow. We stress that the Shimura varieties ${\\rm Sh}(G, h)$ and ${\\rm Sh}(G^\\bullet, h)$ from the previous sections will not reappear. \n\nWe consider the functor\n\\begin{displaymath}\n {\\rm RZ}_{p,\\mathbf{K}^{\\bullet}_p} = {\\rm RZ}_{\\mathfrak{p}_0} \\times \\prod_{i=1}^s\n {\\rm RZ}_{\\mathfrak{p}_i,\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}, \n\\end{displaymath}\ncf. Definitions \\ref{RZ4d} and \\ref{RZ2d}. \nEach of these factors is defined by a choice of a framing object\nwhich we denote by $(\\mathbb{X}_i, \\iota_{\\mathbb{X}_i}, \\lambda_{\\mathbb{X}_i})$, for $i=0,\\ldots,s$.\nWe choose the framing objects as follows. We fix a point\n\\begin{equation}\\label{uniform2e} \n (A_{o}, \\iota_{o}, \\bar{\\lambda}_{o},\n \\bar{\\eta}^p_{o}, (\\bar{\\eta}_{\\mathfrak{q}_j,o})_j, \n ( \\xi_{\\mathfrak{p}_j, o})_j) \\in\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(\\bar{\\kappa}_{E_{\\nu}}). \n \\end{equation}\nThe last two data of the point are for $j = 1, \\ldots, s$. Indeed, by $\\mathbf{M}^{\\bullet}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$, the choice of $\\xi_{\\mathfrak{p}_0}$ is redundant. We choose\nan element $\\eta_{o}^p \\in \\bar{\\eta}_{o}^p$. \nWe denote by $\\mathbb{X}$ the $p$-divisible group of $A_{o}$. We set\n$\\mathbb{X}_i = \\mathbb{X}_{\\mathfrak{p}_i}$, with its action\n$\\iota_{\\mathbb{X}_i}$ from $\\iota_{o}$ and a polarization\n$\\lambda_{\\mathbb{X}_i}$ from some element of $\\bar{\\lambda}_{o}$. \n\n\nWe denote by $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet} \/ \\Spf O_{E_{\\nu}}}$, \nresp. $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet} \/ \\Spf O_{\\breve{E}_{\\nu}}}$ the\nrestriction of the functor $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ to\nthe category of schemes over $\\Spf O_{E_{\\nu}}$, resp. $\\Spf O_{\\breve{E}_{\\nu}}$. \nWe define the uniformization morphism of functors on the category of schemes\n$S$ over $\\Spf O_{\\breve{E}_{\\nu}}$ (the definition depends on the choice of\nthe tuple \\eqref{uniform2e}),\n\\begin{equation}\\label{unimorph1e} \n\\tilde\\Theta^\\bullet : {\\rm RZ}_{p,\\mathbf{K}^{\\bullet}_p} \\times\n G^{\\bullet}(\\mathbb{A}_f^p)\/(\\mathbf{K}^{\\bullet})^p \\rightarrow \n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet} {\/ \\Spf O_{\\breve{E}_{\\nu}}}}. \n \\end{equation}\nFor the definition we recall that a point with values in $S$ of the functor\non the left hand side consists the following data\n\\begin{enumerate}\n\\item[({1})] a point $(Y_0, \\iota_0, \\bar{\\lambda}_0, \\rho_0)$ of \n ${\\rm RZ}_{\\mathfrak{p}_0}(S)$, cf. Definition \\ref{RZ4d}, \n\\item[({2})] a point\n$(Y_j,\\iota_j, \\bar{\\lambda}_j, \\bar{\\eta}_{\\mathfrak{q}_j},\\bar{\\xi}_{\\mathfrak{p}_j}, \\rho_j)$\n of ${\\rm RZ}_{\\mathfrak{p}_j, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_j}}(S)$ for\n $j = 1, \\ldots s$, cf. Definition \\ref{RZ2d}, \n\\item[({3})] an element $g \\in G^{\\bullet}(\\mathbb{A}_f^p)$. \n\\end{enumerate}\nHere \n\\begin{equation}\\label{RZ7e}\n\\rho_i: (Y_i,\\iota_i)_{\\bar{S}} \\rightarrow (\\mathbb{X}_i, \\iota_{\\mathbb{X}_i})\n \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S} \n\\end{equation}\nis a quasi-isogeny for $i = 0, \\ldots, s$ which respects the polarizations on\nboth sides up to a factor in $F_{\\mathfrak{p}_i}^{\\times}$. \nWe define as follows an abelian scheme $(\\bar{A}, \\bar{\\iota}_A)$ over $\\bar{S}$ and\nan isogeny\n\\begin{equation}\\label{RZ8e}\n (\\bar{A}, \\iota_{\\bar{A}}) \\rightarrow (A_{o}, \\iota_{o})\n \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S} .\n\\end{equation}\n Let us denote by $\\bar{Y}$ the $p$-divisible group of $\\bar{A}$.\nThen $(\\bar{Y}_{\\mathfrak{p}_i}, \\iota_i)$ is identified with\n$(Y_{i, \\bar{S}}, \\iota_{i,\\bar{S}})$ and the quasi-isogeny (\\ref{RZ8e}) induces\non the $p$-divisible groups the quasi-isogenies (\\ref{RZ7e}). We choose \n$\\lambda_{o} \\in \\bar{\\lambda}_{o}$ and consider the inverse\nimage $\\theta$ of $\\lambda_{o}$ on $\\bar{A}$ by (\\ref{RZ8e}). Let\n$\\bar{\\theta} = F^{\\times} \\theta$ be the $F$-homogeneous polarization it generates.\nBy the definition of the $RZ$-spaces, the polarization induced by $\\theta$ on\nthe $p$-divisible group $Y_{i, \\bar{S}}$ differs from $\\lambda_{i, \\bar{S}}$ by\na factor in $F_{\\mathfrak{p}_i}^{\\times}$. We then define $\\bar{\\lambda}_{\\bar{A}}$ to be the\n$U_p(F)$-homogeneous polarization on $\\bar{A}$ which consists of all elements\nof $\\bar{\\theta}$ which on the $p$-divisible groups $\\bar{Y}_i$ differ\nfrom $\\lambda_{i, \\bar{S}}$ by a factor in $O_{F_{\\mathfrak{p}_i}}^{\\times}$. \n\nSince a lifting\nof the $p$-divisible group of $\\bar{A}$ with these extra structures is given\nby the data $({ 1})$ and $({ 2})$ above, we obtain by the Serre-Tate theorem a lifting $A$ of $\\bar A$ over $S$ with extra\nstructures $\\iota_A$ and $\\bar{\\lambda}_A$. We obtain a point\n\\begin{displaymath}\n (A, \\iota_A, \\bar{\\lambda}_A, \\bar{\\eta}^p_A, (\\bar{\\eta}_{\\mathfrak{q}_j,A})_j,\n (\\bar{\\xi}_{\\mathfrak{p}_j})_j) ,\n\\end{displaymath}\nwhere the last three items are defined as follows. We take the inverse image\nof $\\eta^p_{o}$ by (\\ref{RZ8e}) to obtain $\\eta^p_{\\bar{A}}$. Since we\nhave \\'etale sheaves, this gives $\\eta^p_A$ and then its class $\\bar{\\eta}^p_A$.\nThe last two items are deduced directly from the item $(2)$ above.\nIndeed $T_{\\mathfrak{q}_j}(A) = T_p(Y_{j, \\mathfrak{q}_j})$ and a rigidification in the\nsense of Definition \\ref{RZ1d} is equivalent to\n\\begin{displaymath}\n \\eta_{\\mathfrak{q}_j}: \\Lambda_{\\mathfrak{q}_j} \\isoarrow T_p(Y_{j, \\mathfrak{q}_j}) \n \\mod \\mathbf{K}^{\\bullet}_{\\mathfrak{q}_j}, \n\\end{displaymath}\nand a function \n\\begin{displaymath}\n \\xi_{\\mathfrak{p}_j}: \\bar{\\lambda}_A \\rightarrow\n O_{F_{\\mathfrak{p}_j}}^{\\times} \\mod \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_j}. \n\\end{displaymath}\nThis last function is induced by the injection\n$\\bar{\\lambda}_A\\rightarrow\\bar{\\lambda}_j$ from $({ 2})$. \n\nWe define the image under $\\tilde\\Theta^\\bullet$ in (\\ref{unimorph1e}) of the point given by\nthe data $({1})$, $({ 2})$, $({ 3})$ to be\n\\begin{equation}\\label{unimorph4e}\n (A, \\iota_A, \\bar{\\lambda}_A, \\eta^p_A g, (\\bar{\\eta}_{\\mathfrak{q}_j,A})_j,\n (\\bar{\\xi}_{\\mathfrak{p}_j})_j) .\n\\end{equation}\nFor this we have used our choice of $\\eta_{o}^p$. \n\n\\begin{proposition}\n Recall from (\\ref{Heckep0e}) the Hecke operator action of $G^{\\bullet}_{\\mathfrak{p}_0}$ on\n ${\\rm RZ}_{\\mathfrak{p}_0}$ and from (\\ref{Hecke11e}) the Hecke operator action of\n $G^{\\bullet}_{\\mathfrak{p}_i}$ on \n ${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}$. Together we obtain an action of $G^\\bullet(\\mathbb{A}_f)$ by\n Hecke operators on the left hand side of (\\ref{unimorph1e}). \n \n There is an extension of the Hecke operators $G^{\\bullet}(\\mathbb{A}_f)$\n from the tower $\\mathcal{A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ to the tower \n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ such that the uniformization\n morphism $\\tilde\\Theta^\\bullet$ is compatible with the actions of Hecke operators\n on both sides. \n\\end{proposition}\n\\begin{proof}\n This is trivial for the action of $G^{\\bullet}(\\mathbb{A}_f^{p})$.\n The proof for elements in $G^\\bullet(\\ensuremath{\\mathbb {Q}}\\xspace_p)$ is based on the description of the Hecke operators after the \n proof of Proposition \\ref{BZ11p}, comp. Remark \\ref{remHecke}. For $j=1,\\ldots, s$, the local component at $\\mathfrak{p}_j$ of a Hecke correspondence is described after (\\ref{RZ2e}). \n The local component at $\\mathfrak{p}_0$ of a Hecke correspondence is described by \\eqref{HOat0}. We write here the argument only for the action of \n $g \\in G^{\\bullet}_{\\mathfrak{p}_0} \\subset G(\\mathbb{A}_f)$. We consider a point\n on the left hand side of (\\ref{unimorph1e}) defined by the data\n (1), (2), (3). We may assume that the element of\n (3) is $1$. We take the image by $\\tilde\\Theta^\\bullet$, \n \\begin{equation}\\label{Hecke13e}\n (A, \\iota, \\bar{\\lambda}, \\bar{\\eta}^p, (\\bar{\\eta}_{\\mathfrak{q}_j})_j,\n (\\bar{\\xi}_{\\mathfrak{p}_j})_j) \\in\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(S), \n \\end{equation}\n cf. (\\ref{unimorph4e}). Let $(Y_0, \\iota_0, \\bar{\\lambda}_0, \\rho_0)$\n be the datum ({1}). Then $Y_{\\mathfrak{p}_0}$ is the $\\mathfrak{p}_0$-part\n of the $p$-divisible group of $A$ with the induced action $\\iota_0$ \n and polarization $\\bar{\\lambda}_0$. Let\n $u= (u_1, u_2)\\in B_{\\mathfrak{q}_0}\\times B_{\\bar{\\mathfrak{q}}_0}=B_{\\mathfrak{p}_0}$ \n be the element which corresponds to $g$ by the anti-isomorphism\n (\\ref{Hecke12e}). The Hecke operator $g$ on the left hand side of \\eqref{unimorph1e} is\n given by $(Y^u_0, \\iota^u_0, \\bar{\\lambda}^u_0, \\rho_0 \\circ\\iota_0(u))$. Note that for the underlying polarized $p$-divisble groups we have \n $(Y^u_0, \\lambda^u_0) = (Y_0,\\lambda_0)$. We consider the quasi-isogeny\n \\begin{equation}\\label{Hecke15e}\n \\iota_0(u) : (Y^u_0, \\iota^u_0, \\bar{\\lambda}^u_0) \\rightarrow\n (Y_0, \\iota_0, \\bar{\\lambda}_0).\n \\end{equation}\n We note that\n $\\iota_0(u)^{*}(\\lambda) =\\lambda^u\\circ\\iota_0(u_2^{\\star} u_1) =\\lambda^u\\circ\\mu(g)$. \n Therefore, applying $g$ on the left hand side of (\\ref{unimorph1e}) leads\n on the right hand side to the following point. There is a quasi-isogeny of\n abelian varieties \n \\begin{displaymath}\n\\alpha: (A', \\iota') \\rightarrow (A, \\iota) \n \\end{displaymath}\n which induces on the $\\mathfrak{p}_0$-parts of the $p$-divisible groups the map \n (\\ref{Hecke15e}) and is an isomorphism on the $\\mathfrak{p}_j$-parts\n for $j > 0$. Looking at the $p$-divisible groups, we see that, for the given \n $p$-principal polarization $\\lambda$ on $A$, the polarization\n $\\mu_{\\mathfrak{p}_0}(g)^{-1} \\alpha^{*} (\\lambda)$ on the $\\mathfrak{p}_0$-part\n of the $p$-divisible group is principal and for $j > 0$ the\n $\\mathfrak{p}_j$-parts of $\\alpha^{*} (\\lambda)$ is principal. We define\n $\\bar{\\lambda}'$ on $A'$ as the class of all polarizations in\n $F^{\\times} \\alpha^{*} (\\lambda)$ which are $p$-principal. We define\n all other data $(\\bar{\\eta}')^p$, $\\bar{\\eta}'_{\\mathfrak{q}_j}$, $\\bar{\\xi}_j$\n for $j > 0$ by pull back via $\\alpha$. Then $A'$ with the extra structure\n just introduced is a point of\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(S)$. To see that this point\n represents for an $E_{\\nu}$-scheme $S$ the Hecke operator, we need the\n functions $\\xi_{\\mathfrak{p}_0}$ and $\\xi'_{\\mathfrak{p}_0}$ with values in\n $F^{\\times}_{\\mathfrak{p}_0}\/ O_{F_{\\mathfrak{p}_0}}^{\\times}$. So far it was not necessary\n to mention them because they have value $1$ for a polarization which is\n principal in $p$. Therefore we have\n \\begin{displaymath}\n 1 = \\xi_{\\mathfrak{p}_0}(\\lambda) =\n \\xi'_{\\mathfrak{p}_0}(\\mu_{\\mathfrak{p}_0}(g)^{-1} \\alpha^{*} (\\lambda)). \n \\end{displaymath}\n But then (\\ref{BZGdot5e}) shows that $A'$ gives the Hecke operator of \n $g \\in G^{\\bullet}_{\\mathfrak{p}_0}$.\n \\end{proof}\n\nBy \\cite[6.29]{RZ} the following maps are isomorphisms,\n\\begin{equation}\\label{uniform1e}\n \\begin{array}{ccc} \n\\End^{o}_B(A_{o}) \\otimes_{\\mathbb{Q}} \\mathbb{A}_f^p & \n \\overset{\\sim}{\\longrightarrow} & \n \\End^{o}_{B \\otimes \\mathbb{A}_f^{p}} \\mathrm{V}^p(A_{o})\\\\[2mm] \n \\End^{o}_B(A_{o}) \\otimes_{\\mathbb{Q}} \\mathbb{Q}_p & \n \\overset{\\sim}{\\longrightarrow} & \n \\End^{o}_{B \\otimes \\mathbb{Q}_p} \\mathbb{X} .\\\\ \n \\end{array}\n\\end{equation}\nHere, as above, $\\mathbb{X} = \\prod_{i=0}^{s} \\mathbb{X}_i$ is the $p$-divisible group\nof $A_{o}$. We obtain\n\\begin{displaymath}\n\\End^{o}_{B \\otimes \\mathbb{Q}_p} \\mathbb{X} \\cong \\prod_{i=0}^{s}\n\\End^o_{B_{\\mathfrak{p}_i}} \\mathbb{X}_i\n \\end{displaymath}\nWe denote by $\\mathbf{I}$ the algebraic group of all $B$-linear quasi-isogenies\n$A_{o} \\rightarrow A_{o}$ which respect the polarization\n$\\lambda_{o}$ up to a constant in $F^{\\times}$. Let\n$\\gamma \\mapsto \\gamma'$ be the Rosati involution on \n$\\End_B^o A_{o}$ induced by $\\lambda_{o}$. We can write\n\\begin{equation}\\label{groupI1e}\n \\mathbf{I}(\\mathbb{Q}) = \\{\\alpha \\in \\End_B^o A_{o} \\; | \\;\n \\alpha' \\alpha \\in F^{\\times} \\}. \n\\end{equation}\nIn the framing object (\\ref{uniform2e}) we choose\n$\\eta^p_{o} \\in \\bar{\\eta}^p_{o}$. This isomorphism\n$\\eta^p_{o}:\\mathrm{V}(A_{o})\\isoarrow V\\otimes\\mathbb{A}_f^p$\ninduces by (\\ref{uniform1e}) an isomorphism\n\\begin{equation}\n \\mathbf{I}(\\mathbb{A}_f^p) \\overset{\\sim}{\\longrightarrow}\n\\{\\gamma \\in \\End^{o}_{B\\otimes \\mathbb{A}_f^p} \\mathrm{V}^p(A_{o}) \n\\; | \\; \\gamma' \\gamma \\in (F\\otimes\\ensuremath{\\mathbb {A}}\\xspace_f)^\\times \\}\n \\overset{\\sim}{\\longrightarrow}\n G^{\\bullet}(\\mathbb{A}_f^p). \n \\end{equation}\nWe also denote by $\\gamma \\mapsto \\gamma'$ the involution on\n$\\End^{o}_{B \\otimes \\mathbb{Q}_p} \\mathbb{X}$ induced by the polarization\n$\\lambda_{\\mathbb{X}}$. We define\n\\begin{displaymath}\n \\mathbf{I}_{\\mathfrak{p}_0} = \\{\\gamma \\in \\End^{o}_{B_{\\mathfrak{p}_0}}\n\\mathbb{X}_0 \\; | \\; \\gamma' \\gamma \\in F_{\\mathfrak{p_0}}^{\\times} \\}, \n\\end{displaymath}\ncf. (\\ref{RZ17e}). If $j> 0$ we can take\n$(\\Lambda_{\\mathfrak{p}_j}^{pd}, \\lambda_{\\psi})$ for the framing object\n$\\mathbb{X}_j$, cf. (\\ref{RZ1e}). Using\nthe definition (\\ref{RZ21e}) of $\\lambda_{\\psi}$ we obtain an isomorphism\n\\begin{equation}\\label{groupI2e}\n \\begin{array}{ll} \n \\mathbf{I}_{\\mathfrak{p}_j} & \\cong \\{\\gamma\\in\n \\End_{B_{\\mathfrak{p}_j}} V_{\\mathfrak{p}_i}\n\\; | \\; \\psi(\\gamma v_1, \\gamma v_2) = \\psi (f v_1, v_2), \\;\n\\text{for some} \\; f \\in F_{\\mathfrak{p}_j}^{\\times} \\} \\\\ \n& = G^{\\bullet}_{\\mathfrak{p}_j} . \n \\end{array}\n\\end{equation}\n\n\nBy (\\ref{uniform1e}) we obtain\n\\begin{displaymath}\n\\mathbf{I}(\\mathbb{Q}_p) = \\prod_{i=0}^{s} \\mathbf{I}_{\\mathfrak{p}_i}. \n \\end{displaymath}\nThe group $\\mathbf{I}_{\\mathfrak{p}_i}$ acts from the left on the functor\n${\\rm RZ}_{\\mathfrak{p}_i}$, for $i = 0, \\ldots, s$ by\n\\begin{displaymath}\n \\alpha_{\\mathfrak{p}_i} \\in \\mathbf{I}_{\\mathfrak{p}_i}:\\quad (Y_{i}, \\iota_{i}, \\bar{\\lambda}_{i}, \\bar{\\eta}_{\\mathfrak{q}_i},\n \\bar{\\xi}_{\\mathfrak{p}_i} , \\rho_{i}) \\mapsto (Y_{i}, \\iota_{i},\n \\bar{\\lambda}_{i} \\bar{\\eta}_{\\mathfrak{q}_i},\n \\bar{\\xi}_{\\mathfrak{p}_i} , \\alpha_{\\mathfrak{p}_i} \\rho_{i}) .\n \\end{displaymath}\n This makes sense\nbecause\n$\\alpha_{\\mathfrak{p}_i}: (\\mathbb{X}_i,\\iota_i)\\rightarrow (\\mathbb{X}_i,\\iota_i)$\nis a quasi-isogeny which respects the polarization $\\lambda_{\\mathbb{X}_i}$ up to\nconstant. Note that for $i = 0$ the data\n$\\bar{\\eta}_{\\mathfrak{q}_i}, \\bar{\\xi}_{\\mathfrak{p}_i}$ are absent. \n\nThe group $\\mathbf{I}(\\mathbb{Q})$ acts on the left hand side of\n(\\ref{unimorph1e}). If $\\alpha \\in \\mathbf{I}(\\mathbb{Q})$, with components\n$\\alpha_{\\mathfrak{p}_i} \\in \\mathbf{I}_{\\mathfrak{p}_i}$ and\n$\\alpha^p \\in \\mathbf{I}(\\mathbb{A}^p_f) \\cong G^{\\bullet}(\\mathbb{A}^p_f)$, then \na point of the left hand side given by the data {1}, {2}, {3}\nis mapped to the data\n\\begin{equation}\\label{unimorph2e}\n\\big((Y_{i}, \\iota_{i}, \\bar{\\lambda}_{i} \\bar{\\eta}_{\\mathfrak{q}_i},\n \\bar{\\xi}_{\\mathfrak{p}_i} , \\alpha_{\\mathfrak{p}_i} \\rho_{i}), \\,\\, \\alpha^p g\\big). \n \\end{equation}\nWith respect to this action the morphism (\\ref{unimorph1e}) is equivariant.\nTo see this, we consider the morphism derived from (\\ref{RZ8e}) \n\\begin{equation}\n(\\bar{A}, \\iota_{\\bar{A}}) \\rightarrow (A_{o}, \\iota_{o})\n \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S} \\overset{\\alpha}{\\longrightarrow}\n (A_{o}, \\iota_{o}) \\times_{\\Spec \\bar{\\kappa}_{E_{\\nu}}} \\bar{S} \n \\end{equation}\nThis composite may be used to compute the image of the data (\\ref{unimorph2e}) \nby (\\ref{unimorph1e}). One can easily see that this gives the same point in \n$\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}(S)$ as for the original data. \n\\begin{proposition}\n Let $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}, where\n $\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. \n Let $\\tilde{{\\mathsf A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ be the coarse moduli\n scheme of $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$. \n The morphism (\\ref{unimorph1e}) induces an isomorphism of formal schemes \n \\begin{equation}\\label{unimorph5e} \n \\Theta^\\bullet: \\mathbf{I}(\\mathbb{Q}) \\backslash ({\\rm RZ}_{p,\\mathbf{K}^{\\bullet}_p}\n \\times G^{\\bullet}(\\mathbb{A}_f^p)\/(\\mathbf{K}^{\\bullet})^p) \\rightarrow \n \\tilde{{\\mathsf A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}{\/ \\Spf O_{\\breve{E}_{\\nu}}}}. \n \\end{equation}\n The morphism is compatible with the Weil descent data on both sides as\n spelled out in the proof. \n\\end{proposition}\n\\begin{proof}\n This is a variant of the general uniformization theorem \\cite[6.30]{RZ}. By Proposition \\ref{BZ8p}, if\n $\\mathbf{K}^{\\bullet}$ is small enough, the morphism\n \\begin{displaymath}\n \\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\rightarrow\n \\tilde{{\\mathsf A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \n \\end{displaymath}\n is the etale sheafification. Therefore we can use deformation theory as\n in \\cite[6.23]{RZ} to show that $\\Theta^\\bullet$ is \\'etale. For this one needs that\n the action of $\\mathbf{I}(\\ensuremath{\\mathbb {Q}}\\xspace)$ is fixpoint free, if $\\mathbf{K}^{\\bullet, p}$ is\n small enough. We refer to the argument in loc.cit. for details. \n In addition to the arguments given in \\cite{RZ}, one needs that $\\Theta^\\bullet$ is\n surjective on the $\\bar{\\kappa}_{E_{\\nu}}$-valued points. This follows from \n the Hasse principle for $G^{\\bullet}$ as explained in \\cite[Prop. 7.1.11, Prop. 7.3.2]{KRZ}. We omit the details.\n If we drop the smallness assumption on $\\mathbf{K}^{\\bullet, p}$, it follows from Proposition \\ref{BZ8p}\n that $\\Theta^\\bullet$ is an isomorphism for the normal subgroup\n $\\mathbf{K}^{\\bullet}_U \\subset \\mathbf{K}^{\\bullet}$. Dividing by the\n action of $\\mathbf{K}^{\\bullet}$ we obtain that $\\Theta^\\bullet$ is an isomorphism. \n\n\nBoth sides of (\\ref{unimorph5e}) are endowed with a Weil descent datum\nrelative to the extension $O_{\\breve{E}_{\\nu}} \/ O_{E_{\\nu}}$. Because the right\nhand side is obtained by a base change\n$\\Spf O_{\\breve{E}_{\\nu}} \\rightarrow \\Spf O_{E_{\\nu}}$, we have there the\nnatural Weil descent datum. On the left hand side the Weil descent\ndatum is induced by a Weil descent datum of ${\\rm RZ}_{p,\\mathbf{K}^{\\bullet}_p}$. It is\ndefined on each ${\\rm RZ}_{\\mathfrak{p}_i}$ as follows. Let\n$\\tau \\in \\Gal(\\breve{E}_{\\nu}\/E_{\\nu})$ be the Frobenius. Let $R$ be\n$O_{\\breve{E}_{\\nu}}${\\rm -Alg}ebra such that $p$ is nilpotent in $R$. We give\nthe descent datum as a functorial map \n\\begin{equation}\\label{uniform3e}\n \\omega_{\\mathfrak{p}_i}(R): {\\rm RZ}_{\\mathfrak{p}_i}(R) \\rightarrow\n {\\rm RZ}_{\\mathfrak{p}_i}(R_{[\\tau]}). \n\\end{equation}\nFor this we write $\\bar{R} = R \\otimes_{O_{\\breve{E}_{\\nu}}} \\bar{\\kappa}_{E_{\\nu}}$ and\nwe denote by $\\varepsilon: \\bar{\\kappa}_{E_{\\nu}} \\rightarrow \\bar{R}$ the\nstructure morphism. Then (\\ref{uniform3e}) maps a point \n$(Y_i,\\iota_i, \\bar{\\lambda}_i, \\bar{\\eta}_{\\mathfrak{q}_i},\\bar{\\xi}_{\\mathfrak{p}_i}, \\rho_i)$\nfrom the right hand side of (\\ref{uniform3e}) to the point given by the\nsame data except the $\\rho_i$ is replaced by the composite \n\\begin{displaymath}\nY_i \\times_{\\Spec R} \\Spec \\bar{R} \\overset{\\rho_i}{\\longrightarrow}\n\\varepsilon_{*} \\mathbb{X}_i\n\\overset{\\varepsilon_{*} F_{\\mathbb{X}, \\tau}}{\\longrightarrow}\n\\varepsilon_{*} \\tau_{*} \\mathbb{X}_i. \n\\end{displaymath}\nWe denote here by $F_{\\mathbb{X}, \\tau}$ the Frobenius morphism of the\n$p$-divisible group relative to $\\kappa_{E_{\\nu}}$.\nThe compatibility of the Weil descent data is explained in the proof\nof \\cite[Lem. 7.3.1]{KRZ}. \n\\end{proof}\nFor the following one should keep in mind that in (\\ref{uniform3e}) we have\nused $\\tau$ to describe the Weil descent datum and not $\\tau^{-1}$ as e.g.\nin Proposition \\ref{Sh_D1p}. \nRecall from Definition \\ref{tildeSh_D1d} that the scheme $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$\nover $O_{E_{\\nu}}$ is a Galois twist of\n$\\tilde{{\\mathsf A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ according to the following\ndiagram \n\\begin{equation}\\label{uniform5e}\n\\begin{aligned}\n\\xymatrix{\n \\tilde{\\mathsf A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}}\n \\ar[d]_{\\dot{z}_{\\mid \\xi} \\times \\tau_c} \\ar[r] & \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})\n \\times_{\\Spec O_{E_{\\nu}}} \\Spec O_{E^{nr}_{\\nu}} \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\\\\\n \\tilde{\\mathsf A}^{\\bullet t}_{\\mathbf{K}^{\\bullet}} \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} \\ar[r] &\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \\times_{\\Spec O_{E_{\\nu}}}\n \\Spec O_{E^{nr}_{\\nu}} ,\n }\n \\end{aligned}\n\\end{equation}\nwhere the horizontal arrows are isomomorphisms. \nIndeed, the morphism (\\ref{BZGdot10e}) becomes an isomorphism if we replace on the\nleft hand side $\\mathcal{A}$ by $\\mathsf A$. Therefore it follows\nfrom Proposition \\ref{Sh_D1p} that\n\\begin{equation}\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \\times_{\\Spec O_{E_{\\nu}}} \n \\Spec E_{\\nu} \\cong {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})_{E_{\\nu}}.\n \\end{equation}\n\n\\begin{theorem}\\label{4epeg1t}\n Let $\\mathbf{K}^{\\bullet}=\\mathbf{K}^{\\bullet}_p\\mathbf{K}^{\\bullet,p}\\subset G^\\bullet(\\ensuremath{\\mathbb {A}}\\xspace_f)$, with $\\mathbf{K}^{\\bullet}_p$ as in \\eqref{BZKpPkt1e}, where\n $\\mathbf{M}_{\\mathfrak{p}_0} = O_{F_{\\mathfrak{p}_0}}^{\\times}$. \n Let $\\Pi \\in D_{\\mathfrak{p}_0}$ be a prime element of this division\n algebra. It acts on\n $V_{\\mathfrak{p}_0} = D^{{\\rm opp}}_{\\mathfrak{p}_0} \\otimes_{F_{\\mathfrak{p}_0}} K_{\\mathfrak{p}_0}$ \n by multiplication with $\\Pi \\otimes 1$ from the right. This defines an\n element of $G^{\\bullet}_{\\mathfrak{p}_0}$ which we denote simply by $\\Pi$ and\n we use this notation also for its image by\n $G^{\\bullet}_{\\mathfrak{p}_0} \\subset G^{\\bullet}(\\mathbb{A}_f)$. The action of\n the Hecke operator is denoted by $\\mid_{\\Pi}$. \n\n Let $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ be the integral model over \n $O_{E_{\\nu}}$ of Definition \\ref{tildeSh_D1d}\n of the Shimura variety ${\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$. We denote by\n $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{E_{\\nu}}}$ the\n $p$-adic completion and we set \n \\[ \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}} = \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{E_{\\nu}}} \\times_{\\Spf O_{E_{\\nu}}} \\Spf O_{\\breve{E}_{\\nu}}.\n \\]\nThen there is an isomorphism of formal schemes\n \\begin{equation}\\label{unimorph6e} \n \\mathbf{I}(\\mathbb{Q}) \\backslash (\\hat{\\Omega}^2_{E_{\\nu}} \\times\n G^{\\bullet}(\\mathbb{A}_f)\/\\mathbf{K}^{\\bullet})\n \\times_{\\Spf O_{E_{\\nu}}} \\Spf O_{\\breve{E}_{\\nu}} \\overset{\\sim}{\\longrightarrow}\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}} \n \\end{equation}\n For varying $\\mathbf{K}^{\\bullet}$ this morphism is compatible with the action\n of $G^{\\bullet}(\\mathbb{A}_f)$ by Hecke operators on both sides. \n \n Let $\\tau \\in \\Gal(\\breve{E}_{\\nu}\/ E_{\\nu})$ the Frobenius and \n $\\tau_c = \\Spf\\tau^{-1}: \\Spf O_{\\breve{E}_{\\nu}}\\rightarrow\\Spf O_{\\breve{E}_{\\nu}}$.\n The canonical Weil descent datum on the right hand side of (\\ref{unimorph6e})\n is given on the left hand side by the commutative diagram \n \\begin{displaymath}\n\\xymatrix{\n \\mathbf{I}(\\mathbb{Q}) \\backslash (\\hat{\\Omega}^2_{E_{\\nu}} \\times\n G^{\\bullet}(\\mathbb{A}_f)\/\\mathbf{K}^{\\bullet})\n \\times_{\\Spf O_{E_{\\nu}}} \\Spf O_{\\breve{E}_{\\nu}}\n \\ar[d]_{ \\mid_{\\Pi^{-1}}\\times\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c} \\ar[r] & \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}}\n \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\\\\\n \\mathbf{I}(\\mathbb{Q}) \\backslash (\\hat{\\Omega}^2_{E_{\\nu}} \\times\n G^{\\bullet}(\\mathbb{A}_f)\/\\mathbf{K}^{\\bullet})\n \\times_{\\Spf O_{E_{\\nu}}} \\Spf O_{\\breve{E}_{\\nu}}\n \\ar[r] & \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}}\n }\n \\end{displaymath}\n\\end{theorem}\n\\begin{proof}\n From the morphism $\\Theta^\\bullet$ in (\\ref{unimorph5e}) and the definition\n and the horizontal line of the diagram (\\ref{uniform5e}) \nwe obtain an isomorphism of formal schemes over $\\Spf O_{\\breve{E}_{\\nu}}$ \n \\begin{equation}\\label{4epeg1e} \n\\mathbf{I}(\\mathbb{Q}) \\backslash ({\\rm RZ}_{p,\\mathbf{K}^{\\bullet}_p}\n\\times G^{\\bullet}(\\mathbb{A}_f^p)\/(\\mathbf{K}^{\\bullet})^p)\n\\overset{\\sim}{\\longrightarrow}\n\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}}. \n \\end{equation} \nWe obtain the isomorphism (\\ref{unimorph6e}) if we rewrite the left hand side\nusing the Propositions \\ref{RZ6p} and \\ref{RZ7p}. \nWe consider an $R$ and $\\varepsilon$\n as in the definition (\\ref{uniform3e}) of $\\omega_{\\mathfrak{p}_i}$. \n Let us denote the functor on the left hand side of (\\ref{4epeg1e})\n by $\\mathcal{F}$. It is endowed with its natural Weil descent datum\n $\\mathcal{F}(R) \\rightarrow \\mathcal{F}(R_{[\\tau]})$ given by the \n $\\omega_{\\mathfrak{p}_i}$, $i = 0, \\ldots, s$. By \\eqref{uniform5e}, the \n morphism (\\ref{4epeg1e}) becomes compatible with the Weil descent \n data if we multiply the natural Weil descent datum on $\\mathcal{F}$ by the\n operator \n $\\dot{z}_{\\mid \\xi}^{-1}$. The exponent $-1$ appears because we use here\n $\\tau$ instead of $\\tau^{-1}$ as in the statement of the theorem. By the\n explanation after (\\ref{xi-action1e}) \n this means that we have to replace $\\omega_{\\mathfrak{p}_o}$ by\n $\\omega_{\\mathfrak{p}_0}(1, \\pi_{\\mathfrak{p}_0}p^{-f_{\\nu}})$, resp.\n $\\omega_{\\mathfrak{p}_j}$ by $\\omega_{\\mathfrak{p}_j}(1, p^{-f_{\\nu}})$, where\n $(1, \\pi_{\\mathfrak{p}_0}p^{-f_{\\nu}}) \\in G^{\\bullet}_{\\mathfrak{p}_0}$ and\n $(1, p^{-f_{\\nu}}) \\in G^{\\bullet}_{\\mathfrak{p}_j}$, cf. Proposition \\ref{Sh_D1p}.\n \n We first check what this modified Weil descent datum does on\n ${\\rm RZ}_{\\mathfrak{p}_0}$. Let \n $(Y,\\iota, \\bar{\\lambda}, \\rho) \\in {\\rm RZ}_{\\mathfrak{p}_0}(R)$. The\n action of the Hecke operator $(1, \\pi_{\\mathfrak{p}_0}p^{-f_{\\nu}})$ is the\n same as $\\mathfrak{h}((1, \\pi_{\\mathfrak{p}_0}p^{-f_{\\nu}}))$, where we regard\n $u:= (1,\\pi_{\\mathfrak{p}_0}p^{-f_{\\nu}})$ as an element of\n $\\mathcal{H}_{\\mathfrak{p}_0}$, cf. \\eqref{HOat0}. We note that\n $\\iota^u = \\iota$ because $u$ lies in\n the center. Therefore the action of the Hecke operator $\\mathfrak{h}(u)$ is\n \\begin{displaymath}\n(Y,\\iota, \\bar{\\lambda}, \\rho) \\mapsto (Y,\\iota, \\bar{\\lambda}, \\rho\\circ\\iota(u)). \n \\end{displaymath}\n The image of $(Y,\\iota, \\bar{\\lambda}, \\rho)$ by the map\n\n \\begin{displaymath}\n \\omega_{\\mathfrak{p}_0} \\mathfrak{h}(u): {\\rm RZ}_{\\mathfrak{p}_0}(R) \\rightarrow\n {\\rm RZ}_{\\mathfrak{p}_0}(R[\\tau]) \n \\end{displaymath}\n is $(Y,\\iota, \\bar{\\lambda}, \\rho')$, where $\\rho'$ is given by\n \\begin{displaymath}\n\\xymatrix{\n (Y_{\\mathfrak{q}_0})_{\\bar{R}} \\times (Y_{\\bar{\\mathfrak{q}}_0})_{\\bar{R}}\n \\ar[r]^{\\rho_{\\mathfrak{q}_0} \\times \\rho_{\\bar{\\mathfrak{q}}_0}} & \n \\varepsilon_{*} \\mathbb{X}_{\\mathfrak{q}_0} \\times \n \\varepsilon_{*} \\mathbb{X}_{\\bar{\\mathfrak{q}}_0} \n \\ar[rrr]^{\\varepsilon_{*} F_{\\mathbb{X}_{\\mathfrak{q}_0},\\tau} \\times\n \\varepsilon_{*} \\pi_{\\mathfrak{p}_0}\n p^{-f_{\\nu}} F_{\\mathbb{X}_{\\bar{\\mathfrak{q}}_0},\\tau}}\n & & & \\qquad \n \\varepsilon_{*} \\tau_{*} \\mathbb{X}_{\\mathfrak{q}_0} \\times\n \\varepsilon_{*} \\tau_{*} \\mathbb{X}_{\\bar{\\mathfrak{q}}_0} .\n}\n \\end{displaymath}\nWe note that the Weil descent datum commutes with all Hecke operators.\nIt is straightforward to compute the following heights,\n\\begin{displaymath}\n \\height_{F_{\\mathfrak{p}_0}} F_{\\mathbb{X}_{\\mathfrak{q}_0},\\tau} = 2, \\quad\n \\height_{F_{\\mathfrak{p}_0}} \n \\pi_{\\mathfrak{p}_0} p^{-f_{\\nu}} F_{\\mathbb{X}_{\\bar{\\mathfrak{q}}_0},\\tau} = 2. \n \\end{displaymath}\nTherefore $\\omega_{\\mathfrak{p}_0} \\mathfrak{h}(u)$ is of degree $(1,1)$, \n\\begin{displaymath}\n\\omega_{\\mathfrak{p}_0} \\mathfrak{h}(u): {\\rm RZ}_{\\mathfrak{p}_0}(a, b) \\rightarrow\n {\\rm RZ}_{\\mathfrak{p}_0}(a + 1, b +1) .\n \\end{displaymath}\nThe Hecke operator $\\mid_{\\Pi}$ has also degree $(1,1)$. We write \n$\\omega_{\\mathfrak{p}_0} \\mathfrak{h}(u) = \\mid_{\\Pi} (\\mid_{\\Pi})^{-1} \\omega_{\\mathfrak{p}_0} \\mathfrak{h}(u)$. \nThe Weil descent datum $(\\mid_{\\Pi})^{-1} \\omega_{\\mathfrak{p}_0} \\mathfrak{h}(u)$\nis of degree $(0,0)$ and defines therefore a Weil descent datum on\n${\\rm RZ}_{\\mathfrak{p}_0}(0,0)$. By the isomorphism of Lemma \\ref{RZ7l}, it\ninduces a Weil descent datum on\n\\begin{equation}\\label{uniform10e} \n\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\\Spf O_{\\breve{E}_\\nu}.\n \\end{equation}\n\nBut this isomorphism is just the projection to the $\\mathfrak{q}_0$-part.\nTherefore the induced Weil descent datum maps an $R$-valued point\n$(Y_{\\mathfrak{q}_0},\\iota_{\\mathfrak{q}_0}, \\rho_{\\mathfrak{q}_0})$ to the point\n$(Y_{\\mathfrak{q}_0}^{\\Pi^{-1}},\\iota_{\\mathfrak{q}_0}^{\\Pi^{-1}}, \\rho'_{\\mathfrak{q}_0})$ \nwhere $\\rho'_{\\mathfrak{q}_0}$ is the following composite \n\\begin{displaymath}\n (Y_{\\mathfrak{q}_0}^{\\Pi^{-1}}) \\overset{\\rho_{\\mathfrak{q}_0}}{\\longrightarrow}\n \\varepsilon_{*} \\mathbb{X}_{\\mathfrak{q}_0}^{\\Pi^{-1}} \n \\overset{\\varepsilon_* F_{\\mathbb{X}_{\\mathfrak{q}_0},\\tau}}{\\longrightarrow}\n \\varepsilon_{*} \\tau_{*} \\mathbb{X}_{\\mathfrak{q}_0}^{\\Pi^{-1}} \n \\overset{\\iota(\\Pi^{-1})}{\\longrightarrow}\n \\varepsilon_{*} \\tau_{*} \\mathbb{X}_{\\mathfrak{q}_0}. \n\\end{displaymath}\nBut by the proof of \\cite[Prop. 5.1.7]{KRZ}, this is exactly the natural\nWeil descent datum on the base change \n$\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\\Spf O_{\\breve{E}_\\nu}$.\nTherefore the Weil descent datum on this factor is as claimed.\n\nNow we consider the Weil descent data $\\omega_{\\mathfrak{p}_j} (1,p^{-f_{\\nu}})$ on\n${\\rm RZ}_{\\mathfrak{p}_j, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_j}}$.\nWe have to show that under the isomorphism of Proposition \\ref{RZ6p} \n\\begin{equation}\\label{uniform8e}\n {\\rm RZ}_{\\mathfrak{p}_j, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_j}}\n \\overset{\\sim}{\\longrightarrow}\n G^{\\bullet}_{\\mathfrak{p}_j}\/\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_j}\n \\times_{\\Spf O_{E_{\\nu}},\\varphi_0}\\Spf O_{\\breve{E}_\\nu} ,\n\\end{equation}\nthe Weil descent datum $\\omega_{\\mathfrak{p}_j}(1, p^{-f_{\\nu}})$ induces on the\nright hand side the natural descent datum.\nThe formal group $(\\mathbb{X}_j, \\iota_j)$ which was used to define\n(\\ref{uniform8e}) is defined over\nthe field $\\kappa_{E_{\\nu}}$, as we see from (\\ref{RZ1e}). Therefore\n${\\rm RZ}_{\\mathfrak{p}_i, \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}}$ is defined over\n$\\kappa_{E_{\\nu}}$ and Proposition \\ref{RZ6p} holds over that field. \nWe obtain a natural isomorphism $\\mathbb{X}_i \\cong \\tau_{*} \\mathbb{X}_i$.\nWith this identification, the morphism\n\\begin{displaymath}\n\\xymatrix{\n \\mathbb{X}_{\\mathfrak{q}_i} \\times \\mathbb{X}_{\\bar{\\mathfrak{q}}_i} \n \\ar[rrr]^{ F_{\\mathbb{X}_{\\mathfrak{q}_i},\\tau} \\times\n p^{-f_{\\nu}} F_{\\mathbb{X}_{\\bar{\\mathfrak{q}}_i},\\tau} \\qquad}\n & & & \\quad \n \\tau_{*} \\mathbb{X}_{\\mathfrak{q}_i} \\times\n \\tau_{*} \\mathbb{X}_{\\bar{\\mathfrak{q}}_i} \\cong \n \\mathbb{X}_{\\mathfrak{q}_i} \\times \\mathbb{X}_{\\bar{\\mathfrak{q}}_i} \n}\n\\end{displaymath}\nbecomes the identity. We see that $\\omega_{\\mathfrak{p}_i}(1, p^{-f_{\\nu}})$ induces\non the right hand side of (\\ref{uniform8e}) the natural descent datum. \nThis proves the commutativity of the last diagram in the theorem. \n\\end{proof}\nWe now turn to the $p$-adic uniformization of the quaternionic Shimura curve. We first recall the following well-known fact. \n\\begin{lemma}\n Let $K\/F$ be separable quadratic extension of fields. Let $L$ be a\n quaternion algebra with center $K$ and let $l \\mapsto l'$ be an involution\n of the second kind of $L$. Then there exist a quaternion algebra $C$ with\n center $F$ and an isomorphism of $K$-algebras\n \\begin{displaymath}\nL \\cong C \\otimes_F K, \n \\end{displaymath}\n such that the involution $l \\mapsto l'$ induces on the right hand side the map $c \\otimes k \\mapsto c^{\\iota} \\otimes \\sigma(k)$, where\n $c^{\\iota}$ is the main involution and $\\sigma$ is the nontrivial element in\n $\\Gal(K\/F)$. \n\\end{lemma}\n\\begin{proof}\n We consider the main involution $l \\mapsto l^{\\iota}$ of $L$ over $K$. It is\n characterized by $l + l^{\\iota} = \\Trace^{o}_{L\/K}$. Since the reduced trace\n is respected by an isomorphism of $F$-algebras, one verifies easily that \n \\begin{displaymath}\n(l')^{\\iota} = (l^{\\iota})'. \n \\end{displaymath}\n Therefore $\\rho(l) := (l^{\\iota})'$ is a $\\sigma$-linear isomorphism\n $L \\rightarrow L$ such that $\\rho^2 = \\ensuremath{\\mathrm{id}}\\xspace_{L}$.\n The invariants define $C$ by Galois descent. \n\\end{proof}\nAs an example we consider\n$(A_{o}, \\iota_{o}, \\bar{\\lambda}_{o})$, cf. \\eqref{uniform2e}. The ring $L = \\End^{o}_B A_{o}$ is by\n(\\ref{uniform1e}) a quaternion algebra with center $K$. Let $l \\mapsto l'$\nbe the Rosati involution induced by $\\lambda_{o}$. We define\nthe Cherednik twist of $D$, \n\\begin{equation}\\label{Cheredniktwist} \n\\check{D} = \\{l \\in \\End^{o}_B A_{o}\\; | \\; l' = l^{\\iota}\\}. \n \\end{equation}\nThis is a quaternion algebra over $F$. Since $l l^{\\iota} \\in F$, we obtain\nby (\\ref{groupI1e}) that\n\\begin{equation}\\label{breveD1e}\n\\check{D}^{\\times} \\subset \\mathbf{I}(\\mathbb{Q}). \n \\end{equation} \nBecause the Rosati involution is\npositive, the main involution is positive on $\\check{D}$. It follows that\nat each infinite place of $F$, the localization of $\\check{D}$ is isomorphic\nto the Hamilton quaternions. By (\\ref{uniform1e}) we find\n\\begin{displaymath}\n\\End^{o}_B A_{o} \\otimes \\mathbb{A}^p_f \\cong\nB^{{\\rm opp}} \\otimes \\mathbb{A}^p_f \\cong D \\otimes_{F} K. \n \\end{displaymath}\nSince the Riemann form of $\\lambda_{o}$ induces via\n$\\eta^p_{o}$ on $V \\otimes \\mathbb{A}_f^p$ the form $\\psi$\n(up to a constant), we see that the induced involution on\n$B^{{\\rm opp}} \\cong D \\otimes_{F} K$ is by (\\ref{BZpsi3e}) the involution\n$d \\otimes k \\mapsto d^{\\iota} \\otimes \\bar{k}$. We obtain the isomorphism\n\\begin{equation}\\label{breveD2e} \n\\check{D} \\otimes \\mathbb{A}^p_f \\cong D \\otimes \\mathbb{A}^p_f. \n\\end{equation}\nThe data $\\eta_{\\mathfrak{q}_j, o}$ for $j = 1, \\ldots, s$ in\n Definition \\ref{BZsApkt4d} provide isomorphisms\n\\begin{displaymath}\n (\\Lambda_{\\mathfrak{p}_j}^{pd},\\bar{\\lambda}_{\\psi}) \\isoarrow\n (\\mathbb{X}_j, \\iota_{\\mathbb{X}_j}, \\bar{\\lambda}_{\\mathbb{X}_j}). \n\\end{displaymath}\nFrom this we obtain\n\\begin{displaymath}\n\\End^{o}_{B} A_{o} \\otimes_{F} F_{\\mathfrak{p}_j} =\n\\End_{B_{\\mathfrak{p}_j}} V_{\\mathfrak{p}_j} =\nB_{\\mathfrak{q}_j}^{{\\rm opp}} \\times B_{\\bar{\\mathfrak{q}}_j}^{{\\rm opp}}. \n\\end{displaymath}\nThe Rosati involution of $\\lambda_{o}$ induces on the right hand side the map \n$(b_1, b_2) \\mapsto (b'_2, b'_1)$. We obtain\n\\begin{equation}\\label{breveD3e}\n \\check{D}_{\\mathfrak{p}_j} =\n \\{(b_1,b_2)\\in B_{\\mathfrak{q}_j}^{{\\rm opp}} \\times B_{\\bar{\\mathfrak{q}}_j}^{{\\rm opp}} \\; | \\;\n (b'_2, b'_1) = (b_1^{\\iota}, b_2^{\\iota})\\} \\cong B_{\\mathfrak{q}_j}^{{\\rm opp}} =\n D_{\\mathfrak{p}_j} ,\n\\end{equation}\nwhere the last isomorphism is given by the projection. Comparing with \n(\\ref{BZGpi2e}) gives the embedding\n\\begin{equation}\\label{uniform18e}\n \\check{D}^{\\times}_{\\mathfrak{p}_j} \\subset G^{\\bullet}_{\\mathfrak{p}_j}, \\quad\n \\; j = 1, \\ldots, s, \n\\end{equation}\nwhich coincides with the inclusion \n$\\check{D}^{\\times}_{\\mathfrak{p}_j}\\subset\\mathbf{I}_{\\mathfrak{p}_j} = G^{\\bullet}_{\\mathfrak{p}_j}$ via (\\ref{groupI2e}). \n\nWe obtain in the same way \n\\begin{displaymath}\n\\End^{o}_{B} A_{o} \\otimes_{F} F_{\\mathfrak{p}_0} \\cong \n\\End_{B_{\\mathfrak{p}_0}} \\mathbb{X}_0 \\cong\n\\End_{B_{\\mathfrak{q}_0}} \\mathbb{X}_{\\mathfrak{q}_0} \\times \n\\End_{B_{\\bar{\\mathfrak{q}}_0}} \\mathbb{X}_{\\bar{\\mathfrak{q}}_0} ,\n\\end{displaymath}\nand therefore \n\\begin{equation}\\label{uniform20e}\n \\check{D}_{\\mathfrak{p}_0} = \\{(\\gamma_1, \\gamma_2) \\in\n \\End_{B_{\\mathfrak{q}_0}} \\mathbb{X}_{\\mathfrak{q}_0} \\times \n\\End_{B_{\\bar{\\mathfrak{q}}_0}} \\mathbb{X}_{\\bar{\\mathfrak{q}}_0}\\mid\n\\gamma'_2 = \\gamma_1^{\\iota} \\; \\}. \n \\end{equation}\nThe projection to the first factor defines an isomorphism\n\\begin{equation}\\label{uniform30e} \n \\check{D}_{\\mathfrak{p}_0} \\cong \\End_{B_{\\mathfrak{q}_0}} \\mathbb{X}_{\\mathfrak{q}_0}\n\\cong {\\rm M}_2(F_{\\mathfrak{p}_0}). \n\\end{equation}\nAltogether we find that the quaternion algebras $D$ and $\\check{D}$ over\n$F$ have the same invariants for all places except for $\\mathfrak{p}_0$\nand the infinite place $\\chi_0 ={\\varphi_0}_{\\mid F}$. For the last two places\nthey have {\\rm opp}osite invariants. \n\nWe denote by $H$ the multiplicative group of $D$ considered as an algebraic\ngroup over $\\mathbb{Q}$. The Shimura curve ${{\\mathrm{Sh}}}(H, h_{D})$ is \ndefined over $E(H, h_D) = \\chi_0(F)$. We have $E(H, h_D) \\subset E$ \nand $E(H, h_D) \\rightarrow E_{\\nu}$ is a $p$-adic place of $E(H, h_D)$, cf. (\\ref{BZE1e}), resp. (\\ref{BZ4e}).\n\n\nLet\n$\\mathbf{K}_{\\mathfrak{p}_0} = O_{D_{\\mathfrak{p}_0}}^{\\times}\\subset D_{\\mathfrak{p}_0}^{\\times}$\nbe the unique maximal compact subgroup. We choose for $i = 1,\\ldots, s$ arbitrary open compact\nsubgroups $\\mathbf{K}_{\\mathfrak{p}_i} \\subset D^\\times_{\\mathfrak{p}_i}$ and set $\\mathbf{K}_{p}=\\prod_{i=0}^s\\mathbf{K}_{\\mathfrak{p}_i}$. Finally, we choose an arbitrary open compact subgroup \n $\\mathbf{K}^{p}\\subset (D\\otimes \\mathbb{A}_f^p)^{\\times}$ and set\n\\begin{equation}\\label{uniform25e} \n \\mathbf{K} = \\mathbf{K}_{p} \\mathbf{K}^p \\subset\n H(\\mathbb{A}_f). \n \\end{equation}\n(Note that, since the group $G$ plays no role anymore, we can recycle the notation used in \\eqref{BZ7e}.) The natural embedding $D \\rightarrow D \\otimes_F K = B^{{\\rm opp}}$ induces by\n(\\ref{Gpunkt2e}) an embedding $H \\subset G^{\\bullet}$. In terms of the exact\nsequence of Lemma \\ref{BZ1l} this embedding is obtained from\n$D^{\\times}\\rightarrow D^{\\times}\\times K^{\\times}\\rightarrow G^{\\bullet}(\\mathbb{Q})$.\n\nWe find isomorphisms\n\\begin{equation}\\label{uniform21e}\n \\begin{aligned} \n D_{\\mathfrak{p}_i}^{\\times} \\cong (B^{{\\rm opp}}_{\\mathfrak{q}_i})^{\\times} &=\n \\Aut_{B_{\\mathfrak{q}_i}} V_{\\mathfrak{q}_i} \\\\ \n D^{\\times}_{\\mathfrak{p}_i} \\cong (B^{{\\rm opp}}_{\\bar{\\mathfrak{q}}_i})^{\\times} &= \n \\Aut_{B_{\\bar{\\mathfrak{q}}_i}} V_{\\bar{\\mathfrak{q}}_i} .\n \\end{aligned}\n \\end{equation}\nWe define $\\mathbf{K}_p^{\\bullet} \\subset G^{\\bullet}(\\mathbb{Q}_p)$ in the form\n(\\ref{BZKpPkt1e}) using the isomorphism above such that\n\\begin{equation}\\label{uniform27e}\n \\mathbf{K}^{\\bullet}_{\\mathfrak{q}_i} = \\mathbf{K}_{\\mathfrak{p}_i}, \\quad\n \\mathbf{M}^{\\bullet}_{\\mathfrak{p}_i} = O_{F_{\\mathfrak{p}_i}}^{\\times}, \\quad\n i= 0, \\cdots, s. \n \\end{equation}\nThen $\\mathbf{K}_p = H(\\mathbb{Q}_p) \\cap \\mathbf{K}^{\\bullet}_p$. \nBy Proposition \\ref{Chevalley2p} below, there is an open compact\nsubgroup $\\mathbf{K}^{\\bullet, p} \\subset G^{\\bullet}(\\mathbb{A}_f^p)$ such that\nfor $\\mathbf{K}^{\\bullet} = \\mathbf{K}^{\\bullet}_p \\mathbf{K}^{\\bullet, p}$ we have \n\\begin{displaymath}\n\\mathbf{K} = H(\\mathbb{A}_f) \\cap \\mathbf{K}^{\\bullet} ,\n \\end{displaymath}\nand the natural morphism\n\\begin{displaymath}\n {\\mathrm{Sh}}_{\\mathbf{K}}(H, h_{D}) \\times_{\\Spec E(H, h_D)} \\Spec E\n \\rightarrow {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \n \\end{displaymath}\nis an open and closed immersion. \nIt follows for example by Theorem \\ref{4epeg1t} that\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$ is a flat\nand integral scheme over $O_{E_{\\nu}}$. Therefore the inclusion of the generic fiber \n\\begin{equation}\\label{uniform22e} \n {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \\times_{\\Spec E} \\Spec E_{\\nu}\n \\subset \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})\n\\end{equation}\ninduces a bijection between the sets of connected components of these schemes. \nThe connected components of the right hand side are the closures of the\nconnected components of the left hand side.\n\\begin{definition}\\label{uniform3d}\n We define the scheme $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})$ over\n $\\Spec O_{E_{\\nu}}$ as the Zariski closure of the\n open and closed subscheme\n ${\\mathrm{Sh}}_{\\mathbf{K}}(H, h_{D}) \\times_{\\Spec E(G, h_D)} \\Spec E_{\\nu}$ of the\n left hand side of (\\ref{uniform22e}) in \n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$. \nHence $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})$ is a union of connected\ncomponents of $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D})$. \n\\end{definition}\nWe consider the diagonal embedding \n\\begin{displaymath}\n D_{\\mathfrak{p}_0}^{\\times} \\subset G^{\\bullet}_{\\mathfrak{p}_0} \\subset\n B^{{\\rm opp},\\times}_{\\mathfrak{q}_0} \\times B^{{\\rm opp}, \\times}_{\\bar{\\mathfrak{q}}_0}, \n \\end{displaymath}\ncf. (\\ref{BZGpi2e}), (\\ref{uniform21e}). It defines an open and closed\nembedding \n \\begin{equation*}\n \\begin{aligned}\n (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\n \\Spf O_{\\breve{E}_\\nu}) \\times D^{\\times}_{\\mathfrak{p}_0}\/\\mathbf{K}_{\\mathfrak{p}_0}\n \\subset \n (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\n \\Spf O_{\\breve{E}_\\nu}) \\times\n G^{\\bullet}_{\\mathfrak{p}_0} \/ \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_0} =\n {\\rm RZ}_{\\mathfrak{p}_0} .\n \\end{aligned}\n \\end{equation*}\nIn terms of the decomposition (\\ref{RZ12e}), the left hand side is\n\\begin{displaymath}\n\\coprod_{a \\in \\mathbb{Z}} {\\rm RZ}_{\\mathfrak{p}_0}(a,a). \n\\end{displaymath}\nThis is invariant by the action of\n$\\check{D}^{\\times}_{\\mathfrak{p}_0} \\subset \\mathbf{I}_{\\mathfrak{p}_0}$\non ${\\rm RZ}_{\\mathfrak{p}_0}$. Therefore $\\check{D}^{\\times}_{\\mathfrak{p}_0}$\nacts on\n\\begin{equation}\\label{uniform23e}\n(\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0}\n \\Spf O_{\\breve{E}_\\nu}) \\times D^{\\times}_{\\mathfrak{p}_0}\/\\mathbf{K}_{\\mathfrak{p}_0} .\n\\end{equation}\nWe now combine the actions of $\\check{D}^\\times$ in its prime-to-$p$ component (\\ref{breveD2e}), its $\\mathfrak{p}_0$-component (\\ref{uniform23e}) and its $\\mathfrak{p}_j$-components (\\ref{breveD3e}).\n\\begin{proposition}\\label{uniform4l}\n Let $\\mathbf{K}_p \\subset (D \\otimes \\mathbb{Q}_p)^{\\times}$ as in\n (\\ref{uniform25e}).\n By (\\ref{uniform27e}) this defines a subgroup\n $\\mathbf{K}^{\\bullet}_p \\subset G^{\\bullet}(\\mathbb{Q}_p)$. Let\n $\\mathbf{K}^{\\bullet,p} \\subset G^{\\bullet}(\\mathbb{A}_f^p)$ be a sufficiently\n small open compact subgroup. We set\n $\\mathbf{K}^{p} = \\mathbf{K}^{\\bullet, p} \\cap (D\\otimes\\mathbb{A}_f^p)^{\\times}$.\n Let $\\mathbf{K} = \\mathbf{K}_{p} \\mathbf{K}^p$ and\n $\\mathbf{K}^{\\bullet} = \\mathbf{K}^{\\bullet}_{p} \\mathbf{K}^{\\bullet,p}$ as above. \n Then the morphism\n \\begin{displaymath}\n \\begin{array}{l} \n \\check{D}^{\\times} \\backslash ((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu})\\times\n D^{\\times}(\\mathbb{A}_f)\/\\mathbf{K}) \\longrightarrow \\\\[2mm] \\hspace{5cm} \n \\mathbf{I}(\\mathbb{Q}) \\backslash ((\\hat{\\Omega}^2_{E_{\\nu}}\n \\times_{\\Spf O_{E_{\\nu}}} \\Spf O_{\\breve{E}_{\\nu}}) \\times\n G^{\\bullet}(\\mathbb{A}_f)\/\\mathbf{K}^{\\bullet}) \n \\end{array} \n \\end{displaymath}\n is an open and closed immersion.\n\\end{proposition}\n\\begin{proof}\n The obvious sequence of algebraic groups over $\\mathbb{Q}$ which is\n induced by (\\ref{breveD1e}) \n \\begin{equation}\\label{BZ1l-var} \n 0 \\rightarrow F^{\\times} \\rightarrow \\check{D}^{\\times} \\times K^{\\times}\n \\overset{\\kappa}{\\rightarrow}\n \\mathbf{I} \\rightarrow 0 \n \\end{equation}\n is exact. By Hilbert 90 the sequence remains exact if we take the\n $\\mathbb{Q}$-valued points\n $0\\rightarrow F^{\\times}\\rightarrow\\check{D}^{\\times}\\times K^{\\times}\\overset{\\kappa}{\\rightarrow} \\mathbf{I}(\\mathbb{Q}) \\rightarrow 0$. \n To check the exactness, we can make a base change from $\\mathbb{Q}$\nto an algebraically closed field where there is no difference to Lemma \n\\ref{BZ1l}. We will regard (\\ref{BZ1l-var}) also as a sequence of algebraic\ngroups over $F$. Note that\n$\\mathbf{I}$ is the Weil restriction $\\Res_{F\/\\mathbb{Q}} \\tilde{\\mathbf{I}}$\nwhere $\\tilde{\\mathbf{I}}$ is an algebraic group over $F$ which is defined\nin terms of the $F$-algebra $\\End_B^o A_{o}$, cf. (\\ref{groupI1e}). Recall that\n$\\tilde{\\mathbf{I}}(F) = \\mathbf{I}(\\mathbb{Q})$. \nFor $i = 0, \\ldots s$ we write the group $\\tilde{\\mathbf{I}}(F_{\\mathfrak{p}_i})$\nas follows\n\\begin{equation}\\label{Le66-1e} \n \\begin{array}{ccccc} \n\\check{D}_{\\mathfrak{p}_i}^{\\times} \\times \\check{D}_{\\mathfrak{p}_i}^{\\times} & \\supset \n& \\check{D}_{\\mathfrak{p}_i}^{\\times} \\times^{F_{\\mathfrak{p}_i}} (K_{\\mathfrak{q}_i} \\times\n K_{\\bar{\\mathfrak{q}}_i}) & \\overset{\\sim}{\\longrightarrow} & \n \\check{D}_{\\mathfrak{p}_i}^{\\times} \\times F_{\\mathfrak{p}_i}^{\\times} = \n\\tilde{\\mathbf{I}}(F_{\\mathfrak{p}_i}) \\\\\n (df_1, d f_2) & \\hookleftarrow & (d, (f_1,f_2)) &\\longmapsto &\n (df_1, f_1^{-1}f_2)\\\\\n\\end{array} \n\\end{equation}\nThe canonical embedding\n$\\check{D}_{\\mathfrak{p}_i}^{\\times} \\subset \\tilde{\\mathbf{I}}(F_{\\mathfrak{p}_i})$\nbecomes\n$\\check{D}_{\\mathfrak{p}_i}^{\\times} \\rightarrow \\check{D}_{\\mathfrak{p}_i}^{\\times} \\times F_{\\mathfrak{p}_i}^{\\times}$, \n$d \\mapsto (d,1)$. \n\nFor $j \\geq 1$, \n$\\mathbf{K}_{\\mathfrak{p}_j}\\subset D^{\\times}_{\\mathfrak{p}_j} =\\check{D}^{\\times}_{\\mathfrak{p}_j}$\nis the arbitrary given compact open subgroup and\n\\begin{equation}\\label{KPJ}\n \\mathbf{K}_{\\mathfrak{p}_j}^{\\bullet} = \\mathbf{K}_{\\mathfrak{p}_j} \\times\n O_{F_{\\mathfrak{p}_j}}^{\\times} \\subset\n D_{\\mathfrak{p}_j}^{\\times} \\times F_{\\mathfrak{p}_j}^{\\times} = \n \\check{D}_{\\mathfrak{p}_j}^{\\times} \\times F_{\\mathfrak{p}_j}^{\\times} =\n \\tilde{\\mathbf{I}}(F_{\\mathfrak{p}_j}). \n\\end{equation}\nWe see again that\n$\\mathbf{K}_{\\mathfrak{p}_j}^{\\bullet} \\cap \\check{D}_{\\mathfrak{p}_j}^{\\times} = \\mathbf{K}_{\\mathfrak{p}_j}$.\nWe introduce the groups\n$C_{F, \\mathfrak{p}_j} = \\mathbf{K}_{\\mathfrak{p}_j} \\cap F_{\\mathfrak{p}_j}^{\\times}$ \nand $C_{K, \\mathfrak{p}_j} = C_{F, \\mathfrak{p}_j} \\times O_{\\mathfrak{p}_j}^{\\times} \\subset K_{\\mathfrak{p}_j}^{\\times}$ \nin the sense of the right hand side of \\eqref{KPJ}. Then we obtain\n\\begin{equation}\\label{Le66-2e}\n \\mathbf{K}_{\\mathfrak{p}_j}^{\\bullet} = \\mathbf{K}_{\\mathfrak{p}_j}\n \\times^{C_{F, \\mathfrak{p}_j}} C_{K, \\mathfrak{p}_j} \\subset\n \\check{D}^{\\times}_{\\mathfrak{p}_j} \\times^{F_{\\mathfrak{p}_j}^{\\times}}\n K_{\\mathfrak{p}_j}^{\\times}. \n \\end{equation}\n\nFor the proof we may assume that $\\mathbf{K}^{\\bullet,p}$ is of the following\ntype. Let $C_F^p = \\mathbf{K}^p \\cap (\\mathbb{A}^p_{F,f})^{\\times}$. We choose\nan arbitrary open compact subgroup\n$C_K^{p} \\subset (K \\otimes_F \\mathbb{A}^p_{F,f})^{\\times}$ such that\n$C_K^{p} \\cap (\\mathbb{A}^p_{F,f})^{\\times} = C_F^p$. We define\n$\\mathbf{K}^{\\bullet,p}_C$ as the image of $\\mathbf{K}^p \\times C^p_K$\nby the homomorphism\n\\begin{equation}\\label{Le66-3e}\n (\\check{D} \\otimes_F \\mathbb{A}^p_{F,f})^{\\times} \\times\n (K \\otimes_F \\mathbb{A}^p_{F,f})^{\\times} \\rightarrow\n \\tilde{\\mathbf{I}}(\\mathbb{A}^p_{F,f}). \n \\end{equation}\nWe set\n\\begin{displaymath}\n \\mathbf{K}^{\\mathfrak{p}_0} = (\\prod_{j=1}^s \\mathbf{K}_{\\mathfrak{p}_j})\n \\mathbf{K}^p \n \\subset (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}, \\quad\n \\mathbf{K}^{\\bullet, \\mathfrak{p}_0}_C =\n (\\prod_{j=1}^s \\mathbf{K}^{\\bullet}_{\\mathfrak{p}_j}) \\mathbf{K}_C^{\\bullet, p}\n \\subset \\tilde{\\mathbf{I}}(\\mathbb{A}^{\\mathfrak{p}_0}_{F,f}). \n \\end{displaymath}\nMoreover we set\n$\\mathcal{Z} = \\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu}$ \nand $\\Lambda = \\{(a,b) \\in \\mathbb{Z}^2 \\mid a+b \\equiv 0 \\mod 2 \\}$. \nThen we may write the morphism of the Proposition as follows,\n\\begin{equation}\\label{DIPfeil1e} \n \\check{D}^{\\times} \\backslash (\\mathcal{Z} \\times \\mathbb{Z}) \\times\n (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}\/\\mathbf{K}^{\\mathfrak{p}_0}\n \\longrightarrow\n \\tilde{\\mathbf{I}}(F) \\backslash (\\mathcal{Z} \\times \\Lambda) \\times\n \\tilde{\\mathbf{I}}(\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})\n \/\\mathbf{K}_C^{\\bullet, \\mathfrak{p}_0}. \n\\end{equation}\nThe group $\\check{D}^{\\times}$ acts on $\\mathcal{Z}$ via\n$\\check{D}^{\\times} \\rightarrow \\check{D}^{\\times}_{\\mathfrak{p}_0} \\rightarrow {\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$, \nas we have explained at the end of section 5. We will denote the image in\n${\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$ of an element $g$ by the last map by $\\bar{g}$.\nAn element $g \\in \\check{D}^{\\times}_{\\mathfrak{p}_0}$ acts on\n$\\mathcal{Z} \\times \\mathbb{Z}$ by\n\\begin{displaymath}\n g[\\omega, m] = [\\bar{g} \\omega, m + \\ord_{\\mathfrak{p}_0} \\det g], \\quad\n [\\omega, m] \\in \\mathcal{Z} \\times \\mathbb{Z}. \n\\end{displaymath}\nAn element\n$(g,f) \\in \\check{D}_{\\mathfrak{p}_0}^{\\times} \\times F_{\\mathfrak{p}_0}^{\\times} = \\tilde{\\mathbf{I}}(F_{\\mathfrak{p}_0})$ acts on $\\mathcal{Z} \\times \\Lambda$ by\n\\begin{displaymath}\n (g,f)[\\omega, (a,b)] = [\\bar{g} \\omega, (a + \\ord_{\\mathfrak{p}_0} \\det g,\n b + \\ord_{\\mathfrak{p}_0} \\det g + 2 \\ord_{\\mathfrak{p}_0} f], \\quad\n [\\omega, (a,b)] \\in \\mathcal{Z} \\times \\Lambda. \n\\end{displaymath} \nThe morphism (\\ref{DIPfeil1e}) induces the diagonal\n$\\mathbb{Z} \\rightarrow \\Lambda \\subset \\mathbb{Z}^2$.\n\nWe fix an element\n$m \\times g \\in \\mathbb{Z} \\times (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}$.\nThe image of $\\mathcal{Z} \\times m \\times g$ in the left hand side of\n(\\ref{DIPfeil1e}) is of the form \n\\begin{displaymath}\n\\bar{\\Gamma}_g \\backslash\\mathcal{Z}, \n\\end{displaymath}\nwhere $\\bar{\\Gamma}_g$ is the image of the group\n$\\{d \\in \\check{D} \\cap g\\mathbf{K}^{\\mathfrak{p}_0}g^{-1} \\; | \\; \\ord_{\\mathfrak{p}_0} \\det d = 0 \\}$\nin ${\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$. \nHere $\\det d \\in F^{\\times}$ denotes the reduced norm of $d$.\n\nLet $J$ be the projective group of inner automorphisms of the $F$-algebra\n$\\check{D}$ considered as an algebraic group over $F$.\nWe denote by $\\bar{\\mathbf{K}}^{\\mathfrak{p}_0}$ the image of\n$\\mathbf{K}^{\\mathfrak{p}_0}$ in $J(\\mathbb{A}_{F,f}^{\\mathfrak{p}_0})$. This is an open\nand compact subgroup. To see this one notes that for each place $w$ of $F$\nthe map $(\\check{D} \\otimes_F F_w)^{\\times} \\rightarrow J(F_w)$ is open because\n$\\check{D}^{\\times} \\rightarrow J$ is a smooth morphism of algebraic varieties. \nBecause $J$ is compact at each archimedian places of $F$ the subgroup\n$J(F) \\subset J(\\mathbb{A}_{F,f})$ is discrete. It follows that\n$\\Gamma'_g := J(F) \\cap \\bar{g} \\bar{\\mathbf{K}}^{\\mathfrak{p}_0} \\bar{g}^{-1} \\subset J(F_{\\mathfrak{p}_0})$\nis a discrete subgroup. If $\\mathbf{K}^p$ is sufficiently small, $\\Gamma'_g$ acts \nwithout fixed points on the Bruhat-Tits building of ${\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$.\nThen each point of $\\mathcal{Z}(\\bar{\\kappa}(\\mathfrak{p}_0))$ has a Zariski \nneighbourhood $U$ such that $\\gamma U \\cap U = \\emptyset$ for\n$\\gamma \\in \\Gamma'_g$ and $\\gamma \\neq 1$. This also holds for\n$\\bar{\\Gamma}_g \\subset \\Gamma'_g$.\nBy our considerations, we can write the left hand side of (\\ref{DIPfeil1e}) as \n\\begin{displaymath}\n\\coprod_{i} \\bar{\\Gamma}_{g_i} \\backslash \\mathcal{Z} ,\n\\end{displaymath}\nfor a suitable choice of elements\n$g_i \\in (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}$. In the\nsame way we can write the right hand side of (\\ref{DIPfeil1e}) as \n\\begin{displaymath}\n\\coprod_{j} \\bar{\\Gamma}_{h_j} \\backslash \\mathcal{Z}, \n \\end{displaymath}\nwith a suitable choice of elements \n$h_j \\in \\tilde{\\mathbf{I}}(\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})$.\nTo show the Proposition, it is sufficient to show that (\\ref{DIPfeil1e}) \ninduces an injection on the $\\bar{\\kappa}(\\mathfrak{p}_0)$-valued points. \nIndeed, the restriction of (\\ref{DIPfeil1e}) to\n$\\bar{\\Gamma}_{g_i} \\backslash \\mathcal{Z}$ induces a morphism\n\\begin{equation}\\label{Le66-7e} \n \\bar{\\Gamma}_{g_i} \\backslash \\mathcal{Z} \\rightarrow\n \\bar{\\Gamma}_{h_j} \\backslash \\mathcal{Z} \n \\end{equation}\nfor a suitable $j$, which is injective on \n$\\bar{\\kappa}(\\mathfrak{p}_0)$-valued points. Up to isomorphism we obtain the\nsame map if we replace $h_j$ by the image \n\\begin{displaymath}\n g_i \\times 1 \\in (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}\n \\times (K \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times} \n \\end{displaymath}\nin $\\tilde{\\mathbf{I}}(\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})$. By the injectivity of\n(\\ref{Le66-7e}) and from the fact that the actions of the $\\Gamma$-groups\non both sides of (\\ref{Le66-7e}) are fixed point free on the\nsets of $\\bar{\\kappa}(\\mathfrak{p}_0)$-valued points, one obtains that the groups\n$\\bar{\\Gamma}_{g_i}$ and $\\bar{\\Gamma}_{h_j}$ coincide\non both sides of (\\ref{Le66-7e}) and that this morphism is an isomorphism. \nFrom this the Proposition easily follows. \n\nIt remains to prove the injectivity. We consider two elements\n$[\\omega_i, h_i] \\in \\mathcal{Z} \\times \\mathbb{Z} \\times (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}$, \n$i = 1,2$, with $\\omega_i \\in \\mathcal{Z}(\\bar{\\kappa}(\\mathfrak{p}_0))$ and\n$h_i \\in \\mathbb{Z} \\times (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}$, \nwhich represent the same element on the right hand side of (\\ref{DIPfeil1e}).\nWe will show that they also represent the same element on the left hand\nside. \n\nBy assumption there exists $g^{\\bullet} \\in \\tilde{\\mathbf{I}}(F)$ and\n$k^{\\bullet} \\in \\mathbf{K}^{\\bullet, \\mathfrak{p}_0}_C$ such that\n\\begin{displaymath}\n [\\omega_1, h_1] = g^{\\bullet} [\\omega_2, h_2] k^{\\bullet}. \n\\end{displaymath}\nWe write\n$g^{\\bullet} = g \\lambda \\in \\check{D}^{\\times} \\times^{F^{\\times}} K^{\\times}$ with\n$g \\in \\check{D}^{\\times}$ and $\\lambda \\in K^{\\times}$. We set\n$C_K^{\\mathfrak{p}_0} = (\\prod_{j=1}^s C_{K,\\mathfrak{p}_j}) C_{K}^{p}$. By\n(\\ref{Le66-2e}) and (\\ref{Le66-3e}) we can write\n$k^{\\bullet} = k c$ with $k \\in \\mathbf{K}^{\\mathfrak{p}_0}$ and\n$c \\in C_K^{\\mathfrak{p}_0}$. Replacing $[\\omega_2, h_2]$ by $g[\\omega_2, h_2]k$\nwe may assume that\n\\begin{equation}\\label{Le66-8e} \n [\\omega_1, h_1] = \\lambda [\\omega_2, h_2] c. \n\\end{equation}\nThis implies $\\omega_1 = \\omega_2$ and\n\\begin{equation}\\label{Le66-4e}\nh_2^{-1}h_1 = \\lambda c.\n \\end{equation}\nThis equation takes place in\n$\\Lambda \\times \\tilde{\\mathbf{I}}(\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})$.\nThe $\\Lambda$-part of (\\ref{Le66-4e}) is equivalent with\n\\begin{equation}\\label{Le66-5e} \n\\ord_{\\mathfrak{q}_0} \\lambda = \\ord_{\\bar{\\mathfrak{q}}_0} \\lambda. \n \\end{equation}\nNext we consider the $\\tilde{\\mathbf{I}}(\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})$-part\nof (\\ref{Le66-4e}). We obtain\n\\begin{equation}\\label{Le66-6e}\n \\lambda c = h_2^{-1} h_1 \\in\n (\\check{D} \\otimes_F \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times} \\cap\n (K \\otimes_{F} \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times} =\n (\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times}. \n\\end{equation}\nWe consider the torus $S = K^{\\times}\/F^{\\times}$ over $F$. This torus is\ncompact at all infinite places of $F$. Therefore\n$S(F) \\subset S(\\mathbb{A}_{F,f})$ is discrete and the group of units of $S$\nis finite. The equation (\\ref{Le66-6e}) tells us that $\\lambda$ is a unit\nin $S(F_w)$ for all finite places $w \\neq \\mathfrak{p}_0$ of $F$, because\n$\\lambda$ is in the image $\\bar{C}_K^{\\mathfrak{p}_0}$\nof the compact open subgroup $C_K^{\\mathfrak{p}_0}$ \nby the morphism\n$(K \\otimes_{F} \\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times} \\rightarrow S(\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})$.\nOn the other the equation (\\ref{Le66-5e}) tells us that $\\lambda$ is\na unit in $S(F_{\\mathfrak{p}_0})$ because by this equation there exists an\nelement $\\alpha \\in F_{\\mathfrak{p}_0}^{\\times}$ such that $\\alpha \\lambda$\nis a unit in $K_{\\mathfrak{p}_0}^{\\times}$. Therefore the image of $\\lambda$ in\n$S(F)$ is a unit. If we choose $C_K^{p}$ sufficiently small,\nthe Theorem of Chevalley implies that the image is $1$. We conclude that\n$\\lambda \\in F^{\\times}$. Going back to (\\ref{Le66-6e}) we obtain that\n\\begin{displaymath}\n c \\in C_K^{\\mathfrak{p}_0} \\cap (\\mathbb{A}^{\\mathfrak{p}_0}_{F,f})^{\\times} \\subset\n \\mathbf{K}^{\\mathfrak{p}_0}. \n\\end{displaymath}\nThis shows that the right hand side of (\\ref{Le66-8e}) represents the same \nelement on the left hand side of (\\ref{DIPfeil1e}) as $[\\omega_2,h_2]$. \n\\end{proof}\nWe can now prove our main result, the Cherednik uniformization of quaternionic Shimura curves.\n\\begin{theorem}\\label{4epeg2t} \n Let $\\mathbf{K} \\subset D^{\\times}(\\mathbb{A}_f)$ be of the form $\\mathbf{K}=\\mathbf{K}_p\\mathbf{K}^p$, where $\\mathbf{K}_p$ is chosen as in\n (\\ref{uniform27e}). Let $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})$ be the model\n over $\\Spec O_{E_{\\nu}}$ of the Shimura curve associated to $D$, cf. Definition \\ref{uniform3d}. Then there\n is a uniformization isomorphism of formal schemes\n \\begin{equation}\\label{unimorph7e}\n \\Theta\\colon \\check{D}^{\\times} \\backslash ((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu})\\times\n D^{\\times}(\\mathbb{A}_f)\/\\mathbf{K}) \n\\overset{\\sim}{\\longrightarrow}\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})^\\wedge_{\\, \/ \\Spf O_{\\breve{E}_{\\nu}}}\n \\end{equation}\n For varying $\\mathbf{K}$ this uniformization isomorphism is compatible\n with the action of the Hecke operators in $D^{\\times}(\\mathbb{A}_f)$ on both sides.\n \n \n Let $\\Pi \\in D_{\\mathfrak{p}_0}$ be a prime element in this division algebra\n over $F_{\\mathfrak{p}_0}$. We denote also by $\\Pi$ the image by the canonical\n embedding $D_{\\mathfrak{p}_0} \\subset (D \\otimes \\mathbb{A}_f)^{\\times}$.\n Let $\\tau \\in \\Gal(\\breve{E}_{\\nu}\/ E_{\\nu})$ be the Frobenius automorphism and \n $\\tau_c = \\Spf\\tau^{-1}\\colon \\Spf O_{\\breve{E}_{\\nu}}\\rightarrow\\Spf O_{\\breve{E}_{\\nu}}$. \n The natural Weil descent datum with respect to \n $O_{\\breve{E}_\\nu}\/O_{E_{\\nu}}$ on the right hand side of (\\ref{unimorph7e})\n induces on the\n left hand side the Weil descent datum given by the following commutative diagram\n \\begin{displaymath}\n\\xymatrix{\n \\check{D}^{\\times} \\backslash ((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu})\\times\n D^{\\times}(\\mathbb{A}_f)\/\\mathbf{K})\n \\ar[d]_{ \\ensuremath{\\mathrm{id}}\\xspace \\times \\mid_{\\Pi^{-1}} \\times \\tau_c} \\ar[r] & \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})^\\wedge_{\\, \/ \\Spf O_{\\breve{E}_{\\nu}}}\n \\ar[d]^{\\ensuremath{\\mathrm{id}}\\xspace \\times \\tau_c}\\\\\n \\check{D}^{\\times} \\backslash ((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu})\\times\n D^{\\times}(\\mathbb{A}_f)\/\\mathbf{K})\n \\ar[r] & \n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})^\\wedge_{\\, \/ \\Spf O_{\\breve{E}_{\\nu}}}\n }\n \\end{displaymath}\n\\end{theorem} \nThe following Corollary provides us with an intrinsic characterization of\nthe integral model $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})$ of\n${{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})$. \n\\begin{corollary}\\label{remstab}\n If $\\mathbf{K}^p$ is sufficiently small, the integral model\n $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})$ is a stable relative curve over\n $\\Spec O_{E_\\nu}$, in the sense of \\cite{DM}. In addition, it has semi-stable\n reduction, i.e., it is regular and the special fiber is a reduced divisor\n with normal crossings.\n\\end{corollary}\n\\begin{proof} By the Theorem the formal scheme\n $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})_{\/ \\Spf O_{\\breve{E}_{\\nu}}}$ a union of\n connected components which are isomorphism to\n $\\bar{\\Gamma}\\backslash (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu})$\n where $\\bar{\\Gamma} \\subset {\\mathrm{PGL}}_2(F_{\\mathfrak{p}_0})$ is a disrete subroup.\n It is known \\cite{Mum} that $\\mathbf{K}^p$ can be chosen such that\n $\\bar{\\Gamma}$ acts without fixed point and such that \n \\begin{equation}\\label{remstab1e} \n \\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu} \\rightarrow\n \\bar{\\Gamma}\\backslash (\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2\n \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu})\n \\end{equation} \n is a local isomorphisms for the Zariski topology. We denote by\n $\\Omega_{\\bar{\\kappa}_\\nu}$ the special fibre over $\\Spf O_{\\breve{E}_{\\nu}}$ of\n the left hand side.\n All components of $\\bar{\\Gamma} \\backslash \\Omega_{\\bar{\\kappa}_\\nu}$ are\n rational curves. We show that each of these components $\\bar{C}$ is\n met by other components in at least $3$ different points. This proves\n that the right hand side of (\\ref{remstab1e}) is a stable curve.\n\n $\\bar{C}$ is the image of a component $C \\subset \\Omega_{\\bar{\\kappa}_\\nu}$\n Let $E_1, \\ldots, E_t$ all different components of $\\Omega_{\\bar{\\kappa}_\\nu}$\n which meet $C$ properly. Each $E_i$ meets $C$ in a single point $z_i$.\n The points $z_1, \\ldots, z_t$ are all different. We know that $t \\geq 3$.\n We denote by $\\bar{E}_1, \\ldots, \\bar{E}_t$ the images in\n $\\bar{\\Gamma} \\backslash \\Omega_{\\bar{\\kappa}_\\nu}$. We note the that\n $\\bar{E}_i \\neq \\bar{C}$ for $i=1, \\ldots t$. Indeed, let $\\bar{z}_i$ be\n the image of $z_i$ in $\\bar{\\Gamma} \\backslash \\Omega_{\\bar{\\kappa}_\\nu}$. The\n inequality follows because a neighbourhood of $z_i$ is isomorphically\n mapped to a neighbourhood of $\\bar{z}_i$.\n We will show that the points $\\bar{z}_1, \\ldots, \\bar{z}_t$ are all different.\n If not we find an element $\\gamma \\in \\bar{\\Gamma}$ such that for example\n $\\gamma z_1 = z_2$. Then all three components $C$, $\\gamma C$, $\\gamma E_1$\n contain the point $z_2$. Therefore two of these components must be equal.\n By what we said above $C = \\gamma C$ follows. But this implies that $\\gamma$\n has a fixpoint on $C$ which is excluded by assumption. We conclude that\n $\\gamma z_1 = z_2$ is impossible. This proves that the points\n $\\bar{z}_1, \\ldots, \\bar{z}_t$ are different. \n\\end{proof} \n \n\\begin{proof}[Proof of Theorem \\ref{4epeg2t}]\n We consider only open compact subgroups\n $\\mathbf{K} \\subset H(\\mathbb{A}_f) = D^{\\times}(\\mathbb{A}_f)$ of\n the type as in the statement of the theorem. For the proof it will suffice to consider those\n $\\mathbf{K}$ where $\\mathbf{K}^p$ is small enough. We choose a chain of\n open compact subgroups of this type \n \\begin{equation}\n \\mathbf{K}_1 \\supset \\mathbf{K}_2 \\supset \\ldots \\supset \\mathbf{K}_t\n \\supset \\ldots ,\n \\end{equation}\n which is cofinal to all subgroups of this type.\n\n We consider open and compact subgroups\n $\\mathbf{K}^{\\bullet} \\subset G^{\\bullet}(\\mathbb{A}_f)$ as in Proposition\n \\ref{BZ8p} with the following properties:\n \\begin{enumerate}\n \\item[(a)] $\\mathbf{K}^{\\bullet} \\cap H(\\mathbb{A}_f) = \\mathbf{K}_t$ for some\n $t \\in \\mathbb{N}$.\n \\item[(b)] The groups $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$ and\n $\\mathbf{K}_{t, \\mathfrak{p}_i}$ are related as in (\\ref{uniform27e}) for $i=0,\\ldots,s$.\n \\item[(c)] The natural morphism\n \\begin{displaymath}\n {\\mathrm{Sh}}_{\\mathbf{K}_t}(H, h_{D}) \\times_{\\Spec E(H, h_D)} \\Spec E\n \\rightarrow {\\mathrm{Sh}}_{\\mathbf{K}^{\\bullet}}(G^{\\bullet}, h^{\\bullet}_{D}) \n \\end{displaymath}\n is an open and closed immersion. Here $\\mathbf{K}^{\\bullet}$ is chosen such that $(a)$ is satisfied. \n \\end{enumerate}\n We find a chain of open and compact subgroups $\\mathbf{K}^{\\bullet}$ with the\n properties $(abc)$\n \\begin{equation}\n \\mathbf{K}^{\\bullet}_1 \\supset \\mathbf{K}^{\\bullet}_2 \\supset \\ldots \\supset\n \\mathbf{K}^{\\bullet}_s \\supset \\ldots ,\n \\end{equation}\n which has the following properties. For each $\\mathbf{K}_t$ there is\n a group $\\mathbf{K}^{\\bullet}_{s}$ which satifies $(abc)$ with respect to\n $\\mathbf{K}_t$. Moreover, for an arbitrary $\\mathbf{K}^{\\bullet}$ satisfying\n $(abc)$, there is a group $\\mathbf{K}^{\\bullet}_s$ such that\n $\\mathbf{K}^{\\bullet}_s \\subset \\mathbf{K}^{\\bullet}$ and such that\n $\\mathbf{K}^{\\bullet}_s \\cap H(\\mathbb{A}_f) = \\mathbf{K}_{t'}$ for some\n $t' > t$. We set\n \\begin{equation}\\label{uniform28e} \n {\\mathrm{Sh}}^{pro}(H, h_{D})_{\\breve{E}_{\\nu}} =\\varprojlim\\nolimits_{\\mathbf{K}_t}\n {\\mathrm{Sh}}_{\\mathbf{K}_t}(H, h_{D})_{\\breve{E}_{\\nu}}. \n \\end{equation} \n We remark that the connected components of\n ${\\mathrm{Sh}}_{\\mathbf{K}_t}(H, h_{D})_{\\breve{E}_{\\nu}}$ are geometrically connected. This\n follows from \\cite[(2.7.1) and (3.9.1)]{D-TS} because\n $\\mathbf{K}_{\\mathfrak{p}_0} \\in D^\\times_{\\mathfrak{p}_0}$ is maximal.\n We choose a connected component $Z$ of the left hand side of\n (\\ref{uniform28e}). This induces a connected component \n $Z_{\\mathbf{K}_t}$ of ${\\mathrm{Sh}}_{\\mathbf{K}_t}(H, h_{D})_{\\breve{E}_{\\nu}}$ for each\n $t$. The closure $\\tilde{Z}_{\\mathbf{K}_t}$ of\n $Z_{\\mathbf{K}_t}$ in $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}_t}(H, h_{D})$\n is a connected component there. Since the last schemes are proper over\n $\\Spec O_{\\breve{E}_{\\nu}}$, the natural restriction morphisms\n $\\tilde{Z}_{\\mathbf{K}_{t+1}} \\rightarrow \\tilde{Z}_{\\mathbf{K}_t}$ are surjective.\n We choose points\n $z_{\\mathbf{K}_t} \\in \\tilde{Z}_{\\mathbf{K}_t}(\\bar{\\kappa}_{E_{\\nu}})$ such that\n $z_{\\mathbf{K}_{t+1}}$ is mapped to $z_{\\mathbf{K}_t}$ for all $t$. Let $\\mathbf{K}^{\\bullet}_{s}$\n be a subgroup such that\n $\\mathbf{K}^{\\bullet}_{s}$ induces $ \\mathbf{K}_{t}$ as in $(abc)$. Then by the open\n and closed immersion of Definition \\ref{uniform3d}, $\\tilde{Z}_{\\mathbf{K}_t}$\n is also a connected component of\n $\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}_s}(G^{\\bullet}, h^{\\bullet}_{D})$.\n We consider $z_{\\mathbf{K}_t}$ as a point of\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}_s^{\\bullet}}(\\bar{\\kappa}_{E_{\\nu}})$. We\n denote this point by $z^{\\bullet}_{\\mathbf{K}^{\\bullet}_s}$. It is represented by the\n isomorphism class of a tuple\n \\begin{equation}\n z^{\\bullet}_{\\mathbf{K}^{\\bullet}_s} =\n (A(\\mathbf{K}^{\\bullet}_{s}),\\iota(\\mathbf{K}^{\\bullet}_{s}),\n \\bar{\\lambda}(\\mathbf{K}^{\\bullet}_{s}),\n \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{s}), ( \\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{s}))_j). \n \\end{equation}\n We note that no datum $(\\xi_{\\mathfrak{p}_i})_i$ appears because of our choice\n (\\ref{uniform27e}). By construction $z^{\\bullet}_{\\mathbf{K}^{\\bullet}_{s+1}}$ is\n mapped to $z^{\\bullet}_{\\mathbf{K}^{\\bullet}_s}$ for all $s$. The triples\n \\begin{displaymath}\n(A(\\mathbf{K}^{\\bullet}_{s}),\\iota(\\mathbf{K}^{\\bullet}_{s}),\n \\bar{\\lambda}(\\mathbf{K}^{\\bullet}_{s})) \n \\end{displaymath}\n are all isomorphic. Therefore we may choose them independent of $s$. The\n classes $\\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{{s+1}})$ and\n $\\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{{s+1}})$ generate classes modulo\n $\\mathbf{K}^{\\bullet}_{s}$. We denote these classes by\n $\\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{{s+1}})_{\\mid s}$, resp.\n $\\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{{s+1}})_{\\mid s}$. Since\n $z^{\\bullet}_{\\mathbf{K}^{\\bullet}_{s+1}}$ is mapped to $z^{\\bullet}_{\\mathbf{K}^{\\bullet}_s}$,\n we obtain an isomorphism of tuples\n \\begin{displaymath}\n (A,\\iota,\\bar{\\lambda}, \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{{s+1}})_{\\mid s}, \n (\\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{{s+1}})_{\\mid s})_j\n ) \\overset{\\sim}{\\rightarrow}\n (A,\\iota,\\bar{\\lambda}, \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{s}), (\\bar{\\eta}_{\\mathfrak{q}_i}(\\mathbf{K}^{\\bullet}_{s})\n)_j). \n \\end{displaymath}\n By this isomorphism the data\n $\\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{{s+1}})$ and $\\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{{s+1}})$ on the left hand side induce on the\n right hand side data\n $\\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{{s+1}})'$ and $\\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{{s+1}})'$ such that\n \\begin{displaymath}\n \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{{s+1}})'_{\\mid s} =\n \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{s}),\\quad \\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{{s+1}})'_{\\mid s} =\n \\bar{\\eta}_{\\mathfrak{q}_i}(\\mathbf{K}^{\\bullet}_{s}) .\n \\end{displaymath}\n Therefore we may assume that \n \\begin{equation} \n \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{{s+1}})_{\\mid s} =\n \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{s}), \\quad \\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{{s+1}})_{\\mid s} =\n \\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{s}). \n \\end{equation}\nNow $\\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{s}) \\subset \\mathrm{Isom}_{B \\otimes \\mathbb{A}_f^p}(V \\otimes \\mathbb{A}^p_f, \\mathrm{V}^p(A))$\nis a compact subset. Therefore the intersection\n$\\cap_s \\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{s})$ is not empty. We choose an\nelement $\\eta^p$ in this intersection. It generates the class\n$\\bar{\\eta}^p(\\mathbf{K}^{\\bullet}_{s})$ for each $s$. Similiarly, we find\nfor each $j = 1, \\ldots, s$ an isomorphism\n$\\eta_{\\mathfrak{q}_i}: \\Lambda_{\\mathfrak{q}_j} \\isoarrow T_{\\mathfrak{q}_i}$\nwhich induces all classes $\\bar{\\eta}_{\\mathfrak{q}_j}(\\mathbf{K}^{\\bullet}_{s})$.\n\nThe tuple \n\\begin{equation}\n(A, \\iota, \\bar{\\lambda}, \\eta^p, ( \\eta_{\\mathfrak{q}_j})_j) \n \\end{equation}\nmakes it possible to define the uniformization morphism of the theorem.\nFor this we consider the morphism (\\ref{unimorph1e}) defined by \nsubstituting $(A, \\iota, \\bar{\\lambda}, \\eta^p, ( \\eta_{\\mathfrak{q}_j})_j) $ for the choice of \\eqref{uniform2e}\nused there. Let $\\mathbb{X} = \\prod_{i=0}^s \\mathbb{X}_{\\mathfrak{p}_i}$ be the\n$p$-divisible group of $A$. In the Definition \\ref{RZ4d} we take\n$\\mathbb{X}_0=\\mathbb{X}_{\\mathfrak{p}_0}$ as\nthe framing object. If we take for $\\rho$ the identity, we obtain a point\nof ${\\rm RZ}_{\\mathfrak{p}_0}(0,0) \\subset {\\rm RZ}_{\\mathfrak{p}_0}$ and by\nLemma \\ref{RZ7l} a point \n\\begin{displaymath}\n \\tilde{z} \\in \\hat{\\Omega}^2_{F_{\\mathfrak{p}_0}}(\\bar{\\kappa}_{E_{\\nu}}) =\n \\hat{\\Omega}^2_{E_{\\nu}}(\\bar{\\kappa}_{E_{\\nu}}). \n \\end{displaymath}\nFrom $\\mathbb{X}_j=\\mathbb{X}_{\\mathfrak{p}_j}$, with the $O_{F_{\\mathfrak{p}_i}}^{\\times}$-homogeneous polarization\ninduced from $\\bar{\\lambda}$, the rigidification $\\bar{\\eta}_{\\mathfrak{q}_j}$, \nand the datum $\\rho = \\ensuremath{\\mathrm{id}}\\xspace_{\\mathbb{X}_i}$, we obtain a point\n$\\tilde{z}_j(\\mathbf{K}_{s,\\mathfrak{p}_j}^{\\bullet})$ of\n${\\rm RZ}_{\\mathfrak{p}_j, \\mathbf{K}_{s,\\mathfrak{p}_j}^{\\bullet}}$. If we use for \n(\\ref{RZ1e}) the isomorphism given by $\\eta_{\\mathfrak{q}_j}$, the point \n$\\tilde{z}_j(\\mathbf{K}_{s,\\mathfrak{p}_j}^{\\bullet})$ corresponds to\n$1 \\in G^{\\bullet}_{\\mathfrak{p}_j}\/\\mathbf{K}^{\\bullet}_{s,\\mathfrak{p}_j}$ under the\nisomorphism of Proposition \\ref{RZ6p}. By construction of the uniformization\nmorphism \\eqref{unimorph6e}, the point\n\\begin{displaymath}\n\\tilde{z} \\times 1 \\in (\\hat{\\Omega}^2_{E_{\\nu}} \\times\n G^{\\bullet}(\\mathbb{A}_f)\/\\mathbf{K}^{\\bullet}_s)(\\bar{\\kappa}_{E_{\\nu}}) \n \\end{displaymath}\nis mapped to the point\n\\begin{displaymath}\n z^{\\bullet}_{\\mathbf{K}^{\\bullet}_s} \\in \\tilde{Z}_{\\mathbf{K}_t}(\\bar{\\kappa}_{E_{\\nu}})\n \\subset\n \\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}_s}(G^{\\bullet}, h^{\\bullet}_{D})(\\bar{\\kappa}_{E_{\\nu}}). \n\\end{displaymath}\nThis implies that\n$(\\hat{\\Omega}^2_{E_{\\nu}} \\times_{\\Spf O_{E_{\\nu}}} \\Spf O_{\\breve{E}_{\\nu}})\\times 1$ \nis mapped by (\\ref{unimorph6e}) to the formal completion of the connected component\n$\\tilde{Z}_{\\mathbf{K}_t}$ of\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}_s}(G^{\\bullet}, h^{\\bullet}_{D})_{O_{\\breve{E}_{\\nu}}}$. Now\nwe restrict (\\ref{unimorph6e}) to\n\\begin{equation}\\label{uniform29e}\n\\check{D}^{\\times} \\backslash ((\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \n\\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu})\\times\nD^{\\times}(\\mathbb{A}_f)\/\\mathbf{K}_t) \\rightarrow\n\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}^{\\bullet}_s}(G^{\\bullet}, h^{\\bullet}_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}}, \n \\end{equation}\ncf. Lemma \\ref{uniform4l}. The image of the connected component\n$(\\hat{\\Omega}_{F_{\\mathfrak{p}_0}}^2 \\times_{\\Spf O_{F_{\\mathfrak{p}_0}},\\varphi_0} \\Spf O_{\\breve{E}_\\nu}) \\times 1$\nis mapped to a connected component of the open and closed formal subscheme \n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}}$.\nBut since the Hecke operators $D^{\\times}(\\mathbb{A}_f)$ act transitively on\nthe connected components of the last formal scheme and the morphism\n(\\ref{unimorph6e}) is compatible with Hecke operators, we conclude that \n(\\ref{uniform29e}) is a surjective map onto\n$\\widetilde{{\\mathrm{Sh}}}_{\\mathbf{K}}(H, h_{D})^\\wedge_{\/ \\Spf O_{\\breve{E}_{\\nu}}}$. Since by\nTheorem \\ref{4epeg1t} and Lemma \\ref{uniform4l} the morphism is an open\nand closed immersion we conclude that (\\ref{unimorph7e}) is an isomorphism for\n$\\mathbf{K} = \\mathbf{K}_t$. \n\nNow the tuple $(A, \\iota, \\bar{\\lambda}, \\eta^p, ( \\eta_{\\mathfrak{q}_j})_j) $\ndefines the uniformization morphism for an arbitrary $\\mathbf{K}$. By choosing\n$\\mathbf{K}_t \\subset \\mathbf{K}$, we see that (\\ref{unimorph7e}) is surjective\nand therefore an isomorphism by Lemma \\ref{uniform4l}. \nThe compatibility with the Weil descent data is a consequence of\nTheorem \\ref{4epeg1t}. This completes the proof.\n\\end{proof} \n\n\n\n \n\n\\section{Conventions about Galois descent}\\label{s:desc}\n\nLet $L\/E$ be a Galois extension (possibly infinite) with Galois group\n$G = \\Gal(L\/E)$. For $\\sigma \\in G$ we set \n\\begin{equation}\\label{tau_c1e}\n\\sigma_c = \\Spec \\sigma^{-1}: \\Spec L \\rightarrow \\Spec L. \n\\end{equation}\nIf $\\tau \\in G$ we find $(\\sigma \\circ \\tau)_{c} = \\sigma_{c} \\circ \\tau_{c}$. \nLet $\\pi: X \\rightarrow \\Spec L$ be a scheme over $L$. We recall that a\ndescent datum on $X$ relative to $L\/E$ is a collection of morphisms\n$\\varphi_{\\sigma}: X \\rightarrow X$ for $\\sigma \\in G$, making the\nfollowing diagram commutative\n\\begin{equation*}\n \\xymatrix{\n X \\ar[r]^{\\varphi_{\\sigma}} \\ar[d]^{\\pi} & X \\ar[d]_{\\pi}\\\\\n \\Spec L \\ar[r]^{\\sigma_c} & \\Spec L , \n }\n\\end{equation*}\nsuch that $\\varphi_{\\sigma} \\circ \\varphi_{\\tau} = \\varphi_{\\sigma \\tau}$ for all\n$\\sigma, \\tau \\in G$. In other words, a descent datum is a left action of $G$\non $X$ by semi-linear automorphisms.\nA descent datum $(X, \\varphi_{\\sigma})$ defines a left action of $G$ on\n$X(L) = \\Hom_{\\Spec L}(\\Spec L, X)$, \n\\begin{displaymath}\n \\begin{array}{ccc}\n G \\times X(L) & \\longrightarrow & X(L).\\\\\n (\\sigma , \\alpha) & \\longmapsto & \\varphi_{\\sigma} \\circ \\alpha \\circ\n \\Spec \\sigma\\\\\n \\end{array}\n \\end{displaymath}\nWe denote the right hand side by $\\sigma \\times_{\\varphi} \\alpha$. This is indeed\na point of $X(L)$:\n\\begin{displaymath}\n\\pi \\circ \\varphi_{\\sigma} \\circ \\alpha \\circ \\Spec \\sigma = \\sigma_c \\circ \\pi \n\\circ \\alpha \\circ \\Spec \\sigma = \\sigma_c \\circ \\Spec \\sigma = \\ensuremath{\\mathrm{id}}\\xspace_{\\Spec L}. \n\\end{displaymath}\n\nLet $u: G \\rightarrow \\Aut_{L}((X, \\varphi))$ be an action of $G$ on this\ndescent datum. This means that for each $\\sigma \\in G$ an $L$-morphism\n$u_{\\sigma}: X \\rightarrow X$ is given such that for each $\\sigma, \\tau \\in G$\n\\begin{displaymath}\n u_{\\sigma} \\circ u_{\\tau} = u_{\\sigma \\tau}, \\quad u_{\\sigma} \\circ \\varphi_{\\tau} =\n \\varphi_{\\tau} \\circ u_{\\sigma}.\n\\end{displaymath}\nThen $\\psi_{\\sigma} := u_{\\sigma} \\circ \\varphi_{\\sigma}$ is another descent datum\non $X$. It defines another action $\\sigma \\times_{\\psi} \\alpha$ of $G$ on $X(L)$.\nFrom the definition we obtain\n\\begin{equation}\\label{GalAb1e}\n\\sigma \\times_{\\psi} \\alpha = u_{\\sigma} \\circ (\\sigma \\times_{\\phi} \\alpha). \n\\end{equation}\n\nIf $X_0$ is a scheme over $\\Spec E$, there is the canonical\ndescent datum on $X = X_0 \\times_{\\Spec E} \\Spec L$, \n\\begin{displaymath}\n \\kappa_{\\sigma} = \\ensuremath{\\mathrm{id}}\\xspace_{X_0} \\times \\sigma_c: X_0 \\times_{\\Spec E} \\Spec L\n \\rightarrow X_0 \\times_{\\Spec E} \\Spec L. \n\\end{displaymath}\nThe action of $G$ induced by $\\kappa_{\\sigma}$ on $X(L)$ coincides with the\naction on $X_0(L) = \\Hom_{\\Spec E}(\\Spec L, X_0)$ via $L$, taking into account\nthe identification $X(L) = X_0(L)$. We denote this action by\n$\\sigma \\alpha_0 = \\sigma \\times_{\\kappa} \\alpha_0$,\n($\\sigma \\in G$, $\\alpha_0 \\in X_0(L)$). \n\nA homomorphism $a: G \\rightarrow \\Aut_{E} X_0$ defines an action on the\ncanonical descent datum $u: G \\rightarrow \\Aut_{L}((X, \\kappa))$ via \n$u_{\\sigma} = a_{\\sigma} \\times \\ensuremath{\\mathrm{id}}\\xspace_{\\Spec L}$. We obtain the new descent datum\n\\begin{displaymath}\n \\psi_{\\sigma} = u_{\\sigma} \\circ \\kappa_{\\sigma} = a_{\\sigma} \\times \\sigma_c:\n X_0 \\times_{\\Spec E} \\Spec L \\rightarrow X_0 \\times_{\\Spec E} \\Spec L. \n \\end{displaymath}\nThe action of $G$ on $X(L) = X_0(L)$ defined by this descent datum is \n\\begin{displaymath}\n \\sigma \\times_{\\psi} \\alpha_0 = a_{\\sigma} \\circ \\sigma \\alpha_0, \\quad\n \\alpha_0 \\in X_0(L). \n \\end{displaymath}\n\nAssume that $X_0 = \\coprod_{X_0(E)} \\Spec E$ is the constant scheme. Then the\naction of $G$ on $X_0(L)$ is trivial. Let $a: G \\rightarrow \\Aut X_0(E)$ be\nan action. It induces an action on $X_0$ which we denote by the same letter\n$a$. It defines on $X = X_0 \\times_{\\Spec E} \\Spec L$ the descent datum\n$a_{\\sigma} \\times \\sigma_c$. The action on $X(L) = X_0(L)$ induced by this\ndescent datum is the operation on $X_0(L)$ by $a$ acting on $X_0$. (Note that\n$X_0(L) = X_0(E))$. \n\n\nA descent datum $(X, \\varphi_{\\sigma})$ is effective if there is scheme\n$X_0$ over $E$ such that there is an isomorphism\n$X_0 \\times_{\\Spec E} \\Spec L \\rightarrow X$ \nwhich respects the descent data $\\kappa_{\\sigma}$ resp. $\\varphi_{\\sigma}$. \nAssume that $L\/K$ is a finite field extension and that $X$ is quasiprojective\nover $\\Spec L$. Then any decent datum\n$\\varphi_{\\sigma}$ is effective (cf. SGA1 Exp. VIII). \n\n\n\\section{Conventions about Shimura varieties}\\label{s:shimvar} \n\nLet $(G,h)$ a Shimura datum. We denote by $X$ the $G(\\mathbb{R})$-conjugacy\nclass of $h$. We consider the operation of $G(\\mathbb{R})$ on $X$ from\nthe left\n\\begin{displaymath} \nG(\\mathbb{R}) \\times X \\rightarrow X, \\quad (g,x) \\mapsto gxg^{-1}. \n\\end{displaymath}\nLet $\\mathbf{K}_{\\infty} \\subset G(\\mathbb{R})$ be the stabilizer of $h$.\nThen we have a $G(\\mathbb{R})$-equivariant map\n\\begin{displaymath}\n G(\\mathbb{R})\/\\mathbf{K}_{\\infty} \\overset{\\sim}{\\longrightarrow} X, \\quad\n g \\mapsto ghg^{-1}. \n \\end{displaymath}\nLet $\\mathbf{K} \\subset G(\\mathbb{A}_f)$ be an open and compact subgroup.\nThen we define the complex Shimura variety\n\\begin{equation}\n \\begin{array}{ll} \n \\mathrm{Sh}_{\\mathbf{K}}(G,h)_{\\mathbb{C}} & = G(\\mathbb{Q}) \\backslash \n (X \\times (G(\\mathbb{A}_f)\/\\mathbf{K}))\\\\[2mm] & = G(\\mathbb{Q}) \\backslash \n ((G(\\mathbb{R})\/\\mathbf{K}_{\\infty}) \\times (G(\\mathbb{A}_f)\/\\mathbf{K})).\\\\ \n \\end{array}\n\\end{equation}\nThe group $G(\\mathbb{Q})$ acts from the left via the homomorphisms\n$G(\\mathbb{Q}) \\rightarrow G(\\mathbb{R})$ and\n$G(\\mathbb{Q}) \\rightarrow G(\\mathbb{A}_f)$. \n\nThe group $G(\\mathbb{A}_f)$ acts from the right on the tower\n$\\{ \\mathrm{Sh}_{\\mathbf{K}} \\}_{\\mathbf{K}}$ for varying $\\mathbf{K}$. For\n$a \\in G(\\mathbb{A}_f)$ this action is given by \n\\begin{equation}\n \\begin{array}{lccc} \n \\mid_{a}: & \\mathrm{Sh}_{\\mathbf{K}} & \\rightarrow &\n \\mathrm{Sh}_{a^{-1} \\mathbf{K} a}, \\\\\n & (x, u) & \\mapsto & (x, ua)\\\\ \n \\end{array} \n \\end{equation} \nwhere $x \\in X$ and $u \\in G(\\mathbb{A}_f)$. We call this a Hecke operator.\n \n\\begin{remark}\nIn \\cite{D-TS}, the action of $G(\\mathbb{R})$ from the right on $X$ is considered,\n\\begin{displaymath}\nh \\times g := g^{-1} h g, \\quad h \\in X, \\; g \\in G(\\mathbb{R}). \n \\end{displaymath}\nThe complex Shimura variety is defined as\n\\begin{displaymath}\n \\begin{array}{ll}\n {\\mathrm{Sh}}^{D}_{\\mathbf{K}}(G, h)_{\\mathbb{C}} & =\n (X\\times(\\mathbf{K}\\backslash G(\\mathbb{A}_f))\/G(\\mathbb{Q})\\\\ \n & = ((\\mathbf{K}_{\\infty} \\backslash G(\\mathbb{R})) \\times\n (\\mathbf{K}\\backslash G(\\mathbb{A}_f))\/G(\\mathbb{Q})\n \\end{array}\n \\end{displaymath}\nLet $X_{-}$ be the conjugacy class of $h^{-1}$. Then there is a natural\nisomorphism\n\\begin{equation}\\label{Shh1e}\n {\\mathrm{Sh}}_{\\mathbf{K}}(G, h)_{\\mathbb{C}} \\overset{\\sim}{\\longrightarrow}\n {\\mathrm{Sh}}_{\\mathbf{K}}^{D}(G, h^{-1})_{\\mathbb{C}} ,\n \\end{equation}\n given by \n\\begin{displaymath}\n \\begin{array}{ccc} \n G(\\mathbb{Q}) \\backslash (X \\times (G(\\mathbb{A}_f)\/\\mathbf{K})) &\n \\overset{\\sim}{\\longrightarrow} & \n (X \\times(\\mathbf{K}\\backslash G(\\mathbb{A}_f))\/G(\\mathbb{Q})\\\\\n (x, g) & \\longmapsto & (x^{-1}, g^{-1}) \n \\end{array}\n\\end{displaymath}\n\\end{remark}\nLet $H$ be a torus over $\\mathbb{Q}$ and let\n\\begin{displaymath}\nh: \\mathbb{S} \\rightarrow H_{\\mathbb{R}} \n\\end{displaymath}\nbe a morphism of algebraic groups over $\\mathbb{R}$. It induces a morphism\nof algebraic groups over $\\mathbb{C}$\n\\begin{displaymath}\n\\mu: \\mathbb{G}_{m,\\mathbb{C}} \\rightarrow H_{\\mathbb{C}}. \n \\end{displaymath}\nThe field of definition $E$ of $\\mu$ is the reflex field of $(H, h)$. We consider \nthe composite \n\\begin{displaymath}\n \\mathfrak{r}: \\mathrm{Res}_{E\/\\mathbb{Q}} \\mathbb{G}_{m,E} \n \\overset{\\mathrm{Res}\\, \\mu}{\\longrightarrow} \\mathrm{Res}_{E\/\\mathbb{Q}} H_E \n \\overset{\\Nm_{E\/\\mathbb{Q}}}{\\longrightarrow} H. \n\\end{displaymath}\nThe homomorphism\n\\begin{equation}\\label{Sh-rec3e}\n r(H,h) = \\mathfrak{r}^{-1}: \\mathrm{Res}_{E\/\\mathbb{Q}} \\mathbb{G}_{m,E}\n \\rightarrow H \n \\end{equation}\nis called the reciprocity law of $(H,h)$, cf. \\cite[(3.9.1)]{D-TS}.\nLet $\\mathbf{K} \\subset H(\\mathbb{A}_f)$ be an open\ncompact subgroup. There is an open and compact subgroup\n$C \\subset (E \\otimes \\mathbb{A}_f)^{\\times}$ such that\n$r(H,h)(\\mathbb{A}_f)(C) \\subset \\mathbf{K}$. Therefore $r(H,h)$\ninduces a map\n\\begin{equation}\\label{Sh-rec1e}\n E^{\\times} \\backslash (E \\otimes \\mathbb{A}_f)^{\\times}\/C \\rightarrow\n H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K}\n \\end{equation}\nBy class field theory\n\\begin{displaymath}\n E^{\\times} \\backslash (E \\otimes \\mathbb{A}_f)^{\\times}\/C =\n E^{\\times} \\backslash (E \\otimes \\mathbb{A})^{\\times}\/C (E \\otimes\\mathbb{R})^{\\times}\n\\end{displaymath}\ncorresponds to a finite abelian extension $L$ of $E$. We consider the\nhomomorphism\n\\begin{displaymath}\n\\Gal(E^{ab}\/E) \\rightarrow \\Gal(L\/E) =\n E^{\\times} \\backslash (E \\otimes \\mathbb{A}_f)^{\\times}\/C.\n \\end{displaymath}\nIf we compose this with (\\ref{Sh-rec1e}) we obtain the class field version of the reciprocity map,\n\\begin{equation}\\label{Sh-rec2e}\n r^{\\rm cft}(H,h): \\Gal(\\bar{E}\/E) \\rightarrow \\Gal(E^{ab}\/E) \\rightarrow\n H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K}. \n\\end{equation}\nThis Galois action on $H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K}$\ndefines a finite \\'etale scheme over $E$ which we denote by\n$\\mathrm{Sh}_{\\mathbf{K}}(H,h)$.\nThis is called the canonical model of $\\mathrm{Sh}_{\\mathbf{K}}(H,h)$ over $E$.\nBy definition \n\\begin{displaymath}\n \\mathrm{Sh}_{\\mathbf{K}}(H,h)_{\\mathbb{C}} =\n H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K}. \n\\end{displaymath}\nIn other words we can say that the $E$-scheme $\\mathrm{Sh}_{\\mathbf{K}}(H,h)$\nis obtained from the constant scheme\n$H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K}$ over $E$ by the descent\ndatum\n\\begin{equation}\\label{Sh-rec4e}\n \\begin{array}{lr}\n r^{\\rm cft}(H,h)(\\sigma) \\times \\sigma_c: &\n (H(\\mathbb{Q})\\backslash\n H(\\mathbb{A}_f)\/\\mathbf{K})\\times_{\\Spec E}\\Spec\\bar{E}\n \\longrightarrow \\qquad \\quad \\\\[2mm] \n & \\qquad (H(\\mathbb{Q})\\backslash\n H(\\mathbb{A}_f)\/\\mathbf{K})\\times_{\\Spec E}\\Spec\\bar{E}, \n \\end{array}\n \\end{equation}\nfor $\\sigma \\in \\Gal(\\bar{E}\/E)$ and $\\sigma_c := \\Spec \\sigma^{-1}$. We can also\nexpress the last statement by a commutative diagram. There is an isomorphism\nof schemes over $\\bar{E}$ \n\\begin{equation}\\label{r-descent1e}\n H(\\mathbb{Q})\\backslash H(\\mathbb{A}_f)\/\\mathbf{K})\\times_{\\Spec E}\\Spec\\bar{E}\n \\rightarrow\n \\mathrm{Sh}_{\\mathbf{K}}(H,h)\\times_{\\Spec E}\\Spec\\bar{E}\n\\end{equation}\nsuch that for each \n$\\sigma \\in \\Gal(\\bar{E}\/E)$ the following diagram is commutative,\n\\begin{displaymath}\n\\xymatrix{\n (H(\\mathbb{Q})\\backslash H(\\mathbb{A}_f)\/\\mathbf{K})\n \\times_{\\Spec E}\\Spec\\bar{E} \\ar[d]_{r^{\\rm cft}(H,h)(\\sigma) \\times \\sigma_c} \\ar[r] \n & \\mathrm{Sh}_{\\mathbf{K}}(H,h) \\times_{\\Spec E}\\Spec\\bar{E}\n \\ar[d]^{id \\times \\sigma_c}\\\\\n (H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K})\n \\times_{\\Spec E}\\Spec\\bar{E} \\ar[r] & \n \\mathrm{Sh}_{\\mathbf{K}}(H,h) \\times_{\\Spec E}\\Spec\\bar{E} .\n }\n \\end{displaymath}\n For varying $\\mathbf{K}$ the isomorphism\nis compatible with the action of the Hecke operators $H(\\mathbb{A}_f)$.\nIn the considerations above we can replace the field of definition $E$ of\n$\\mu$ by any finite extension $E'$, $E \\subset E' \\subset \\mathbb{C}$.\nThe definition (\\ref{Sh-rec3e}) gives\n\\begin{displaymath}\n r_{E'}(H,h) = \\mathfrak{r}^{-1}: \\mathrm{Res}_{E'\/\\mathbb{Q}} \\mathbb{G}_{m,E'}\n \\rightarrow H.\n\\end{displaymath}\nThis reciprocity law gives the action of $\\Gal(\\bar{E}\/E')$ on\n$H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K}$ \nobtained by restriction from (\\ref{Sh-rec2e}). \n\n\\begin{remark}\nThe map\n\\begin{displaymath}\n \\begin{array}{ccc} \n H(\\mathbb{Q}) \\backslash H(\\mathbb{A}_f)\/\\mathbf{K} &\n \\overset{\\sim}{\\longrightarrow} & \n \\mathbf{K}\\backslash H(\\mathbb{A}_f)\/H(\\mathbb{Q})\\\\\n h & \\longmapsto & h^{-1} \n \\end{array}\n\\end{displaymath}\nis equivariant with respect to the action of $E^{\\times}(\\mathbb{A}_f)$ on the\nleft hand side by $r(H,h)$ and the action of $E^{\\times}(\\mathbb{A}_f)$ on the\nright hand side by $r(H, h^{-1})$.\nTherefore we obtain an isomorphism of canonical models over $E$, \n\\begin{displaymath}\n{\\mathrm{Sh}}_{\\mathbf{K}}(H, h) \\overset{\\sim}{\\longrightarrow}\n {\\mathrm{Sh}}_{\\mathbf{K}}^{D}(H, h^{-1}). \n\\end{displaymath}\n\\end{remark}\nWe go back to a general Shimura datum $(G,h)$. We denote by $E(G,h)$ the\nShimura field. We call a model ${\\mathrm{Sh}}(G,h)$ over $E$ of ${\\mathrm{Sh}}(G,h)_{\\mathbb{C}}$\ncanonical if for each maximal torus $H \\subset G$ and each\n$h': \\mathbb{S} \\rightarrow H_{\\mathbb{R}}$ which is conjugate to $h$ in\n$G(\\mathbb{R})$, the induced morphism\n${\\mathrm{Sh}}(H,h')_{\\mathbb{C}} \\rightarrow {\\mathrm{Sh}}(G,h)_{\\mathbb{C}}$ is defined over the\ncompositum of $E(H,h')$ and $E(G,h)$.\nWith this definition (\\ref{Shh1e}) induces an isomorphism of canonical models,\ncf. \\cite[3.13]{D-TS}.\n\n\\begin{equation}\\label{Shh2e}\n{\\mathrm{Sh}}_{\\mathbf{K}}(G, h) \\overset{\\sim}{\\longrightarrow}\n {\\mathrm{Sh}}_{\\mathbf{K}}^{D}(G, h^{-1}). \n\\end{equation}\n\nWe now consider the situation of \\cite[4.9--4.11]{D-TS}. \nLet $L$ be a semisimple algebra over $\\mathbb{Q}$ with a positive involution\n$*: L \\rightarrow L$. Let $F \\subset L$ be a subfield in the center of $L$\nwhich is invariant by the involution $*$. \nLet $V$ be a faithful $L$-module which is finite-dimensional over $\\mathbb{Q}$. Let $\\psi: V \\times V \\rightarrow \\mathbb{Q}$\nbe an alternating $\\mathbb{Q}$-bilinear form such that\n\\begin{displaymath}\n\\psi(\\ell x, y) = \\psi(x, \\ell^{*} y), \\quad \\ell \\in L, \\; x,y \\in V. \n\\end{displaymath}\nWe consider the algebraic group $G$ over $\\mathbb{Q}$ given by\n\\begin{displaymath}\n G(\\mathbb{Q}) = \\{g \\in \\mathrm{GL}_{L} (V) \\; | \\; \\psi(gx, gy) = \\psi(\\mu(g) x,y),\n \\; \\mu(g) \\in F^{\\times}\\}. \n \\end{displaymath}\nThere is up to conjugation by an element of $G(\\mathbb{R})$ a unique complex\nstucture $J: V \\otimes \\mathbb{R} \\rightarrow V \\otimes \\mathbb{R}$,\n$J^{2} = - \\ensuremath{\\mathrm{id}}\\xspace$ which commutes with the action of $L$ and such that\n\\begin{displaymath}\n\\psi(Jx,y) \n \\end{displaymath}\nis a symmetric and positive definite. Let\n$h: \\mathbb{S} \\rightarrow G_{\\mathbb{R}}$ such that \n$h(z)$ acts on $V \\otimes \\mathbb{R}$ by multiplication by $z$ with respect\nto the complex structure just introduced. Then $(G,h)$ is a Shimura datum.\nWe set\n\\begin{displaymath}\n \\mathbf{t}(\\ell) = {\\rm Tr}_{\\mathbb{C}} (\\ell \\mid V \\otimes \\mathbb{R}), \\quad\n \\ell \\in L. \n \\end{displaymath}\nThe numbers $\\mathbf{t}(\\ell)$ generate over $\\mathbb{Q}$ the Shimura field\n$E(G,h)$. The canonical model ${\\mathrm{Sh}}_{\\mathbf{K}}(G,h)$ is the coarse moduli\nscheme of the following functor $\\mathcal{M}(L,V,\\psi)$ on the category\nof $E$-schemes $S$.\n\\begin{definition}\\label{Shh1d}\n A point of $\\mathcal{M}(L,V,\\psi)(S)$ is given by the following data: \n \\begin{enumerate}\n \\item[(a)] an abelian scheme $A$ over $S$ up to isogeny with an action\n $\\iota: L \\rightarrow \\End^{o} A$, \n\\item[(b)] an $F^{\\times}$-homogeneous polarization $\\bar{\\lambda}$ of $A$,\n\\item[(c)] a class $\\bar{\\eta}$ modulo $\\mathbf{K}$ of $L \\otimes \\mathbb{A}_f$-module\n isomorphisms \n\\begin{displaymath}\n\\eta: V \\otimes \\mathbb{A}_f \\isoarrow \\hat{V}(A). \n\\end{displaymath}\nsuch that for each $\\lambda \\in \\bar{\\lambda}$ there is locally for the\n Zariski topology a constant \n $\\xi(\\lambda) \\in (F \\otimes \\mathbb{A}_f)^{\\times}(1)$\n with\n \\begin{displaymath}\n\\psi(\\xi(\\lambda) v_1, v_2) = E^{\\lambda}(\\eta(v_1), \\eta(v_2)). \n \\end{displaymath} \n\\item[(d)] The $L$-module $H_1(A, \\mathbb{Q})$ with its Riemann form\n defined by $\\lambda$ is isomorphic to $(V, \\psi)$, up to a factor\n in $F^{\\times}$. \n \\end{enumerate}\n\n We require that the following condition holds\n \\begin{displaymath}\n\\Trace (\\iota(\\ell) \\mid \\Lie A) = \\mathbf{t}(\\ell), \\quad \\ell \\in L. \n \\end{displaymath}\n\\end{definition}\n\nWe reformulate \\cite[5.11]{D-TS} with our conventions.\n\\begin{proposition}\\label{zentralerTwist1p}\n Let $(G,h)$ be a Shimura datum. Let $Z \\subset G$ be the connected center\n of $G$. Let $\\delta: \\mathbb{S} \\rightarrow Z_{\\mathbb{R}}$ be a\n homomorphism. Let $E \\subset \\mathbb{C}$ be a finite extension of\n $\\mathbb{Q}$ which contains the Shimura fields $E(G,h)$ and $E(Z,\\delta)$. \n Let $\\mathbf{K} \\subset G(\\mathbb{A}_f)$ be an open and compact subgroup.\n We denote by $\\mathrm{Sh}_{\\mathbf{K}}(G,h)$ and\n $\\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta)$ the quasi-canonical models over $E$. Let\n \\begin{displaymath}\n r_E^{\\rm cft}(Z,\\delta): \\Gal(\\bar{E}\/E) \\rightarrow\n Z(\\mathbb{Q}) \\backslash Z(\\mathbb{A}_f)\/(\\mathbf{K} \\cap Z(\\mathbb{A}_f)) \n \\end{displaymath}\n be the reciprocity law. There is an isomorphism of schemes over $\\bar{E}$\n \\begin{equation}\\label{deltaTwist1e}\n \\mathrm{Sh}_{\\mathbf{K}}(G,h) \\times_{\\Spec E}\\Spec\\bar{E} \\rightarrow\n \\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta) \\times_{\\Spec E}\\Spec\\bar{E}\n \\end{equation}\n such that for each $\\sigma \\in \\Gal(\\bar{E}\/E)$ the following diagram\n is commutative,\n \\begin{displaymath}\n\\xymatrix{\n\\mathrm{Sh}_{\\mathbf{K}}(G,h)\n \\times_{\\Spec E}\\Spec\\bar{E} \\ar[d]_{r^{\\rm cft}_E(Z,\\delta)(\\sigma) \\times \\sigma_c} \\ar[r] \n & \\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta) \\times_{\\Spec E}\\Spec\\bar{E}\n \\ar[d]^{id \\times \\sigma_c}\\\\\n \\mathrm{Sh}_{\\mathbf{K}}(G,h) \\times_{\\Spec E}\\Spec\\bar{E} \\ar[r] & \n \\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta) \\times_{\\Spec E}\\Spec\\bar{E} . \n }\n \\end{displaymath}\n For varying $\\mathbf{K}$ the morphism (\\ref{deltaTwist1e}) is compatible\n with the Hecke operators induced by elements $g \\in G(\\mathbb{A}_f)$. \n \\end{proposition}\n\\begin{proof}\n By the definition of a canonical model one can reduce the question to the\n case when $G$ is an algebraic torus (cf. \\cite[5.11]{D-TS}). Then the proposition\n is a consequence of (\\ref{Sh-rec4e}). \n \\end{proof}\n\nWe formulate a ''local'' version of the last proposition, keeping the notations\nthere. We fix a diagram as in (\\ref{BZ2e})\n\\begin{displaymath}\n\\mathbb{C} \\leftarrow \\bar{\\mathbb{Q}} \\rightarrow \\bar{\\mathbb{Q}}_p.\n \\end{displaymath}\nIt determines a $p$-adic place $\\nu$ of $E$. Let\n$\\mu(\\delta): \\mathbb{G}_{m,\\mathbb{C}} \\rightarrow Z_{\\mathbb{C}}$ the\nhomomorphism associated to $\\delta$ as usual. It is defined over $E_{\\nu}$,\n\\begin{displaymath}\n \\mu_{\\nu}: \\mathbb{G}_{m,E_{\\nu}} \\rightarrow Z_{E_{\\nu}}.\n\\end{displaymath}\nWe consider the homomorphism\n\\begin{displaymath}\n \\mathfrak{r}_{\\nu}: \\mathbb{G}_{m,E_{\\nu}} \\overset{\\mu_{\\nu}}{\\longrightarrow}\n Z_{E_{\\nu}} \\overset{\\Nm_{E_{\\nu}\/\\mathbb{Q}_p}}{\\longrightarrow} Z_{\\mathbb{Q}_p}.\n\\end{displaymath}\nWe define $r_{\\nu}(Z,\\delta)$ as the composite\n\\begin{equation}\\label{local-rec1e}\n r_{\\nu}(Z,\\delta): E_{\\nu}^{\\times}\n \\overset{\\mathfrak{r}_{\\nu}^{-1}}{\\longrightarrow} Z(\\mathbb{Q}_p)\n \\rightarrow Z(\\mathbb{Q})\\backslash Z(\\mathbb{A}_f)\/\n (\\mathbf{K} \\cap Z(\\mathbb{A}_f)), \n\\end{equation}\nwhere the last arrow is induced by the inclusion\n$Z(\\mathbb{Q}_p) \\subset Z(\\mathbb{A}_f)$. By local class field theory, this\ninduces a homomorphism\n\\begin{displaymath}\n r_{\\nu}^{\\rm cft}(Z,\\delta): \\Gal(\\bar{E}_{\\nu}\/E_{\\nu}) \\rightarrow\n Z(\\mathbb{Q})\\backslash Z(\\mathbb{A}_f)\/ (\\mathbf{K} \\cap Z(\\mathbb{A}_f)). \n \\end{displaymath}\n\\begin{corollary}\\label{zentralerTwist1c}\n We denote by $\\mathrm{Sh}_{\\mathbf{K}}(G,h)_{E_{\\nu}}$ and\n $\\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta)_{E_{\\nu}}$ the schemes over $E_{\\nu}$\n obtained by base change from the canonical models. There is an isomorphism of schemes over $\\bar{E}_{\\nu}$ \n \\begin{displaymath}\n \\mathrm{Sh}_{\\mathbf{K}}(G,h)_{E_{\\nu}} \\times_{\\Spec E_{\\nu}} \\Spec \\bar{E}_{\\nu} \n \\rightarrow\n \\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta)_{E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec\\bar{E}_{\\nu}\n \\end{displaymath}\n such that for any $\\sigma \\in \\Gal(\\bar{E}_{\\nu}\/E_{\\nu})$ the following\n diagram is commutative\n \\begin{displaymath}\n\\xymatrix{\n\\mathrm{Sh}_{\\mathbf{K}}(G,h)_{E_{\\nu}} \\times_{\\Spec E_{\\nu}} \\Spec \\bar{E}_{\\nu}\n \\ar[d]_{r^{\\rm cft}_{\\nu}(Z,\\delta)(\\sigma) \\times \\sigma_c} \\ar[r] \n & \\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta)_{E_{\\nu}} \n \\times_{\\Spec E_{\\nu}} \\Spec \\bar{E}_{\\nu} \\ar[d]^{id \\times \\sigma_c}\\\\\n \\mathrm{Sh}_{\\mathbf{K}}(G,h)_{E_{\\nu}} \\times_{\\Spec E_{\\nu}} \\Spec \\bar{E}_{\\nu}\n \\ar[r] & \n \\mathrm{Sh}_{\\mathbf{K}}(G,h\\delta)_{E_{\\nu}}\\times_{\\Spec E_{\\nu}}\\Spec \\bar{E}_{\\nu}.\n }\n \\end{displaymath}\n \\end{corollary}\n\\begin{proof}\nThis follows from the compatibilities of local and global class field theory. \n\\end{proof}\n\nOur final topic is the following variant of \\cite[Prop. 1.15]{D-TS}. A similar variant appears in Kisin \\cite[Lem. 2.1.2]{KisinJAMS}. \n\n\\begin{proposition}\\label{Chevalley2p}\n Let $S$ be a finite set of prime numbers. \n Let $M \\subset G$ be closed immersion of reductive subgroups over\n $\\mathbb{Q}$. We assume that $M$ is the kernel of a homomorphism\n $G \\rightarrow T$ to a torus over $\\mathbb{Q}$. Let\n $h: \\mathbb{S} \\rightarrow M_{\\mathbb{R}} \\rightarrow G_{\\mathbb{R}}$\n be a homomorphism of algebraic groups such that $(M,h)$ and $(G,h)$\n are Shimura data. \n Let $\\mathbf{I} = \\mathbf{I}_S \\mathbf{I}^S$ be an open and compact subgroup\n $M(\\mathbb{A}_f)$, where $\\mathbf{I}_S \\subset \\prod_{p \\in S} M(\\mathbb{Q}_p)$\n and $\\mathbf{I}^S \\subset M(\\mathbb{A}_f^S)$. Let \n $\\mathbf{K}_S \\subset \\prod_{p \\in S} G(\\mathbb{Q}_p)$ be a compact open \n subgroup such that\n \\begin{displaymath}\n\\mathbf{K}_S \\cap (\\prod\\nolimits_{p \\in S} M(\\mathbb{Q}_p)) = \\mathbf{I}_S. \n \\end{displaymath}\nThen there exists an open compact subgroup\n $\\mathbf{K}^S \\subset G(\\mathbb{A}_f^S)$ which contains $\\mathbf{I}^S$ \n such that the induced morphism of schemes over $\\mathbb{C}$ \n \\begin{equation}\\label{Einbettung1e} \n {\\mathrm{Sh}}_{\\mathbf{I}}(M,h)_{\\mathbb{C}} \\rightarrow\n {\\mathrm{Sh}}_{\\mathbf{K}_S \\mathbf{K}^S}(G,h)_{\\mathbb{C}} \n \\end{equation}\nis an open and closed immersion. \n \\end{proposition}\n\\begin{proof}\n Let $Z_G$ be the center of $G$ and let $G^{\\mathrm{der}}$ be the derived group.\n The map $Z_G \\times G^{\\mathrm{der}} \\rightarrow G$ is an isogeny. Since\n $G^{\\mathrm{der}}$ is mapped to $\\{ 1 \\} \\in T$ we see that $G$ and $Z_G$\n have the same image in $T$. We obtain that the homomorphism\n $Z_G \\times M \\rightarrow G$ is surjective. Therefore $Z_M \\subset Z_G$.\n We obtain a morphism $M^{\\mathrm{ad}} \\rightarrow G^{\\mathrm{ad}}$ which is an isomorphism.\n Let $X_G$ be the set of conjugates of $h$ by elements of $G(\\mathbb{R})$\n and define $X_M$ in the same way. Since the adjoint groups are the same, the induced map \n $X_M \\rightarrow X_G$ is an isomorphism onto a union of connected components\n of $X_G$. This implies that the Shimura varieties\n ${\\mathrm{Sh}}_{\\mathbf{I}}(M,h)_{\\mathbb{C}}$ and\n ${\\mathrm{Sh}}_{\\mathbf{K}_S\\mathbf{K}^S}(G,h)_{\\mathbb{C}}$ have the same dimension.\n By \\cite{KisinJAMS} we may choose $\\mathbf{K}^{S}$ in such a way that\n (\\ref{Einbettung1e}) is a closed immersion. But since both varieties\n are normal of the same dimension, the induced morphisms on the local\n rings must be isomorphisms. Therefore (\\ref{Einbettung1e}) is also open. \n \\end{proof}\nThe main ingredient of the proof in \\cite{KisinJAMS} is the following theorem.\nBecause it is needed for other purposes in this paper, we state it here. \n\n\\begin{proposition} (Theorem of Chevalley) \\; \n Let $T$ be an algebraic torus over $\\mathbb{Q}$. Let\n $\\mathcal{E} \\subset T(\\mathbb{Q})$ be a finitely generated subgroup.\n Let $S$ be a finite set of rational primes. We denote by\n $\\mathbb{A}_{f}^{S}$ the restricted product over all $\\mathbb{Q}_{\\ell}$\n where $\\ell$ runs over all prime numbers $\\ell \\notin S$. \n Let $m$ be an integer.\n Then there exists a compact open subgroup\n $C \\subset T(\\mathbb{A}_{f}^{S})$ such that\n $C \\cap \\mathcal{E} \\subset \\mathcal{E}^m$. \n\\end{proposition}\n\nIn the case where $L$ is a number field and\n$T = \\Res_{L\/\\mathbb{Q}} \\mathbb{G}_{m,L}$, this is the first theorem in \\cite{Che}.\nThe general case is easily reduced to this.\n\n\\begin{comment}\n\\newpage \n \n{\\bf INHALT} \\vspace{4mm} \n\n\\begin{tabular}{lcr} \n $\\chi_0: F \\rightarrow \\mathbb{R}$ & unique infinite place where $D$ splits\n & 1\\\\[3mm]\n $G^{\\bullet}$ & (\\ref{Gpunkt1e}) & 1 \\\\[3mm]\n $\\mathfrak{p}_0, \\ldots, \\mathfrak{p}_s$ & the primes of $F$ over $p$\n (\\ref{BZsplit1e}) & 4\\\\[3mm]\n $\\mathfrak{p}_i O_K = \\mathfrak{q}_i \\bar{\\mathfrak{q}}_i$ &\n each $\\mathfrak{p}_i$ splits in $K$\n & 4\\\\[3mm]\n $\\psi: V \\times V \\rightarrow \\mathbb{Q}$ & (\\ref{BZpsi1e}) & 3\\\\\n $\\Lambda_{\\mathfrak{q}_i}, \\Lambda_{\\bar{\\mathfrak{q}}_i}, \\Lambda_{\\mathfrak{p}}$ &\n $\\subset V \\otimes \\mathbb{Q}_p$ & 9\\\\[3mm]\n $r_{\\varphi}$ & generalized CM-type (\\ref{BZgCM1e}), (\\ref{BZEnu1e}) & 1, 4\\\\ \n $\\mathcal{A}_{\\mathbf{K}}$ & Definition \\ref{BZAK1d} & 7\\\\[3mm]\n $\\mathcal{A}^{bis}_{\\mathbf{K}}$ & Definition \\ref{BZsAK2d} & 11\\\\[3mm]\n $\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}$ & Definition \\ref{BZsAtK2d} & 13\\\\[3mm]\n $\\mathcal{A}^t_{\\mathbf{K}}$ & & 14\\\\[3mm]\n $\\mathcal{A}_{\\mathbf{K}^{\\bullet}}^{\\bullet}$ & Definition \\ref{BZApkt3d} & 15\\\\[3mm]\n $\\mathcal{A}^{\\bullet bis}_{\\mathbf{K}^{\\bullet}}$ & Definition \\ref{BZApkt4d} &\n 16\\\\[3mm]\n $\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}}$ & Definition \\ref{BZsApkt4d} &\n 28\\\\[3mm]\n$E_{\\nu}$ & local Shimura field (\\ref{BZEnu1e}), (\\ref{BZ3e}) & 4 \\\\[3mm]\nmonomorphism \n$\\mathcal{A}_{\\mathbf{K}} \\rightarrow \\mathcal{A}^{\\bullet}_{\\mathbf{K}_U^{\\bullet}}$\n& Prop. \\ref{BZ4p} & 21\\\\[3mm]\nmonomorphism\n$\\tilde{\\mathcal{A}}^t_{\\mathbf{K}}\\rightarrow\\tilde{\\mathcal{A}}^{\\bullet t}_{\\mathbf{K}^{\\bullet}_{U}}$ & Prop. \\ref{BZ6p} & 27\\\\[3mm]\n$\\mathbf{K}_{\\mathfrak{q}_i}$, $\\mathbf{K}^{\\bullet}_{\\mathfrak{p}_i}$ & \n(\\ref{BZKpPkt1e}) & 16\\\\[3mm] \n\\end{tabular}\n\\end{comment} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}