数理解析研究所講究録
第 1659 巻 2009 年 27-36

27

Paramodular form on $GSp(2,\mathbb{A})$TAKEO OKAZAKI*
Abstract
We $\dot{g}ve\infty Mructions$ of automorphic fOrms on $GSp(2,A)$ which are flxed by paramodular groupa

Introduction.
Let A be the adele of Q. Let $\Pi(G)$ be the set of equivalenoe classes of admissible representations of( $G$ may local or global). The -litk from GO $(2,2,A)$ to $GSp(2,A)$ provides Slegel modular fom whose
spinor Lfunction (of degee 4) is of the following type A) or B),$G$$\theta$A$)$. aproductfor $\sigma\iota\in\Pi(GL(2,A))$ .

$L(s,\sigma_{1})L(s,\sigma_{2})$.

$\sigma_{1},\sigma z$

have a $\infty mmon$ oentral chara 何冶 r.

B . $L(s,\sigma)$ for $\sigma\in n(GL(2,L_{A}))$is a real quadratic fleld, andthrough the nom map $L^{X}arrow Q^{x}$
$)$

$L$

.

$\sigma$

has a central character which factor

We can identify $GL(2,Q)xGL(2,Q)$ or $GL(2,L)$ with GO $(2,2,Q)$ , roughly. By Hre and PiatetshShapiro [3], it is knwn these -lift ls non-vanishing and generlc, in almost all cases (remark there is a
non-generic case, see Theorem 5). However, nobody gave how to $\infty nstruct$ them and what $\infty ngmenoe$, tn this article, we shall give the method to $\infty nstmct$
sugbroupe fixes the -lifts, as $fu$ as we know.
them. To state our results, we racall some notation and deflne some groups. For an arbitrary $\infty mmutative$$r\dot{\mathfrak{W}}^{A,GSp(n,A)}$ is the group of $g\in GL(2n,A)$ such that for some$\theta$$\Re$$\theta$$\lambda(g)\in A^{x}$$g\{\begin{array}{ll}0_{n} -I_{n}I_{n} 0_{n}\end{array}\}\iota_{g=\lambda(g)}\{\begin{array}{ll}0_{n} -I_{n}I_{n} 0_{n}\end{array}\}$,.

where denotes the transpose of
In the case of $n=2$ , typical unipotent elementS of the maximaJ
parabolic subgroup of $Sp(2)$ are written as$\iota_{g}$$g$$u=\{\begin{array}{lll}1 t* *0 1* \delta 0 01 00 0-t l\end{array}\}$.

(1)

We flx the standard additive characteron. For $\pi_{v}\in n(GSp(2,Q_{v}))$ and $c_{1},q\in Q$, let
be the space of functions satisifying$\psi$$Q\backslash A$$W(\psi_{e_{1^{g}l}},,\pi_{v})\subset\pi_{v}$$W(ug)=\psi(c_{1}t+c_{2}s)W(g)$for $u\in U(Q_{v})$ .
(Gamma nero type$\infty n\Psi^{uenoe}$group)$\Gamma_{0}^{(n)}(N)=[NM(n,Z)M(n,Z)$$M(n_{1}Z)M(n,Z)]\cap GSp(n,Z)$.

is deflned, similarly (and other groups are also).
(Paramodular group (of degree 2)) Paramodular group $K(N)$ of conductor $N$ (i.e., of the polariiation

Locd$\Gamma_{0}^{(n)}(Nl)$$mw\infty k$

vnn supported by$Gran\triangleright in\cdot A\triangleleft$

for JSPS libNows.

28
$($

1, $N))$ is defined by.

$K(N)=\{\begin{array}{llll}Z Z N^{-1}Z ZNZ Z Z ZNZ NZ Z NZNZ Z Z Z\end{array}\}\cap GSp(2,Q)$(Semi-paramodular group) &mi-paramodular $g\infty upK(N,N’)$ of level $(N, ,N’)$ by$K(N,N’)=\{\begin{array}{llll}Z Z N^{-1}Z ZNZ Z Z ZNN,Z NZ Z NZNZ Z Z Z\end{array}\}\cap GSp(2,Q)$

For$\eta_{v}\in\hat{Q_{v}^{x}}$.

, we say a function $f$ on $GSp(2,Q_{v})$ is $\eta,,.aemistable$ on $K(NZ_{v},N’Z_{v})$ , if $f$ satisfies$f(zg)=\eta_{v}(z)f(g)$$brg\in GSp(2,Q_{v}),$, and $f(gu)=\eta_{v}(d_{1})f(g)$$z\in Z(GSp(2,Q_{v}))\simeq Q_{v}^{x}$$u=[****$$****$, and$d_{1}^{*}**$$****]\in K(NZ_{\eta},NZ)$.

Then, $m$ shall describe for the case A).

Theorem 1 –Let $\sigma_{1},\sigma_{2}\in n(GL(2,A))$ be satisfying be folluring .
$\epsilon$$\bullet$.
$\bullet$$\sigma_{2}\in\Pi(GL(2,A))$$\sigma_{1},\sigma_{2}$is curpidaL

have a common central unitavy character$\eta$.

At $eve V^{v}$ both ofhave $m:takker$ modets $us\infty iitd$ tospace of $uchmim_{er}$ modelS).
$\sigma_{1v},\sigma_{2v}$

Take autonwrphic forms
and$f_{1}\in\sigma_{1}^{\vee},f_{2}\in\sigma_{2}$so that$W_{1p}(z_{1}g)=W_{1p}(g[:$W_{1}=\theta_{v}W_{1v}\in W(\sigma_{1}^{v},\psi)(\sigma_{1}^{\vee}\dot{u}$z_{1}*])=\eta_{p}^{-1}(z_{1})W_{1p}(g)$,$W_{2p}(z_{2}g)=W_{2p}(g[z_{2}* :])=ib(a)W_{2p}(g)$ ,$W(\sigma_{lv},\psi_{v})$thethe contragradient of$[:$$z_{1}*]\in\Gamma_{0}^{(1)}(N_{1}Z_{P})$$\{\begin{array}{ll}a ** *\end{array}\}\in\Gamma_{0}^{(1)}(N_{2}Z_{P})$

.

,

1 the
$)$

(we denote by$W_{2}=\otimes_{v}W_{2v}\in W(\sigma_{2},\psi)$$\sigma_{1})$$\mathbb{R}en$$\psi$$\theta$-lift $F=h(f_{1} @f_{2},\varphi)(d\epsilon find in$ (5) has a $gbW$ VVhittaker jfiesnction$)$

as $W_{F}^{1,-1}(1)\neq 0$ .

$W_{F,p}\dot{u}\eta_{p}^{-1}$$W_{F}^{1,-1}=\emptyset_{1},W_{F_{t}^{2}v}^{1-1}$

such-semitttable on$K(c(\sigma_{1p})e(\sigma_{2p}),c(b))$

for the conductors$c(\sigma_{lp}),c(b)$. At archimedean place, if$W_{t}$

has $weig\aleph\sim$ , then the highest weight of $F$$\dot{u}$$( \frac{\kappa_{1}+\kappa_{2}}{2},$(Remark that

2 If both of$)$$\kappa_{1}\pm\alpha\dot{u}$$W_{tp}$even, aince$\sigma_{1},\sigma_{2}$are $ne\psi om$, then$\frac{-|\kappa_{1}-\eta|}{2})$.

have the common central character.)$Z_{N}(s,W_{Fp}^{1,-1})=L(s,\sigma_{1p}^{v})L(s,\sigma_{2p}^{\vee})$

where $Z_{N}(s,W_{Fv}^{1,-1})ls$ Novodvaraky’s zetn integmZ.
$S)F\dot{u}$ curptial unless
is not $m\dot{\mu}dd$ or.
$\sigma_{1}$$\sigma_{1}=\sigma_{2}$$If\sigma_{1}\dot{u}$not $cu\varphi idal$ then the dqenaMe Whitakker

29$\hslash ncuonW_{F}^{1,0}$

4-;

$)$

If

$\sigma_{1p},\sigma_{2p}$is not zerv. ff

are princi$\ovalbox{\tt\small REJECT}$$\sigma_{1}=\sigma_{2}$, then the degenarate VVhitakJCer

representation$fimc\hslash onW_{F}^{0,1}$

is not zero.

$\pi(\mu_{1},\eta_{p}\mu_{1}^{-1}),\pi(\mu_{l},\eta_{p}\mu_{2}^{-1})_{f}$$W_{Fp}^{1.arrow 1}\in\eta_{p}\mu_{l}^{arrow 1}\mu_{1}^{arrow 1}x\mu_{2}\mu_{1}^{\sim 1}x\mu_{1}$.

is the Borel parabolically induced representatio (seefor the $d\ell finiuon.$ ): $Q_{p}\ni xarrow|x|_{p}$ . Suppose, whereis a principal series reperesentation, anda chamcter such asmeans the -twist of for $\pi\in\Pi(GL(2))$ . Then,

Here$[1\theta J$$n$$\sigma_{2p}\rangle b$4-ii) Let$\nu$$\chi\pi(\nu_{1}^{u}\eta’\nu^{-u})$$\sigma_{1p}$$\chi^{2}\eta’=\eta_{p}$$\chi\pi$$\chi\dot{u}$$\pi$$\chi$$W_{Fp}^{1,-1},$ $W_{Fp}^{1_{1}0}\in\chi^{-1}\nu^{-a}\sigma_{2p}r\chi\nu_{2}^{a}$the Siegel parabolically induced representation.
Supposeis a special reperesentation$l)-|ii)$$\sigma_{1p}$$\chi\sigma(\nu^{1/2}, \nu^{-1/2})$. Then,$W_{Fp}^{1,-1},W_{Fp}^{1,0}\in G(\nu^{1/2}\chi^{arrow 1}\sigma_{2p},\nu^{arrow 1/2}\chi)$which is a generic constituent of,$\nu^{1/2}\chi^{-1}\pi x\nu^{-1/2}\chi$(eqlained below).

$\ell-:v)If\sigma_{1p}=\sigma_{2p}$$W_{Fp}^{1,arrow 1},W_{Fp}^{0,1}\in 1_{OL(1)}x\sigma_{2p}$,me Klingen parabolically induoed vepresentation.
Rmark 2 In the case of $\eta_{p}=1,$ we take neuforme $W_{1p},W_{2p}$ , thenparamoduZar vector for $K(N_{1}N_{2}h)(c.f$. $[9J)$ .
$|f$

Remark $SR\alpha pt$ the cases $1- i$), $ii),i|i),$As explained later, we$n\infty 1$

case$W_{Fp}^{1,-1}$$W_{F.p}\dot{u}$the newform, i. e., thebelongs to $\epsilon_{upe1ru\Psi^{idd w\epsilon\epsilon entauon}}$.

the Whitakker models for the nonvanishing of global$W_{F}^{1.-1}$. However, in the$\sigma_{1}=\chi(\det)\in n(GL(2,A))$also, we have the[11].

nonr of$W_{Fp}^{1,0}$. We recall the local Saito-Kurokawa representation due to Schmidt

Proposltlon 4 (local SaSto-Kurokawa repreeentations.(Schmidt [11])) –iiet
be a $lo\Phi$ irredudble, admuSible, :nfinite-dimemional oepnsentation of $PGL(2)$ . Let
be a character of $GL(1)$ .
Then,Assume$\pi$$\pi\neq\pi(\nu^{a/2}, \nu^{arrow\epsilon/2})$.

$\tau$$\nu^{1/2}\pi x\nu^{-1/2}r=Q(\nu^{1/2}\pi,\nu^{arrow 1/2}\tau)+G(\nu^{1/2}\pi,\nu^{arrow 1/2}\tau)$

the unique induclbk subrepresentation, which
unique irreducible quotient, which not generic.

$G(\nu^{1/2}\pi,\nu^{-1/2}\tau)\dot{u}$.

.

$\dot{u}g\epsilon ne\dot{n}e$$Q(\nu^{1/2}\pi,\nu^{-1/2}\tau)\dot{u}$the$\dot{u}$They calllocal Saito-Kurokawa representation. Rmark the elements of $G(\nu^{1/2}\pi)\nu^{-1/2}\tau)$
ihi)
given
by
are
in Theorem 1.
$Q(\nu^{1/2}\pi,\nu^{-1/2}\sigma)$$fr,\varphi)\dot{u}van\dot{u}hing.
ButuudqmmteW_{F}^{1}’\dot{u}t^{and\sigma_{2}\in\Pi(GL(2,A))\triangleright a\iota;nZ\mathfrak{n}\infty itm1_{1}(\infty ur\iota\epsilon ndy\eta=}*^{2}\cdot)$$Th\infty r\circ ml-Let\sigma_{1}--\chi(\det ffl_{Cn},\alpha_{\epsilon}mimerfl\iota
nc\# onW_{F}’arrow 1ofF=\theta_{2}(\chi(\det)$ロ

not vanuhing, and$W_{F^{2}}^{10}\in Q(\nu^{1/2}\chi^{-1}\sigma_{2p},\nu^{-1/2}\chi)$.

Next, we shall doecribe for the case B), i.e., from Hilbert modular forms over real quadratic fleld $L$$Th\infty r\epsilon m6$–We jfix addtiive chamcter on$L\backslash L_{A}$by$\psi_{L}:=\psi oTr_{L/Q}$.

Thke irreducible unitary $cu pidul rp\epsilon senta\hslash om\sigma\in n(GL(2,L_{A}))$ so that.

30
the central character1$\omega$of is written as$\sigma$by some character$\eta\circ N_{L/Q}$$\omega=\eta\circ N_{L/Q}$

At every plaoe

ofTake an auomo phic form$\bullet$

$w$

$L,$

$\sigma_{w}$whose Whitakker functtonof$Q^{x}\backslash A^{x}$:.

has mitakker mdeb associated to$f\in\sigma^{v}$$\eta$$\psi_{Lw}$.

$W=\otimes_{w}W_{w}\in \mathcal{W}(\sigma^{\vee}, \psi_{L})$satisfies,$W_{w}(zg)=W_{w}(g\{\begin{array}{ll}z ** *\end{array}\})=\omega_{p}^{-1}(z)W_{w}(g)$for any$\{\begin{array}{ll}z ** *\end{array}\}\in\{\delta_{L}^{-l} l\}\Gamma_{0}^{(1)}(c(\sigma_{w}))\{\delta_{L} l\}$

where
1 the
$)$

$\delta_{L}\dot{u}$$\theta$-lift

de d;scriminant. fThen,$F=\theta_{2}(\eta f,\varphi)$$W_{F}^{1,arrow 1}(1)\neq 0$.

$W_{F,p}$is$\eta_{p}^{-1}$(defined in (9)) has a $gbW$ miuaker function-semistable on.

$K(N_{L/Q}(c(\sigma)\delta_{L}^{2}),c(\eta))$

At archimedean plaoe, if $W$ has a (muuiple) weight
for the central character), then the highest weight of(iemark that$(\kappa_{1},\kappa_{2})$$)$$W_{w}\dot{u}$$\kappa_{1}\pm\kappa_{2}$is even, by the $oondit\dot{w}n$$F\dot{u}$$( \frac{\kappa_{1}+\kappa_{2}}{2},$ $\frac{arrow|\kappa_{1}-\kappa_{2}|}{2})$2 If$W_{F}^{1,-1}=\otimes_{v}W_{F,\tau/}^{1,arrow 1}su\iota hu$.

the newform, then$L(s,\sigma_{\eta_{1}}^{\vee})L(s,\sigma_{\eta}^{\vee},)$$Z_{N}(s,W_{Fp}^{1,-1})=\{$$ifp\dot{u}de\infty mpsedoAenvis\epsilon$to$aS_{1}\mathfrak{P}_{2}$,$L(s,\sigma_{p}^{\vee})$

$S)$

Let

$\chi\iota$

be the $\Phi^{ udmc}$ character of $Q$ aasoiciated to $L/Q$ . $F$ is $Ct\varphi idd$, unkfs$\sigma$

is a $kse$ change lift of $\sigma_{1}\in n(GL(2,A))$ and$\eta=\omega_{\sigma_{1}}|_{A^{X}}\chi\iota$.

In this cnse, the dqenamte mitakker fUnction
is not zero.
4 VVhen is decomposed, things are similar to Zheorem 1. We mention about the caseinnert.
J-i) $S\varphi poie$ and. Take so $unt\mu_{1}^{2}=\mu$ . llhen,$W_{F}^{1}$

$)$

$p$

$\sigma_{p}=\pi(\mu,\omega_{p}\mu^{-1})$$\sigma_{p}\dot{u}a 0\infty l$base change of $\sigma_{1p}\in n(GL(2))$ , then.

$I\hslash e$

additive character$\psi_{L\tau v}$

has conductor$[\delta_{L_{w}}c(\sigma_{w})O_{\tau tt}$

$D_{w}=(D(Q)\Phi L)_{w}$ splits at$\sigma$$\delta_{\vec{\iota}_{w}}^{1}$.

. Hence the VVhitakker function$\delta_{L_{wO_{w}}}^{-1}O_{w}]\cap GL(2,L_{w})$Take a deinlte quatermion algebra $D(Q)$ defiined overWhen

is not vanishing and$W_{Fp}^{0,1}$$W_{Fp}^{1_{1}-1},W_{Fp}^{0,1}\in\chi_{L}\sim\sigma_{1p}$Remark 7
invariant on

ramified or$\mu_{1}$$W_{Fp}^{1,-1}\in\eta_{p}\chi\iota\mu x\chi_{L}x\mu_{1}$4-iii) If$p\dot{u}$is holomorphic and$\kappa_{1}\geq 2$$Q$

is semi-.

so thatevery finite place of, and$W_{w}$$\kappa_{2}\geq 2$

.

$L$

.

,

there always exists $\sigma^{JL}\in n(D(L_{A}^{x}))$ with $L(s,\sigma^{JL})=L(s,\sigma)$ The main theorem of Roberts [8] is that,tempered at every place $w$ , the Yoshida liftif
(to be explained later) is not vanishing.
Hrever Blasius [1] showed the $\infty ndition\ln$ this case. Hence we have:$\sigma_{w}1s$$\Theta_{2}(\eta,\sigma^{JL})$

31
Corollary 8 –Let $\sigma\in n(GL(2,L_{A}))$ be in the previOus theonm. Assume
that is holomorphic, and is not a base change lifl. Take a definite quaternion algebm
withof multiple weight$D(Q)$ definied over $Q$ so that$fi\iota\hslash her$$(\kappa_{1},\kappa_{2})$$D_{v}$

Then the Yoshida lift$\sigma$$\kappa_{i}\geq 2$

splils at$\Theta_{\#}(\eta,\sigma^{JL})$

$v$

where

$L_{\nu}/Q_{v}$

is not ramified.

(2)

has a (global) semi-paromoduhr vector, too.

Bsmark 9 The Yoshida lift
is always holomophic. Henoe it doesnot have glObal milakker
functions, although it has local ones at every finite place if we take $D(Q)$ such as (2).
$\Theta_{2}(\eta,\sigma^{JL})$, and its $\infty mplex\infty njugate),$trivial,If the oentral character ofconjugate) provide paramodular vectors. In particular, if$\sigma\ddagger s$$\omega$$\Theta_{2}(1,\sigma)$$\kappa_{1},$$($$=4,\kappa_{2}=2$thaee paramodular vectors are $d\Re ntial$ fom ofto the paramodular group.

1$\theta$$\Theta_{2}(1,\sigma^{JL})$(and its$\infty mpl\alpha$and vioe versa,$h^{2,1},h^{1,2}$and$h^{a,0},h^{0,3}$of the Siegel threefold asociated.lift.

First, we recall thefrom GO $(2m)$ to $GSp(n)$ . Let (X, (, )) be a $2m$ dimensional quadratic spaoe
Q.
over
Let
be the discriminant of $X$ . We denote by GO $(X)$ the group of $Q$ linear automorphisims$X$of such that $(h(x),h(y))=\lambda(x,y)$ with a oertain $\lambda=\lambda(h)\in Q^{x}$ for any $x,yEX$ . Deflne the sign
map $sgn(h)$ $;=\det(h)\lambda(h)^{-2}$ on GO $(X)$ . We set$\theta\cdot 1ffl$$d_{X}$$h$$O(X)=\{h\in GO(X)|\lambda(h)=1\},$ $GSO(X)=ker(sgn),$ $SO(X)=GSO(X)\cap O(X)$$Rs_{0}\in O(X)\backslash SO(X)$$\{1, s_{0}\}\simeq GO(X)$. By the actionthat takne$(h,\delta)$to$s_{0}\cdot h:=s_{0}hs_{0}$$h\delta$. Let

on $GSO(X)$ , we have the isomorphism $GSO(X)x$$\mathcal{R}=\{(g,h)\in GSp(n)xGO(X)|\lambda(g)=\lambda(h)\}$The $WeI1$ representationby$r_{v}$.

of $Sp(n)xO(X)$ related to$\psi_{v}$.

is the unitary repreeentation on $L^{2}(X)$ given$r_{v}(\{\begin{array}{ll}a 00 \ell_{a^{-1}}\end{array}\},$$1)\varphi(x)$$=$$\chi x(\det a)|\det a|^{m/2}\varphi(xa)$$r_{v}(\{\begin{array}{ll}I_{n} b0 I_{n}\end{array}\},$$1)\varphi(x)$$=$$\psi(\frac{1}{2}tr(b(x, x)))\varphi(x)$$r_{v}(\{\begin{array}{ll}0 -I_{\hslash}I_{n} 0\end{array}\},$$1)\varphi(x)$$=$$\gamma\varphi^{v}(xa)$$r_{v}(1, h)\varphi(x)$$=$$\varphi(h^{-1}x)$

,

,

,

.

is defined by the Hilbert symbol $\{*r(-1)^{m}d_{X}\}_{v}$ . is the Weil constant depending only on
the anisotropic $\infty m\mu nent$ of $X,n$ and . The fourier transfomationof is defined byHere$\chi_{X}(*)$$\gamma$$\psi$$\varphi^{\vee}$$\varphi$$\varphi^{v}(x)=\int_{X_{l}^{n}}\psi_{v}(tr(x,y))\varphi(y)dy$

where $da$ is a self dual Haar measure. The Weil representation$r_{v}\ddagger s$$r_{v}(g,h)\varphi_{v}(x)=\lambda(h)^{-n}r_{v}(g\{l \lambda(g)^{-1}\},$

extended to$1)\varphi_{v}(\hslash^{-1}x)$Occasionally, in order to indicate the dependenoe of on , we will write
$r$

and

$(g,h)\in \mathcal{R}(A)$$n$, we set a -series$\theta$$\theta(g,h;\varphi)=a\sum_{e\epsilon x(k)^{n}}r(g,h)\varphi(x)$,$r^{n}$$\mathcal{R}(k_{\nu})$

by

.

. For

$\varphi=\prod_{v}\varphi_{v}\in S(X(A)")$32
which $\infty nvergae$ absolutely and is left

invariant. This 9-series gives the following -lifts.

$\mathcal{R}(k)$$\theta$$\theta_{n}(f,\varphi)(g)=\int_{o(X,k)\backslash O(X,A)}\theta(g,h_{1}h;\varphi)f(h_{1}h)dh_{1}$,(3)where $f$ is a cuspfom on GO $(X,A)$ , and $(h,g)\in \mathcal{R}(A)$ . For a cuspidal $\tau\in\Pi(GO(X,A))$ , we denote by
the subepace of automorphic foms generated by
for $f\in\tau$ and $\varphi ES(X(A)^{n})$$X(Q)$
Now, suppose
is adlmensional space $(m=2)$ and $dE(Q^{x})^{2}$ . Then $X$ is isometric toa quatemion $\Phi bra$ $(B(Q), (,))$ defined over $Q$ with $(x,y)=\# tr(xy’)$ . Here indicates the canonicalinvolution and $tr(y)=y+y$ The norm of will be denoted by $n(y)$ . Put$\Theta_{n}(\tau)$.

$\theta_{n}(f,\varphi)$$b\prime r$.

$\iota$$y$$H(Q)=B(Q)^{x}xB(Q)^{x},$The action$\rho$$H^{1}(Q)=\{(b_{1},h)\in H(Q)|n(b_{1})=n(b)\}$.

(4)

of $H$ on $B$ defined by$\rho(b_{1},h)x=b_{1}^{-\iota}x4$

an isomorphism $GSO(B)\simeq H/Q^{x}$ and $SO(B)\simeq H^{1}/Q^{x}$ , whereis embedded tnto $H$
diagonally. If
have a oommon oentral character , thencan be regarded as$\mathbb{I}(GSO(B,A))$ロ.
an element inwith central charoeter If the indueed representation
ロ $\sigma_{2}\in n(GO(X,A))$ . Otherwise, the induced repreeentation isis irrducible, we denote it also byロ. This can happen only when
divided into two $\infty mtituents(\sigma_{1}$ ロ
However,
and
ロ
ロ
sinoe
is always vanishing, we will treat onlyand denote it also by ロ .
ロWhen $B(Q)$ is a deflnite quatmion algebrais the “Yoahida lift of the first type” (c.f. [14]).
Next, suppose $X(Q)$ is four dimensional and. In this case, $:=Q(\sqrt{d_{X}})$ is a quadratic
field. $X(Q)$ is isometric to$indu\varpi$$Q^{x}$$\sigma_{1},\sigma_{l}\in \mathbb{I}(B(A)^{x})$$\sigma_{1}g_{\sigma_{2}}$$\eta$$Ind_{GSO(\acute{X},A)}^{\varpi\langle XA)}\sigma_{1}$$\eta$$\sigma_{1}$$\sigma_{2})^{+}$$\sigma_{2})^{-}$$(\sigma_{1}$$\sigma_{1}=\sigma_{2}$$\sigma_{2})^{-})$$\Theta_{2}((\sigma_{1}$$\sigma_{2})^{+}$$(\sigma_{1}$$\Theta_{2}(\sigma_{1}$.

$\sigma_{1}$$\sigma_{2}$$\sigma_{2})$$L$$d_{X}\not\in(Q^{x})^{2}$$\{b\in B(L)|i=b^{l}\}$$\sigma_{l}$or $\{b\in B(L)|i=-b\}$

for a quaternion algebra $B(L)=B(Q)\theta L$ . Hereof those of . Put$b’\epsilon\infty efflcient\epsilon$

in

$L$

are the algebraic conjugate over

$Q$

$b$

$H’(Q)=Q^{x}xB(L)^{x},$The action$\beta$$H^{\prime 1}(Q)=\{(t,b)\in H’(Q)|n(b)=t^{2}\}$.

of $H’$ on $X$ deflned by$\rho’(t,b)x=t^{-1}b’xb$, whereinduces an isomorphism $GSO(X)\simeq H’/Q^{x}$ andbyis embedded into$t\mapsto(t,t)$ . If $\tau\in n(B(A_{L})^{x})$ has a oentral character, then we can regard
as an element
in $n(GSO(X,A))$ wlth oentral character . If its induced representation to GO $(X,A)$ is irreducible,
we denote lt also by $(\eta,\tau)\in n(GO(X,A))$ . Otherwise, the induced $oepr\infty ntation$ is divided into two$\infty nstituents(\eta,\tau)^{+}$ and. This can happen only when $\tau\in\Pi(B(A_{L})^{x})$ is a base change lift ofsome $\sigma\in n(B(A)^{x})$ . However, since
is always vanishing, we only treat
and denote it. When $d_{X}>0$ and $B(Q)$ is a deMte quaternon algebra,
also byis the $uYoshu_{oM}$ of$SO(X)\simeq H^{\prime\iota}/Q^{x}$$H^{\prime 1}$$Q^{x}$$(\eta,\tau)$$\eta\circ N_{L/Q}$$\eta$$(\eta,\tau)^{-}$$(\eta,\tau)^{+}$$\Theta_{2}((\eta,\tau)^{arrow})$$(\eta,\sigma)$$\Theta_{2}((\eta,\tau))$the seoond type“.

2

Schwartz function.

First, let us treat the case A). Take $W_{1}=@_{v}W_{1v}\in W(\sigma_{\check{1}},\psi)$ and $W_{l}=\Phi_{v}W_{lv}\in W(\sigma_{2},\psi)$ as in
Theorem 1. Corregponding them, we define Schwartz functlon $\varphi_{v}\in S(M_{l}(Q_{v})^{2})$ as follows.
At $aoehimd\alpha n$ place
We choose two polynomials of $M_{2}(R)$$($$\infty)$$P_{1}(x)=i(4-a_{l})-(b_{l}+e_{*}),$$B(x)=i(a_{l}+4)+(c_{-} -\ )$ ,where we write$x=\{\begin{array}{ll}a_{l} b_{r}\ d_{l}\end{array}\}$.

These polynomials have the properties.

$P_{1}(u_{l_{1}}^{-1} t4,)=e^{-i\langle\theta_{1}+*)}P_{1}(x),h(\overline{h_{\iota}}^{1_{Xu,)}}=e^{l(1_{1}-\partial,)}R(x)$

33
for

$u_{\theta}=\{\begin{array}{ll}m\theta 8in\theta-\dot{n}n\theta c\infty\theta\end{array}\}$. Let$\epsilon_{1},s_{2}$$\varphi_{\infty}(x_{1},x_{2})$be indeterminants. Define$=$$\varphi_{\infty_{t}}\in S(M_{2}(R)^{2})\otimes C[\iota_{1},s_{2}]\Psi$$\propto p(-\pi tr(R[x_{1},x_{2}|))P_{1}(s_{1}x_{1}+s_{2}x_{2})^{\alpha}$$x\{\begin{array}{l}P_{2}(\epsilon_{2}x_{1}-\epsilon_{1}x_{2})^{\beta} lf \kappa 1\leq\kappa_{2},F_{2}(s_{2}x_{1}-s_{1}x_{2})^{\beta} otherwise,\end{array}$.

andHere symmetic $m\iota t\dot{m}R\in M_{4}(Q)$ is chosen so that $R[x_{t}]=$and is regarded as a line vedor $(so R[oe_{1},x_{2}]={}^{t}(R[x_{1},x_{2}])\in M(2,R))$ . $R$ is an$m\ddot{m}mal$ majorant of the symmetricmatirix corresponding to the quadratic fom in $M(2)$ ,Hermite’s

where$\alpha=arrow^{\hslash_{1}}\neq$$\beta=1_{\Delta^{-d}}^{n_{1}}\neq$$a_{x_{i}}^{2}+\# oel+d_{r_{l}}+d^{2}oer$i.e., $RQ^{-1}R=Q$$x_{l}$$Q$.

(At finite plaoe ) Let $c_{i}=c(\sigma_{ip})$ be the $\infty nductor$ of$p$$\infty mmon$

oentral character$\eta’$$\varphi_{p}(x_{1},x_{2})=$. In the case of $f=0$, deflne$\sigma_{ip}$the characteristic function of

and $c=c_{1}c_{2}$ . Let$f$be the $\infty nductor$ of the$(\{\begin{array}{ll}c_{2} Z_{p}‘ c_{1}\end{array}\}\oplus M(2,Z_{p}))$.

In the case of $f>0$, we deflne$\varphi_{p}(x_{1},x_{2})=\{\begin{array}{l}\eta_{p}(b_{g_{1}}), if (x_{1},x_{2})\in[c_{2}e z_{C_{1}^{p}}^{x}]\oplus M_{2}(Z_{p}),0, otherwise.\end{array}$

Then (3) is wnitten as$\theta_{2}(f_{1}$ロ $f_{2}, \varphi)(g)=\int_{A^{x}H^{1}(Q)\backslash H^{1}(A)_{g_{l}}}\sum_{\epsilon u_{*(Q)}}(r(g,h)\varphi)(x_{1},x_{2})f_{1}$ ロ $f_{2}(hh’)dh$

Whittaker functlon of$\theta(fi$.

(6)

ロ $fa,\varphi)$ : We selaet a palr of elements$e_{-1}=[$$-1]$$\alpha_{-1}=\{arrow 1 l\}$.

Put
$Z_{0}(Q)$$=$$\{h=(h_{1}, h_{2})\in H^{1}(Q)|\rho(h)e_{arrow 1}=e_{arrow 1},\rho(h)\alpha_{-1}=\alpha_{-1}\}$$=$$\{(\{1 xl\},$ $\{1 -x1\})|x\in Q\}$

.

(6)

(7)

The global VVhittaker function$W_{F}^{1,-1}(g):= \int_{U(k)\backslash U(A)}\overline{\psi_{1,\sim 1}}(u)F(ug)du$of $F=\theta(g;f_{1}$ ロ $fa)$ is calculated as.

(8)$\int_{Z_{Q}(A)\backslash H^{1}(A)}r(g,h)\varphi(e_{-1},a_{-1})W_{1}(h_{1})W_{2}(h_{2})dh$

We can calculate each local$f\infty tors$of (8).

$I_{v}(g)= \int_{Z_{0}(Q.)\backslash H^{1}(Q_{w})}r_{v}(1,h)\varphi_{v}(e_{-1},\alpha_{-1})W_{1v}(h_{1})W_{lv}(h_{2})dh$

at arbitary$g$. In particular,$I_{v}(1)\neq 0$. Henoeロ. By calculating$W_{F}^{1,arrow 1}\neq 0$$W_{Fv}^{1,-1}(g)=I_{v}(g),$$2)$$\sigma_{2})y\mathfrak{R}_{-}^{thed}\%_{(g),W_{p_{v}}^{0,1}(g)}^{enerat\propto
1WNtm_{r}}$$hmctionW_{p}^{1,0}orW_{F},a\dot{m}i_{8}obund.Thenon_{1}c_{i\epsilon n\circ tzero.4)byca1cu1atingW_{Fv}^{1}(g),W_{Fv}^{1}}u\epsilon
pida1ity3)of\Theta_{2}(\sigma_{1}$34
Next, we treat the case B). Write $L=Q(\epsilon)$ by $\epsilon\in O=O_{L}$ , where
and Hilbert modular fom $f$ in Theorem 6, we put

character$f_{\eta}((t,h)):=\eta^{-1}(t)f(h)$

Let

$c$

is squarefree. For the$\epsilon^{2}\in Z$$\eta$.

be the generator of Gal $(L/Q)$ . Let$X(Q)$$=$$\{x\in M_{2}(L)|c(x’)=-x\}$$=$$\{\{\begin{array}{ll}b b_{l}c_{r} -a_{\ae}^{\epsilon}\end{array}\}|b_{l},c_{x}\in Q\}$.

and set the quadratic $bm(x,y)=tr(xy’)$ in $X(Q)$ . Then, we define Schwartz functions, $\infty rr\infty ponding$to the above and $W$ in Theorem 6, as follows.
(At ): With the ume polynoimial
as in the case A), define$\eta$$P_{1},P_{2}$$\infty$$\varphi_{\infty}(x_{1},x_{2})$where $\alpha=\varphi$ andof the quadratic fom$($At$\mathfrak{p}=\mathfrak{P}^{2},$$=$$\propto p(-\pi\sum_{*arrow 1}^{2}(a_{g_{i}}^{2}+(a_{l_{i}}^{c})^{2}+P_{g_{i}}+c_{x_{i}}^{2}))P_{1}(s_{1}x_{1}+\iota_{2}x_{2})^{\alpha}$$x$$\{\begin{array}{l}Ri (s_{2}x_{1}-e_{1}x_{2})^{\beta} if \kappa_{1}\leq\infty,F_{2}(\epsilon_{2}x_{1}-\epsilon_{1}x_{2})^{\beta} otherwise,\end{array}$$\beta=L^{\kappa}\lrcorner^{-}\mu$$($$c(\mu’)=0$

,

$)$

.

. Here$(a_{x\iota}^{2}+(a^{c}ae\iota)^{2}+\#_{x_{i}}+c_{l}^{2})$cue : Se set, for$)$is$\infty rr\infty pond\dot{m}g$

to a minimmal majorant$y\in X_{p}$$\varphi_{p}^{0}(y)=\{\begin{array}{ll}\chi_{L,p}(\ovalbox{\tt\small REJECT}_{V}), if y\in[_{p}o_{h^{\Phi}} \varpi_{O_{\mathfrak{P}}}^{-1}Z_{p}^{x}],0, otherwise.\end{array}$Take the summation$\varphi_{p}^{1}(y)=\varphi_{p}^{0}(y)+\sum_{\dot{*}\epsilon z/pZ}r_{p}^{1}(\{\begin{array}{ll}i -11 \end{array}\},$

which$\ddagger s$

not identically zero. Let$\iota=ord_{\Phi}(c(\sigma_{\mathfrak{P}}))$. When,$\varphi_{p}(x_{1},x_{2})=\varphi_{p}^{1}(p^{-1}c^{-1}[\epsilon^{e}$When${}^{t}b$$\eta_{p}$$1)\varphi_{p}^{0}(y)$,is trivial, deflne$1]x_{1}[(\epsilon^{c})^{-*}$$1])\varphi_{\mathfrak{p}}^{1}(x_{2})$is not trivial, define$\varphi_{p}^{\eta}(y)=\mathfrak{w}(p^{1}c_{y})\varphi_{p}^{0}(y)$for $y\in X_{p}$ and$\varphi_{p}(x_{1},x_{2})=\varphi_{p}^{\eta}(p^{-1}c^{-1}\{\epsilon^{e} 1\}x_{1}[(\epsilon^{c})^{-*}$(At$\mathfrak{p}=\mathfrak{P}$in$1])\varphi_{l}^{1}(x_{2})$in $L,$ $c(\mu_{p})=0$ case): Set$\varphi_{p}^{0}(x)=$ch $(X_{p}\cap M_{2}(O_{p});x)\in S(X_{p})$ ,ch denotes the characteristic function. Deflne$\varphi_{p}(x_{1},x_{2})=\varphi_{p}^{0}(c^{-1}\{p^{*} 1\}x_{1}\{p^{-} l\}\varphi_{p}^{0}(x_{2})$with $e=e(\sigma_{p})$ .
$c(\mu_{p})>0$ case : Let $\mu^{0}(y)=\mu(y)$ . chAt$($

$p=\mathfrak{P},$$)$$(0_{\mathfrak{p}}^{x};y)$. We define$\varphi_{p}^{\mu}(x)=\mu^{0}(c_{r})\varphi_{p}^{0}(x)$

.

.

35
and
$\varphi_{p}(x_{1:}x_{2})=\varphi_{p}^{\mu}(p^{-c}\{p^{\epsilon} 1\}x_{1}\{p^{-\epsilon} 1\})\varphi_{p}^{1}(x_{2})$.

$GSp_{l}(Q)^{N}=\{g\in GSn(Q)|\nu(g)\in N_{L/Q}(L^{x})\}$.

Put

Define$\theta(\eta f,\varphi)(g)=\int_{A^{x}(H’)^{1}(k)\backslash (H’)^{1}(A)_{g_{i}}}\sum_{\in M_{l}(k)}(r(g,h)\varphi)(x_{1},x_{2})f_{\eta}(hh’)dh$.

(9)Here $h’=(1, h_{0}’)\in H(A)$ is choeen so that $\nu(g)=N_{L/k}\det(h_{0}’)^{arrow 1}$ , md we embed $A^{x}\ni trightarrow(t^{2},t)\in$$(H’)^{1}(A)$ . Slnoe
is $1eRGSn(Q)^{N}$ -invariant,canto a hmction on $GSn(Q)\backslash GSn(A)$by insisting that it is left $GSp_{2}(Q)$ -invariant and zero outside of $GSn(Q)GSn(A)^{N}$ .
$\theta(\eta f,\varphi)$$n|e$$\infty\zeta tend\theta(\eta f,\varphi)$

Rmarks. When both of $\sigma_{1},\sigma_{2}\in\Pi(GL(2,A))$ are holomorphic of weight 2 in the case and whenis holomorphic of weight (2,2) in the case $B$ , the -lifts can be the generic Siegel
modular foms corresponding some abelian surface, similar to the Yoshida lift. That is,$A$$\sigma\in n(GL(2,L_{A}))$$\theta$(A) Let $f1,f_{2}\in S_{2}(\Gamma_{0}(N_{i}))$ be elliptic cuspforms of welght 2 with level $N_{1},N_{2}$ . Then there exists$F\in M_{2_{i}0}(K(N_{1}N_{2}))$$\infty rr\infty pondlng$(B) Letto the $GSp(2)$-valued Galois repreeentation$f\in S_{(2_{t}2)}(\Gamma_{0}(\mathfrak{n}))$$\rho f_{1}\oplus\rho_{f},$.

be a Hilbert cuspform of multiple might (2, 2) of level$\mathfrak{n}$. Then there exists$F\in M_{2_{i}0}(K((N_{L/Q}(\mathfrak{n}\delta_{L}^{2})))$$\infty rr\infty ponding$to the $GSp(2)$ -valued Galois representation$Ind_{G\cdot 1(r/L)}^{G\cdot t(\Phi/Q)}\rho f$.

(A) means that all imbian varieties of elliptic modular curves of genus 2 are Siegel modular in the genericsenoe, e.g., product of elliptic curves. (B) means all motives of Hilbert modular forms over a real quadratic
fleld of welght (2,2) are slSo Siegel modular, e.g, jacobian of Shlmura curves obtained by lndefinite
quatemion algebrm, and abelian surfaoe with $\infty mplex$ multiplication of quartic CM-fleld. However,according to Przebinda , the archimedan $\infty mponent$ of $F$ belongs a -principal series repereeentation($c.f$ p.904 of [6]), not a (limit of) discrete series representation.
is an imaginary quadratic fleld, we can also
-lift to $GSp(2)$ from oertain classes in
II $(GL(2,L))$ . In this case, we identify $GL(2,L)$ with GO $(3,1,Q)$ . But, dithirent ftom the real quadratic
case, the spueof the imeages of -lift is $de\infty mp\propto ed$ as follows.
$[7|$$P_{1}$$\ddagger fL$$\infty n\epsilon ider\theta$$\Theta_{2}(\sigma)$$\theta$$\Theta_{2}(\sigma)$“$=$$\Theta_{2}(\sigma)^{p}$$+$$\Theta_{2}(\sigma)^{gen}$$+$$\Theta_{2}(\sigma)^{hd}$of highest weight $(N, 1)$of highest weight $(N,0)$of highest weight $(N,2)$.

(More strictly, we have three ways to extend
to $n(GO(3,1,R))hr$ nontrivial -lift. rme \S 3 and table inp. 394 of $[4|)$ . Similar to Theorm 6, we have non-vanishing offor such classes $\sigma\in n(GL(2,\mathcal{L}_{A})$ ,i.e., me can almays generic Siegel modular foms which are semistable on semi-paramodular groups.
But, different $\Re m$ the real quadratic case, this -lift may provides holomorphic Siegel modular foms.
We cannot say. See [4] for some nonvanishlng $\infty nditions$$\theta$$\sigma_{\infty}$$\Theta^{g\epsilon n}(\sigma)$$\theta$$\Theta_{2}^{hd}(\sigma)\neq 0$.

ACKNOmEDGEhIBNT: We thanks to Proffisor T. Ibukiyama, T. Moriyama and H. Yoehida for

their helpful advioe.

36
References[1] D. Blasisus: Hilbert modular foms and Ramaniijan conjecture, arXiv $:math/0611\infty 7vl$ [math.NT](2008)[2 M. Harris, S. Kudla Arithmetic automophic forms for the nonholomorphic discrete series of $GSp(2)$ ,Duke Math 66 (1992), 6$*$ 121.
$|$

[3] R. Howe, I.L Piatetski-Shapiro: Some examples of automorphic forms on$W106$.

$Sp_{4}$, Duke. math 50 (1983),[4] M. Harris, D. Soudry, R. kylor: l-adic representations rciated to modular
quadratic flelds, Invent. math. 112 (1993), 377411.
$[5|$$bm\epsilon$

over imaginary

H. Jacquet, R.P. Langlands: $Aut\sigma m\Phi phie$ forms on $GL(l)$ , L.N.M. 114 (1970), Springer.

[6] T. Moriyama: Entireneee of the spinor L-functions for rtain generic cusp forms on $GSp(2)$ , Amer.
J. Math. 27 (2002), $89k920$ .
[7] T. Przebinda: The oscillator duahty $\infty rraepondenoe$ br the pair $O(2,2),Sp(2,R)$ , Mmoirs of A.M.
79 Number 403 (1989).

$S$[8] B. Roberts: Global $L$-pa&ts for GSp(2) and theta lifts, Docu. math 6 (2001) u7-314.

[9] B. Roberts, R. Schmidt: Local newforma for $GSp(4)$ , L.N.M. 1918 (2007), Spninger.
[10] P J. Sally, M. Thd16: Induoed rerpeeentations and classiflcation for $GSp(2,F)$ and $Sp(2,F),$$Math4matique$ de $rPanoeM4moirel2$ (1993), 76-133.

$So\dot{\alpha}\ell t6$[11] R. Schmidt: The Saito-Kurokawa lift and functoriarity, Amer. J. math $(2\alpha\}6),$ $2\Re 240$ .

[12] G. Shimura: Abelian Varieties with Complex multiplication and Modular Functions, Princeton univ.
(1998).
$pr\infty_{J}$[13] R. $m\infty Bighuh$ : Lfimctions for the $\mu adlc$ group GSp(4), Amer. J. Math. 122 (2000), 1085-1120.

[14] H. Yoeludec Siegel’s modular forms and the arithmetics of quadratic forms, Innnt. math. 00 (1980),

193248.

Takeo OWaki, Department of Mathematics, Fbcidty of Scienoe, Kyoto University,, JAPAN.
Kyoto,
Email: okazaki@math.kyotO-u.ac.jp$\mathfrak{W}- 8602$