barsotti tate groups and p adic representations of the fundamental group scheme on a scheme s over a base scheme b we study the category of locally constant bt groups i e groups over s that are twists in the flat topology of bt groups defined over b these groups generalize p adic local systems and can be interpreted as integral p adic representations of the fundamental group scheme of s b classifying finite flat torsors on the base scheme when such a group exists we generalize to these coefficients the katz correspondence for p adic local systems and show that they are closely related to the maximal nilpotent quotient of the fundamental group scheme

morphisms of f isocrystals and the finite monodromy theorem for unit root f isocrystals we discuss tate type problems for f isocrystals that is the full faithfulness of the natural restriction functors between categories of overconvergent f isocrystals on schemes of positive characteristic we prove it in the cases of unit root f isocrystals using this result we prove that an overconvergent unit root f isocrystal has a finite monodromy

truncations of level 1 of elements in the loop group of a reductive group the aim of this article is to dene and study a new invariant of elements of loop groups that is invariant under conjugation by a hyperspecial maximal open subgroup and that we call the truncation of level 1 we classify truncations of level 1 and describe their specialization behavior furthermore we prove group theoretic conditions for the set of conjugacy classes obtained from elements of a given truncation of level 1 and in particular for the generic conjugacy class in any given truncation stratum in the last section we relate our invariant to the ekedahl oort stratication of the siegel moduli space and to generalizations to other pel shimura varieties

conformal blocks and generalized theta functions let yp be the moduli space of rank r vector bundles with trivial determinant on a riemann surface x this space carries a natural line bundle the determinant line bundle s we describe a canonical isomorphism of the space of global sections of sk with the space of conformal blocks defined in terms of representations of the lie algebra 5 c z it follows in particular that the dimension of h yk is given by the verlinde formula

minimal p divisible groups a p divisible group x can be seen as a tower of building blocks each of which is isomorphic to the same finite group scheme x p clearly if x1 and x2 are isomorphic then x1 p x2 p however conversely x1 p x2 p does in general not imply that x1 and x2 are isomorphic can we give over an algebraically closed field in characteristic p a condition on the p kernels which ensures this converse here are two known examples of such a condition consider the case that x is ordinary or the case that x is superspecial x is the p divisible group of a product of supersingular elliptic curves in these cases the p kernel uniquely determines x

words avoiding repetitions in arithmetic progressions carpi constructed an infinite word over a 4 letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference we show a connection between carpi s construction and the paperfolding words we extend carpi s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference we also construct infinite words avoiding overlaps and infinite words avoiding all sufficiently large squares in arithmetic progressions of odd difference we use these words to construct labelings of the 2 dimensional integer lattice such that any line through the lattice encounters a squarefree resp overlapfree sequence of labels

the newton stratification on deformations of local g shtukas bounded local g shtukas are function field analogs for p divisible groups with extra structure we describe their deformations and moduli spaces the latter are analogous to rapoport zink spaces for p divisible groups the underlying schemes of these moduli spaces are a ne deligne lusztig varieties for basic newton polygons the closed newton stratum in the universal deformation of a local g shtuka is isomorphic to the completion of a corresponding a ne deligne lusztig variety in that point this yields bounds on the dimension and proves equidimensionality of the basic a ne deligne lusztig varieties

transcendence of formal power series with rational coefficients we give algebraic proofs of transcendence over q x of formal power series with rational coefficients by using inter alia reduction modulo prime numbers and the christol theorem applications to generating series of languages and combinatorial objects are given

on the lp norm of spectral clusters for compact manifolds with boundary we use microlocal and paradifferential techniques to obtain l8 norm bounds for spectral clusters associated with elliptic second order operators on two dimensional manifolds with boundary the result leads to optimal lq bounds in the range 2 q for l2 normalized spectral clusters on bounded domains in the plane and more generally for two dimensional compact manifolds with boundary we also establish new sharp lq estimates in higher dimensions for a range of exponents qn q

displays and formal p divisible groups over p adic nagata rings formal p divisible groups are classified by nilpotent displays according to t zink we extend this result to arbitrary p adic rings the proof uses the grothendieck illusie deformation theory of truncated p divisible groups

appendix on parahoric subgroups we give the proofs of some simple facts on parahoric subgroups and on iwahori weyl groups used in t haines the base change fundamental lemma for central elements in parahoric hecke algebras preprint 2008 g pappas m rapoport twisted loop groups and their affine flag varieties adv math 219 2008 118 198 m rapoport a guide to the reduction modulo p of shimura varieties asterisque 298 2005 271 318

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

stratifications of newton polygon strata and traverso s conjectures for p divisible groups the isomorphism number resp isogeny cutoff of a p divisible group d over an algebraically closed field of characteristic p is the least positive integer m such that d p m determines d up to isomorphism resp up to isogeny we show that these invariants are lower semicontinuous in families of p divisible groups of constant newton polygon thus they allow refinements of newton polygon strata in each isogeny class of p divisible groups we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers which is shown to be optimal in the isoclinic case in particular the latter disproves a conjecture of traverso as an application we answer a question of zink on the liftability of an endomorphism of d p m to d

newton polygons and formal groups conjectures by manin and grothendieck we consider p divisible groups also called barsotti tate groups in characteristic p their deformations and we draw some conclusions for such a group we can define its newton polygon abbreviated np this is invariant under isogeny for an abelian variety in characteristic p the newton polygon of its p divisible group is symmetric in 1963 manin conjectured that conversely any symmetric newton polygon is algebroid i e it is the newton polygon of an abelian variety this conjecture was shown to be true and was proved with the help of the hondaserre tate theory we give another proof in section 5 grothendieck showed that newton polygons go up under specialization no point of the newton polygon of a closed fiber in a family is below the newton polygon of the generic fiber in 1970 grothendieck conjectured the converse any pair of comparable newton polygons appear for the generic and special fiber of a family this was extended by koblitz in 1975 to a conjecture about a sequence of comparable newton polygons in section 6 we show these conjectures to be true these results are obtained by deforming the most special abelian varieties or p divisible groups we can think of in describing deformations we use the theory of displays this was proposed by mumford and has been developed in 17 18 and recently elaborated in 32 and 33 also see 11 31 having described a deformation we like to read off the newton polygon of the generic fiber in most cases it is difficult to determine the newton polygon from the matrix defined by f on a basis for the deformed dieudonne module in general i have no procedure to do this e g in case we deform away from a formal group where the dieudonne module is not generated by one element however in the special case we consider here a go 1 a noncommutative version of the theorem of cayley hamilton every matrix satisfies its own

foliations in moduli spaces of abelian varieties in this paper we study abelian varieties and p divisible groups in characteristic p even though a non trivial deformation of an abelian variety can produce a non trivial galois representation say on the tate group of the generic fiber in any characteristic the geometric generic fiber has a constant tate group we en counter the same phenomenon for the p structure in positive characteristic any two ordinary abelian varieties of the same dimension over an algebraically closed field have isomorphic p divisible groups however for non ordinary abelian varieties this seems to break down the fas cinating structure which comes out of this is that the maximal locus where a given geometric isomorphism class of a p divisible group is realized e g in a family of abelian varieties is a locally closed set the locus defined by the geometric iso morphism type of a p divisible group will be called a central leaf see 2 1 and 3 4 this gives rise to a foliation of the open stratum attached to a newton polygon the dimension of any leaf in the same newton polygon stratum solely depends on in extreme cases either the leaf is the whole stratum as in the ordinary case or in the almost ordinary case the p rank equals g 1 or a leaf is zero dimensional as in the supersingular case in intermediate cases a leaf can be a proper subset and still be positive dimensional we have worked out the example of g 4 in 8 1

newton polygons as lattice points let m be the reduction modulo p of a shimura variety and let n m be the partially ordered set predicted to be the set of all types occurring in the stratification of m by newton polygons in the context of f isocrystals with additional structures we prove that n m is ranked or catenary in the sense that any two maximal chains have the same length we also give a lie theoretic formula for the distance between two comparable elements in n m this formula can be used to give the expected dimension of the newton strata of m

tate s conjecture for k3 surfaces of finite height this paper extends the proof 16 of the tate conjecture for ordinary k3 surfaces over a finite field to the more general case of all k3 s of finite height as in 16 our method is to find a lifting of the k3 to characteristic zero with sufficiently many hodge cycles in the ordinary case the so called canonical lifting of deligne and illusie 7 did the job and a study of the galois action on p adic etale cohomology revealed the hodge cycles here we use more general quasi canonical liftings and the action of the crystalline weil group on

simple p kernels of p divisible groups a p divisible group x determines its p kernel x p g we show that g determines x uniquely if g is minimal and that there are infinitely many possibilities for x if g is not minimal the indecomposable minimal g are precisly those which are simple

overconvergent witt vectors let a be a finitely generated algebra over a field k of characteristic p 0 we introduce a subring w a w a which we call the ring of overconvergent witt vectors and prove its basic properties in a subsequent paper we use the results to define an overconvergent de rham witt complex for smooth varieties over k whose hypercohomology is the rigid cohomology

ray normal mode and hybrid analysis of transient waves in a finite beam abstract general solutions for wave propagation in a beam with semi infinite or finite length based on the timoshenko theory are presented the solutions are derived on the basis of the ray theory the normal mode theory and a hybrid formulation which combines these two theories to generate a solution valid for short time and near field as well as long time and far field all three formulations can be formulated rigorously from a unified treatment of the inverse integral transforms finite beams with fixed hinged or free boundary conditions are considered and a numerical example is shown to illustrate the comparison of three methods

codimensions of newton strata for sl3 f in the iwahori case we study the newton stratification on sl 3 f where f is a laurent power series field we provide a formula for the codimensions of the newton strata inside each component of the affine bruhat decomposition on sl 3 f these calculations are related to the study of certain affine deligne lusztig varieties in particular we describe a method for determining which of these varieties is non empty in the case of sl 3 f

serre tate theory for shimura varieties of hodge type we study the formal neighbourhood of a point in the mu ordinary locus of an integral model of a hodge type shimura variety we show that this formal neighbourhood has a structure of a shifted cascade moreover we show that the cm points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms analogous to the serre tate canonical lift of an ordinary abelian variety

hybrid ray mode fields in inhomogeneous waveguides and ducts high frequency fields radiated by sources in transversely inhomogeneous waveguides or ducts are conventionally analyzed by ray or normal mode techniques when the duct width is very large the excessive number of relevant rays or modes introduces computational difficulties an alternative approach is provided by a hybrid representation that involves ray fields and modal fields in well defined combinations with respect to the angular domain centered at the source point the modes fill those angular intervals left vacant by rays and vice versa the mode angles being defined as the angles of the upgoing and downgoing modal ray congruences as seen from the source the hybrid mix is dependent on the position of the observer and an additional remainder field assures the smooth transition from one hybrid combination to another the analytical formulation involving partial poisson summation of the formal ray series is applied to various ducting environments that cause wave trapping by refraction reflection from boundaries or both for the special case of monotonically varying refractive index in a parallel plane waveguide the results from the asymptotic method are verified by rigorous green s function methods special examples supported by numerical comparisons with exact reference solutions are presented to demonstrate the validity and utility of the hybrid scheme

on the slope stratification of certain shimura varieties in this paper we study the slope stratification on the good reduction of the type c family shimura varieties we show that there is an open dense subset u of the moduli space such that any point in u can be deformed to a point with a given lower admissible newton polygon for the siegel moduli spaces this is obtained by f oort which plays an important role in his proof of the strong grothendieck conjecture concerning the slope stratification we also investigate the p divisible groups and their isogeny classes arising from the abelian varieties in question

the transformation between the mode representation and the generalized ray representation of a sound field abstract the mode generation function and the generalized ray generation function b u are introduced for a stratified wave guide it is proved that and b u are a fourier transform pair when lateral waves do not appear mode representation and the generalized ray representation satisfy the poisson summation formula furthermore the local conversion relation between the mode and the ray is considered from the point of view of member selection it is found that local conversion is just the result of using the stationary phase approximation in the fourier transformation of and b u this is the same result as obtained by selective processing in which the constructive interference between adjoining members of the ray or mode families is used

ekedahl oort and newton strata for shimura varieties of pel type we study the ekedahl oort stratification for good reductions of shimura varieties of pel type these generalize the ekedahl oort strata defined and studied by oort for the moduli space of principally polarized abelian varieties the siegel case they are parameterized by certain elements w in the weyl group of the reductive group of the shimura datum we show that for every such w the corresponding ekedahl oort stratum is smooth quasi affine and of dimension ell w and in particular non empty some of these results have previously been obtained by moonen vasiu and the second author using different methods we determine the closure relations of the strata we give a group theoretical definition of minimal ekedahl oort strata generalizing oort s definition in the siegel case and study the question whether each newton stratum contains a minimal ekedahl oort stratum as an interesting application we determine which newton strata are non empty this criterion proves conjectures by fargues and by rapoport generalizing a conjecture by manin for the siegel case we give a necessary criterion when a given ekedahl oort stratum and a given newton stratum meet

conditional entropy of some automatic sequences burrows and sulston 1991 have introduced conditional block entropies hn from information theory in order to give a quantitative measure of disorder for sequences and if possible a characterization of quasicrystalline sequences we give here some properties concerning these block entropies and give an explicit formula for the sequences hn n in n corresponding to the thue morse sequence the rudin shapiro sequence and the paperfolding sequence we deduce from these computations that this measure of disorder does not enable us to distinguish between deterministic sequences even if they have different spectral properties

about the p paperfolding words abstract let p be an integer greater than or equal to 2 the aim of this paper is to study the language associated to a p paperfolding sequence it is known that the number of factors of length n of a 2 paperfolding sequence i e its complexity function is p n 4n for n 7 it is also known that the language of all the factors of all 2 paperfolding sequences is not context free and that its generating function is transcendental we show that the complexity function of a p paperfolding sequence is either strictly subaffine or ultimately linear the first case never happens if p 2 or 3 in the second case the complexity function is either p n 2n or p n 4n for n large enough we give a simple necessary and sufficient condition for the number of special factors to be p automatic we finally show that for any given p the language of all factors of all p paperfolding sequences is not context free and that the associated generating series is not algebraic

local monodromy of p divisible groups a p divisible group over a field k admits a slope decomposition associated to each slope is an integer m and a representation gal k gl m d where d is the p division algebra with brauer invariant we call m the multiplicity of in the p divisible group let go be a p divisible group over a field k suppose that is not a slope of g 0 but that there exists a deformation of g in which appears with multiplicity one assume that s 1 s for any natural number s 1 we show that there exists a deformation g r of g 0 k such that the representation gal fracr gl 1 d has a large image

a p adic local monodromy theorem we produce a canonical filtration for locally free sheaves on an open p adic annulus equipped with a frobenius structure using this filtration we deduce a conjecture of crew on p adic differential equations analogous to grothendieck s local monodromy theorem also a consequence of results of andre and of mebkhout namely given a finite locally free sheaf on an open p adic annulus with a connection and a compatible frobenius structure the module admits a basis over a finite cover of the annulus on which the connection acts via a nilpotent matrix

generalization of deuring reduction theorem abstract in this paper we generalize the deuring theorem on a reduction of an elliptic curve with complex multiplication more precisely for an abelian variety a arising after reduction of an abelian variety with complex multiplication by a cm field k over a number field at a place of good reduction we establish a connection between a decomposition of the first truncated barsotti tate group scheme a p and a decomposition of p o k into prime ideals in particular we produce these explicit relationships for abelian varieties of dimensions 1 2 and 3

purity of the stratification by newton polygons in this paper we prove four theorems one on surface singularities two on fcrystals and one on moduli of p divisible groups the reason we put together these results in one paper is that the proofs as given here show how these theorems are related let us first describe our results let s 0 be a normal surface singularity over an algebraically closed field of characteristic p let s s be a resolution of singularities our first result is theorem 3 2 1 any qp cohomology class on the link of the singularity extends to the resolution more precisely

monodromy and irreducibility of leaves we show that non supersingular newton polygon strata in the principally polarized case are irreducible consider the theory of foliations of an open newton polygon stratum in the moduli space of abelian varieties in positive characteristic we show that any non supersingular leaf is irreducible a nd that the monodromy on such a leaf is maximal note that in the final result degrees of polarizations are arbitrary the irreducibility of leaves as proved here is the discrete part of a proof of the hecke orbit conjecture which will be published in 7 for a survey of this proof and for the terminology discrete part see 2

the almost product structure of newton strata in the deformation space of a barsotti tate group with crystalline tate tensors in this paper we construct the almost product structure of the minimal newton stratum in deformation spaces of barsotti tate groups with crystalline tate tensors similar to oort s and mantovan s construction for shimura varieties of pel type it allows us to describe the geometry of the newton stratum in terms of the geometry of two simpler objects the central leaf and the isogeny leaf this yields the dimension and the closure relations of the newton strata in the deformation space in particular their nonemptiness shows that a generalisation of grothendieck s conjecture of deformations of barsotti tate groups with given newton polygon holds as an application we determine analogous geometric properties of the newton stratification of shimura varieties of hodge type and prove the equidimensionality of rapoport zink spaces of hodge type

fundamental elements of an affine weyl group fundamental elements are certain special elements of affine weyl groups introduced by gortz haines kottwitz and reuman they play an important role in the study of affine deligne lusztig varieties in this paper we obtain characterizations of the fundamental elements and their natural generalizations we also derive an inverse to a version of newton hodge decomposition in affine flag varieties as an application we obtain a group theoretic generalization of oort s results on minimal p divisible groups and we show that in certain good reduction of pel shimura datum each newton stratum contains a minimal ekedahl oort stratum this generalizes a result of viehmann and wedhorn

foliations in deformation spaces of local g shtukas abstract we study local g shtukas with level structure over a base scheme whose newton polygons are constant on the base we show that after a finite base change and after passing to an etale covering such a local g shtuka is isogenous to a completely slope divisible one generalizing corresponding results for p divisible groups by oort and zink as an application we establish a product structure up to finite surjective morphism on the closed newton stratum of the universal deformation of a local g shtuka similarly to oort s foliations for p divisible groups and abelian varieties this also yields bounds on the dimensions of affine deligne lusztig varieties and proves equidimensionality of affine deligne lusztig varieties in the affine grassmannian

on the cohomology of certain pel type shimura varieties in this article we study the local geometry at a prime p of pel type shimura varieties for which there is a hyperspecial level subgroup we consider the newton polygon stratification of the special fiber at p of shimura varieties and show that each newton polygon stratum can be described in terms of the products of the reduced fibers of the corresponding pel type rapoport zink spaces with certain smooth varieties which we call igusa varieties and of the action on them of a p adic group that depends on the stratum we then extend our construction to characteristic zero and in the case of bad reduction at p use it to compare the vanishing cycle sheaves of the shimura varieties to those of the rapoport zink spaces as a result of this analysis in the case of proper shimura varieties we obtain a description of the l adic cohomology of the shimura varieties in terms of the l adic cohomology with compact supports of the igusa varieties and of the rapoport zink spaces for any prime l p

the ring of k regular sequences ii in this paper we continue our study of k regular sequences begun in 1992 we prove some new results give many new examples from the literature and state some open problems

on central leaves of hodge type shimura varieties with parahoric level structure kisin and pappas publ math inst hautes etudes sci 2018 constructed integral models of hodge type shimura varieties with parahoric level structure at p 2 such that the formal neighbourhood of a mod p point can be interpreted as a deformation space of p divisible group with some tate cycles generalising faltings construction in this paper we study the central leaf and the closed newton stratum in the formal neighbourhoods of mod p points of kisin pappas integral models with parahoric level structure namely we obtain the dimension of central leaves and the almost product structure of newton strata in the case of hyperspecial level structure i e in the good reduction case our main results were already obtained by hamacher math z 287 3 4 1255 1277 2017 and we show that the result of this paper holds for ramified groups as well

the geometry of newton strata in the reduction modulo p of shimura varieties of pel type in this thesis we study the newton stratification on the reduction of shimura varieties of pel type with hyperspecial level structure the main result is a formula for the dimension of newton strata and the description of their closure a key ingredient of its proof is the calculation of the dimension of some rapoport zink spaces as an interesting application to deformation theory we determine the dimension and closure of newton strata on the algebraisation of the deformation space of a barsotti tate group with p el structure

some combinatorial properties of the thue morse sequence and a problem in semigroups abstract some combinatorial properties of the special factors of the thue morse sequence in a two letter alphabet are considered an application of this analysis to a problem in semigroups is given

integral motives and special values of zeta functions for each field k we define a category of rationally decomposed mixed motives with z coefficients when k is finite we show that the category is tannakian and we prove formulas relating the behaviour of zeta functions near integers to certain ext groups

computational algorithm for reduction type of cm abelian varieties abstract let a be an abelian variety over a number field with a good reduction at a prime ideal containing a prime number p denote a as an abelian variety over a finite field of characteristic p which is obtained by the reduction of a at the prime ideal in this study we derive an algorithm that allows us to decompose the group scheme a p into indecomposable quasi polarized bt 1 group schemes up to isomorphism this can be achieved for the unramified p based on its decomposition into prime ideals in the endomorphism algebra of a we also compute all types of these correspondences for abelian varieties with dimensions up to 5 as a consequence we establish the relationship between the decompositions of prime p and the corresponding pairs of p rank and a number for an abelian variety a

canonical positions for the factors in paperfolding sequences it has been proved by the first author that the number of factors of length k in any paperfolding sequence is equal to 4k once k 7 we prove here that for every integer k there exists a set of integers pk of cardinality 4k for k 7 such that the factors of length k of any paperfolding sequence are exactly the factors beginning at the places indexed by the integers in pk this gives a bijective proof of the result mentioned above then we give an explicit uniform linear upper bound for the recurrence function of paperfolding sequences finally we study the set of all factors of all paperfolding sequences evaluating their number for a given length and studying the language of these words

every ordinary symplectic isogeny class in positive characteristic is dense in the moduli we prove that any ordinary symplectic separable isogeny class in the moduli space of principally polarized abelian varieties over a field of positive characteristic is dense in the zariski topology

on the factors of the sturmian sequences resume cet article a pour objet l etude d une construction associant a toute droite de pente p q p et q premiers entre eux et q n un mot de longueur n sur l alphabet 0 1 nous montrons que nous obtenons par cette construction le langage constitue de tous les facteurs des suites de sturm nous formulons apres avoir obtenu une equation fonctionelle dont la solution est la serie generatrice de ce langage une conjecture reliant cette serie generatrice a la fonction d euler

schrodinger operators with rudin shapiro potentials are not palindromic we prove a conjecture of a hof o knill and b simon commun math phys 174 149 159 1995 by showing that the rudin shapiro sequence is not palindromic i e does not contain arbitrarily long palindromes we prove actually this property for all paperfolding sequences and all rudin shapiro sequences deduced from paperfolding sequences as a consequence and as guessed by the above authors their method cannot be used for establishing that discrete schrodinger operators with rudin shapiro potentials have a purely singular continuous spectrum

the iwasawa mu invariant of p adic hecke l functions for an odd prime p we compute the invariant of the anticyclotomic katz p adic l function of a p ordinary cm field if the conductor of the branch character is a product of primes split over the maximal real subfield except for rare cases where the root number of the p adic functional equation is congruent to 1 modulo p the invariant vanishes

some studies on the p q type sequences in this paper we have introduced the p q type sequences their arithmetic and analytic properties are studied and their relations with finite automata are established the results obtained extend those of classical paperfolding sequences and some new informations are obtained

l adic etale cohomology of shimura varieties of hodge type with non trivial coefficients let mathsf g mathsf x be a shimura datum of hodge type let p be an odd prime such that mathsf g mathbb q p splits after a tamely ramified extension and p not mid pi 1 mathsf g mathrm der under some mild additional assumptions that are satisfied if the associated shimura variety is proper and mathsf g mathbb q p is either unramified or residually split we prove the generalisation of mantovan s formula for the l adic cohomology of the associated shimura variety on the way we derive some new results about the geometry of the newton stratification of the reduction modulo p of the kisin pappas integral model

abelian varieties isogenous to a jacobian we dene a notion of weyl cm points in the moduli space ag 1 of g dimensional principally polarized abelian varieties and show that the andr e oort conjecture or the grh implies the following statement for any closed subvariety x ag 1 over q a there exists a weyl special point b 2 ag 1 q a such that b is not isogenous to the abelian variety a underlying any point a 2 x the title refers to the case when g 4 and x is the torelli locus in this case tsimerman has proved the statement unconditionally the notion of weyl special points is generalized to the context of shimura varieties and we prove a corresponding conditional statement with the ambient spaceag 1 replaced by a general shimura variety

newton strata in the loop group of a reductive group we generalize purity of the newton stratification to purity for a single break point of the newton point in the context of local g shtukas respectively of elements of the loop group of a reductive group as an application we prove that elements of the loop group bounded by a given dominant coweight satisfy a generalization of grothendieck s conjecture on deformations of p divisible groups with given newton polygons

period spaces for hodge structures in equal characteristic we develop the analog in equal positive characteristic of fontaine s theory for crystalline galois representations of a p adic eld in particular we describe the analog of fontaine s functor which assigns to a crystalline galois representation an isocrystal with a hodge ltration in equal characteristic the role of isocrystals and hodge ltrations is played by z isocrystals and hodge pink structures the latter were invented by pink our rst main result in this article is the analog of the colmez fontaine theorem that weakly admissible implies admissible next we construct period spaces for hodge pink structures on a xed z isocrystal these period spaces are analogs of the rapoport zink period spaces for fontaine s ltered isocrystals in mixed characteristic and likewise are rigid analytic spaces for our period spaces we prove the analog of a conjecture of rapoport and zink stating the existence of a universal local system on a berkovich open subspace of the period space as a consequence of weakly admissible implies admissible this berkovich open subspace contains every classical rigid analytic point of the period space as the principal tool to demonstrate these results we use the analog of kedlaya s slope filtration theorem which we also formulate and prove here