vertexwise criteria for admissibility of alcoves we give a new description of the set rm adm mu of admissible alcoves as an intersection of certain obtuse cones of alcoves and we show this description may be given by imposing conditions vertexwise we use this to prove the vertexwise admissibility conjecture of pappas rapoport smithling the same idea gives simple proofs of two ingredients used in the proof of the kottwitz rapoport conjecture on existence of crystals with additional structure

dimension formula for the affine deligne lusztig variety x mu b x μ b the study of certain union x mu b of affine deligne lusztig varieties in the affine flag varieties arose from the study of shimura varieties with iwahori level structure in this paper we give an explicit dimension formula for x mu b associated to sufficiently large dominant coweight mu

moduli of finite flat group schemes and modularity we prove that under some mild conditions a two dimensional p adic galois representation which is residually modular and potentially barsotti tate at p is modular this provides a more conceptual way of establishing the shimura taniyama weil conjecture especially for elliptic curves which acquire good reduction over a wildly ramified extension of 3 the main ingredient is a new technique for analyzing flat deformation rings it involves resolving them by spaces which parametrize finite flat group scheme models of galois representations introduction

on the μ ordinary locus of a shimura variety abstract in this paper we study the ordinary locus of a shimura variety with parahoric level structure under the axioms in 12 we show that ordinary locus is a union of certain maximal ekedahl kottwitz oort rapoport strata introduced in 12 and we give criteria on the density of the ordinary locus

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

construction of central elements in the affine hecke algebra via nearby cycles 0 1 overview let g be a connected reductive group over a finite field fq and let g k be the corresponding group over the local field k fq t let g o g k be a maximal compact subgroup of g k here o fq t and let hsph denote the hecke algebra of g k with respect to g o in other words hsph as a vector space consists of compactly supported bi g o invariant functions g k q and the product is defined by

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

on the coherence conjecture of pappas and rapoport we prove the generalized coherence conjecture proposed by pappas and rapoport as a corollary one of their theorems which describes the geometry of the special bers of the local models for ramied unitary groups holds unconditionally our proof is based on the study of the geometry in particular certain line bundles and adic sheaves of the global schubert varieties which are the equal characteristic counterparts of the local models

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

topological flatness of local models in the ramified case local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain shimura varieties with parahoric level structure we investigate the local models for groups of the form open image in new window is a totally ramified extension as defined by pappas and rapoport and show that they are topologically flat in the linear case flatness can be deduced from this

the supersingular locus in siegel modular varieties with iwahori level structure we study moduli spaces of abelian varieties in positive characteristic more specifically the moduli space of principally polarized abelian varieties on the one hand and the analogous space with iwahori type level structure on the other hand we investigate the ekedahl oort stratification on the former the kottwitz rapoport stratification on the latter and their relationship in this way we obtain structural results about the supersingular locus in the case of iwahori level structure for instance a formula for its dimension in case g is even

optimal lower bound of the grow up rate for a supercritical parabolic equation we study the behavior of solutions of the cauchy problem for a supercritical semilinear parabolic equation which approach a singular steady state from below as t it is known that the grow up rate of such solutions depends on the spatial decay rate of initial data we give an optimal lower bound on the grow up rate by using a comparison technique based on a formal asymptotic analysis

global unbounded solutions of the fujita equation in the intermediate range we consider the fujita equation u t vardelta u u p on mathbb r n with n ge 3 we prove the existence of global unbounded solutions for p s p p jl where p s n 2 n 2 and begin aligned p jl left begin array l quad l frac n 2 2 4n 8 sqrt n 1 n 2 n 10 text if n 10 infty text if n le 10 end array right end aligned previously it was known that global unbounded solutions exist for p ge p jl whereas for p p s there are no such solutions

refined asymptotics for entire solutions of a biharmonic equation with a supercritical nonlinearity we consider positive classical radially symmetric solutions of the prototype of a fourth order nonlinear elliptic equation delta 2 u u p quad x in mathbb r n quad star where n 13 and p pc here p c frac n 4 n 4 is a critical exponent that is known to mark a borderline in respect of the asymptotic behavior in this problem gazzola and grunau in math ann 334 905 936 2006 for p frac n 4 n 4 all positive radial solutions of approach an explicitly known singular solution u in the sense that frac u x u infty x to 1 as x if p pc and a rate involving a logarithmic correction when p pc these results parallel those known for the second order counterpart of gui et al in comm pure appl math 45 1153 1181 1992 but the technique is completely different

ideal triangles in euclidean buildings and branching to levi subgroups abstract let g denote a connected reductive group defined and split over z and let m g denote a levi subgroup in this paper we study varieties of geodesic triangles with fixed vector valued side lengths in the bruhat tits buildings associated to g along with varieties of ideal triangles associated to the pair m g the ideal triangles have a fixed side containing a fixed base vertex and a fixed infinite vertex such that other infinite side containing has fixed ideal length and the remaining finite side has fixed length we establish an isomorphism between varieties in the second family and certain varieties in the first family the pair and the triple satisfy a certain relation we apply these results to the study of the hecke ring of g and the restriction homomorphism r g r m between representation rings we deduce some new saturation theorems for constant term coefficients and for the structure constants of the restriction homomorphism

uber die lokale zetafunktion von shimuravarietaten monodromiefiltration und verschwindende zyklen in ungleicher charakteristik

a bound for global solutions of semilinear heat equations we show that global positive solutions of the initial boundary value problem foru t u u p are bounded provided thatp 1 is subcritical our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity

topological flatness of orthogonal local models in the split even case i local models are schemes defined in terms of linear algebra that were introduced by rapoport and zink to study the etale local structure of integral models of certain pel shimura varieties over p adic fields a basic requirement for the integral models or equivalently for the local models is that they be flat in the case of local models for even orthogonal groups genestier observed that the original definition of the local model does not yield a flat scheme in a recent article pappas and rapoport introduced a new condition to the moduli problem defining the local model the so called spin condition and conjectured that the resulting spin local model is flat we prove a preliminary form of their conjecture in the split iwahori case namely that the spin local model is topologically flat an essential combinatorial ingredient is the equivalence of admissibility and permissibility for two minuscule cocharacters in root systems of type d

connected components of closed affine deligne lusztig varieties for rese fgln abstract let f be a p adic field and let e f be a finite unramified extension for the weil restriction res e f g l n of the general linear group g l n we determine the connected components of closed affine deligne lusztig varieties of arbitrary parahoric level

growup of solutions for a semilinear heat equation with supercritical nonlinearity abstract we consider a cauchy problem for a semilinear heat equation p u t u u p in r n 0 u x 0 u 0 x 0 in r n let v be the radially symmetric singular steady state of p it is proved that if p n 2 n 1 n 4 2 n 1 and n 11 then for each nonnegative even integer n there exists a radially symmetric global solution u n of p with n intersections with v such that t a n u n t 1 as t for some a n 0 depending on n the exact value of a n is also given we show that a 0 is the optimal upper bound of growup rate for solutions below v

formulae relating the bernstein and iwahori matsumoto presentations of an affine hecke algebra abstract we consider the anti dominant variants of the elements occurring in the bernstein presentation of an affine hecke algebra h we find explicit formulae for in terms of the iwahori matsumoto generators tw w ranging over the extended affine weyl group of the root system r in the case i r is arbitrary and is a minuscule coweight or ii r is attached to gln and mek where ek is a standard basis vector and m 1 in the above cases certain minimal expressions for play a crucial role such minimal expressions exist in fact for any coweight for gln we give a sheaf theoretic interpretation of the existence of a minimal expression for the corresponding perverse sheaf on the affine schubert variety x t is the push forward of an explicit perverse sheaf on the demazure resolution m x t x t this approach yields for a minuscule coweight of any r or for an arbitrary coweight of gln a conceptual albeit less explicit expression for the coefficient w of the basis element tw in terms of the cohomology of a fiber of the demazure resolution

vers le socle localement analytique pour mathrm gl _n ii on conjecture que certaines representations localement mathbb q p analytiques irreductibles de mathrm gl n l pour l mathbb q p infty apparaissent en sous objet dans des espaces hecke isotypiques de formes automorphes p adiques lorsque l mathbb q p on demontre quelques resultats partiels dans la direction de cette conjecture en utilisant des resultats recents sur les varietes de hecke et une nouvelle formule d adjonction pour le foncteur localement analytique de jacquet emerton

convergence rate to singular steady states in a semilinear parabolic equation abstract we study the asymptotic behavior of singular solutions of a semilinear parabolic equation our aim is to determine a convergence rate to singular steady states and to derive a universal lower bound of the convergence rate which implies the optimality of the convergence rate proofs are given by using a super and subsolution method based on matched asymptotic expansion

the jordan holder series for nearby cycles on some shimura varieties and affine flag varieties we study the jordan holder series for nearby cycles on certain shimura varieties and rapoport zink local models and on finite dimensional pieces of beilinson s deformation of the affine grassmannian to the affine flag variety and their p adic analogues we give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups and we also provide an algorithm to compute multiplicities in terms of the affine hecke algebra msc classification 14m15 14g35 20c08

a surface of maximal canonical degree it is known since the 1970s from a paper of beauville that the degree of the rational canonical map of a smooth projective algebraic surface of general type is at most 36 though it has been conjectured that a surface with optimal canonical degree 36 exists the highest canonical degree known earlier for a minimal surface of general type was 16 by persson the purpose of this paper is to give an affirmative answer to the conjecture by providing an explicit surface

intersection cohomology of b b orbit closures in group compactifications an adjoint semi simple group g has a wonderful compactification x which is a smooth projective variety containing g as an open subvariety x is acted upon by g g and b denoting a borel subgroup of g the group b b has finitely many orbits in x the main results of this paper concern the intersection cohomology of the closures of the b b orbits examples of such closures are the large schubert varieties the closures in x of the double cosets bwb in g after recalling some basic results about the wonderful compactification we discuss in section 1 the description of the b b orbits and establish some basic results in section 2 the bruhat order of the set v of orbits is introduced and described explicitly as an application we obtain cellular decompositions of the large schubert varieties let h be the hecke algebra associated to g it is a free module over an algebra of laurent polynomials z u u 1 as a particular case of results of ms the spherical g g variety x defines a representation of the hecke algebra associated to g g i e h z u u 1 h in a free module m over an extension of z u u 1 with a basis mv indexed by v the definition of m is sheaftheoretical working over the algebraic closure of a finite field this is discussed in section 3 on the model of lv a duality map is introduced on m coming from verdier duality in sheaf theory the matrix coefficients of relative to the

on the μ admissible set in the extended affine weyl groups of e6 and e7 giving explicit examples we show that the admissible set and the permissible set do not agree for non trivial minuscule coweights in the cases of e6 and e7

connected components of closed affine deligne lusztig varieties for split reductive algebraic groups we determine the sets of connected components of closed affine deligne lusztig varieties with parahoric level structure

alcoves associated to special fibers of local models the special fiber of the local model of a pel shimura variety with iwahori type level structure admits a cellular decomposition the set of strata is in a natural way a finite subset of the affine weyl group determined by the shimura data in this paper we generalize and give a new proof of a theorem of kottwitz and rapoport concerning the description of this set for the case of linear and symplectic groups for higher rank groups not of type an we produce counterexamples showing the situation is not as nice in general we also recover a theorem of deodhar characterizing the bruhat order on sn the local geometry can be approached using the rapoport zink local model m j 13 this protective op scheme is a local model for the singularities in the special fiber of sh g x9 k 9 but it is defined in terms of the pair g p using linear algebra and is somewhat easier to deal with than the shimura variety itself in this paper we address certain combinatorial questions which arise in the study of the special fiber m f of m to fix ideas consider the case where g is split it turns out that m f can be regarded as a finite dimensional union of iwahori orbits in the affine flag variety for g fp t see 5 and 8 it follows that m f has a stratification indexed by a finite set perm p of the extended affine weyl group w g for g equivalently perm p can be regarded as a finite set of alcoves in the affine coxeter complex determined by a splitting for g let wo denote the finite weyl group of g for each translation in the wo orbit

blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity abstract this paper is concerned with blowup phenomena of solutions for the cauchy and the cauchy dirichlet problem of p u t u u p with p in the supercritical range in the sense of sobolev s embedding we first show that if p 1 7 n 11 and n 12 then there are no radially symmetric bounded positive solutions of w y 2 w 1 p 1 w w p 0 in r n which intersect the radially symmetric singular solution at least twice using the above result the existence of a blowup solution of type ii for the cauchy dirichlet problem for p in a ball is proved where a solution u is said to exhibit the type ii blowup at t t if lim sup t t t t 1 p 1 u t

topological flatness of local models for ramified unitary groups i the odd dimensional case abstract local models are certain schemes defined in terms of linear algebraic moduli problems which give etale local neighborhoods of integral models of certain p adic pel shimura varieties defined by rapoport and zink when the group defining the shimura variety ramifies at p the local models and hence the shimura models as originally defined can fail to be flat and it becomes desirable to modify their definition so as to obtain a flat scheme in the case of unitary similitude groups whose localizations at q p are ramified quasi split gu n pappas and rapoport have added new conditions the so called wedge and spin conditions to the moduli problem defining the original local models and conjectured that their new local models are flat we prove a preliminary form of their conjecture namely that their new models are topologically flat in the case n is odd

weight elimination in serre type conjectures we prove the weight elimination direction of the serre weight conjectures as formulated by herzig for forms of u n which are compact at infinity and split at places dividing p in generic situations that is we show that all modular weights for a mod p galois representation are contained in the set predicted by herzig under some additional hypotheses we also show modularity of all the obvious weights

grow up rate of solutions for a supercritical semilinear diffusion equation abstract we study solutions of the cauchy problem for a supercritical semilinear parabolic equation which converge to a singular steady state from below as t we show that the grow up rate of such solutions depends on the spatial decay rate of initial data

rates of convergence to zero for a semilinear parabolic equation with a critical exponent abstract we study nonnegative solutions to the cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of joseph and lundgren we establish the rate of convergence to zero of solutions that start from initial data which are near the singular steady state in the critical case this rate contains a logarithmic term which does not appear in the supercritical case and makes the calculations more delicate

linear behaviour of solutions of a superlinear heat equation we give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data this improves a previous result of lee and ni by showing that this behaviour holds for a significantly larger and rather tightly defined class of solutions

local models of shimura varieties and a conjecture of kottwitz we give a group theoretic definition of local models as sought after in the theory of shimura varieties these are projective schemes over the integers of a p adic local field that are expected to model the singularities of integral models of shimura varieties with parahoric level structure our local models are certain mixed characteristic degenerations of grassmannian varieties they are obtained by extending constructions of beilinson drinfeld gaitsgory and the second named author to mixed characteristics and to the case of general tamely ramified reductive groups we study the singularities of local models and hence also of the corresponding integral models of shimura varieties in particular we study the monodromy inertia action and show a commutativity property for the sheaves of nearby cycles as a result we prove a conjecture of kottwitz which asserts that the semi simple trace of frobenius on the nearby cycles gives a function which is central in the parahoric hecke algebra

convergence rate for a semilinear parabolic equation with a singular nonlinearity we study the behavior of solutions of the cauchy problem for a semilinear parabolic equation with a singular power nonlinearity it is known for a supercritical heat equation that if two solutions are initially close enough near the spatial infinity then these solutions approach each other in fact given a specific decay rate of initial data a sharp estimate of the convergence rate can be obtained explicitly the main purpose of this research is to show a similar result for a semilinear parabolic equation with a singular absorption term we also obtain a universal lower bound of the convergence rate which implies the optimality of the result proofs are given by a comparison method that is based on matched asymptotic expansion

slow convergence to zero for a parabolic equation with a supercritical nonlinearity we study the behaviour of solutions of the cauchy problem for a supercritical semilinear parabolic equation which converge to zero from above as t we show that any algebraic decay rate slower than the self similar one occurs for some initial data

geometry of b b orbit closures in equivariant embeddings abstract let x denote an equivariant embedding of a connected reductive group g over an algebraically closed field k let b denote a borel subgroup of g and let z denote a b b orbit closure in x when the characteristic of k is positive and x is projective we prove that z is globally f regular as a consequence z is normal and cohen macaulay for arbitrary x and arbitrary characteristics moreover in characteristic zero it follows that z has rational singularities this extends earlier results by the second author and m brion

some examples of square integrable representations of semisimple p adic groups we construct irreducible representations of the hecke algebra of an affine weyl group analogous to kilmoyer s reflection representation corresponding to finite weyl groups and we show that in many cases they correspond to a square integrable representation of a simple p adic group

on nonexistence of type ii blowup for a supercritical nonlinear heat equation in this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut 1 u u p 1 u either on r n or on a finite ball under the dirichlet boundary conditions we assume that the exponent p is supercritical in the sobolev sense that is p ps n 2 n 2 we prove that if ps p the above range of exponent p is optimal we will also derive various fundamental estimates for blowup that hold for any p ps and regardless of type of blowup among other things we classify local profiles of type i and type ii blowups in the rescaled coordinates w e then establish useful estimates for the so called incomplete blowup which r eveal that incomplete blowup solutions belong to nice function spaces even after the blowup time c 2004 wiley periodicals inc

critical exponent for the bipolar blowup in a semilinear parabolic equation abstract the blowup of solutions of the initial boundary value problem formula is studied where is a cone inrn we say that a solutionuexhibits bipolar blowup if formula for somet 1 for the bipolar blowup in the following sense if 1 p then the bipolar blowup does not occur for any sufficiently small initial data the value ofp is expressed in terms of the dimensionnand the second dirichlet eigenvalue of the laplace beltrami operator on sn 1 in the case of r we definek polar blowup and determine critical exponents for thek polar blowup fork 1 2 3

pro p iwahori hecke ring and supersingular representations the motivation of this paper is the search for a langlands correspondence modulo p we show that the pro p iwahori hecke ring open image in new window of a split reductive p adic group g over a local field f of finite residue field fq with q elements admits an iwahori matsumoto presentation and a bernstein z basis and we determine its centre we prove that the ring open image in new window is finitely generated as a module over its centre these results are proved in 11 only for the iwahori hecke ring let p be the prime number dividing q and let k be an algebraically closed field of characteristic p a character from the centre of open image in new window to k which is as null as possible will be called null the simple open image in new window modules with a null central character are called supersingular when g gl n we show that each simple open image in new window module of dimension n containing a character of the affine subring open image in new window is supersingular using the minimal expressions of haines generalized to open image in new window and that the number of such modules is equal to the number of irreducible k representations of the weil group wf of dimension n when the action of an uniformizer pf in the hecke algebra side and of the determinant of a frobenius frf in the galois side are fixed i e the number nn q of unitary irreducible polynomials in fq x of degree n one knows that the converse is true by explicit computations when n 2 10 and when n 3 rachel ollivier

computing the alternating trace of frobenius on the sheaves of nearby cycles on local models for gl4 and gl5 abstract consider a shimura variety defined over some number field and assume we have a model over the ring of integers at some prime of bad reduction it is then interesting to know the alternating trace of frobenius on the invariants under the inertia group of the sheaf of nearby cycles since these traces are related to the local factor of the hasse weil zeta function in this article we compute the semi simple alternating trace of frobenius for shimura varieties associated to the groups gu 2 2 and gu 3 2 with level structure of iwahori type by investigating the equations of the local model defined by rapoport and zink and by performing explicit blowing ups