compatibility of local and global langlands correspondences we prove the compatibility of local and global langlands correspondences for gln which was proved up to semisimplification in ht more precisely for the ndimensional l adic representation rl of the galois group of an imaginary cm field l attached to a conjugate self dual regular algebraic cuspidal automorphic representation of gln al which is square integrable at some finite place we show that frobenius semisimplification of the restriction of rl to the decomposition group of a place v of l not dividing l corresponds to v by the local langlands correspondence if v is square integrable for some finite place v 6 l we deduce that rl is irreducible we also obtain conditional results in the case v l

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

multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

on the variety of projectors in this paper we are going to give an answer to the following two questions let k be a field and a4 the space of n x n square matrices with coefficients in k the first question we want to investigate is what are the equations of the subvariety a of ikf consisting of rank h matrices a such that a2 a by equations we mean of course a set of explicit generators of the prime ideal of x the second question is what are the equations of the varieties yh h n 2 which are the closures of the nilpotent conjugacy classes in h4 e consisting of matrices of rank h whose square is the matrix 0 in this case we shall also prove that these varieties y are normal and cohen macaulay in arbitrary characteristic thus generalizing a result of hesselink 3 to arbitrary characteristic note that in characteristic 0 all closures of nilpotent conjugacy classes have been proved to be normal and cohen macaulay by kraft and procesi 4 but our method is not suitable for generalizing their result the method used in this paper is the one of hodge algebras 11 namely we shall exhibit an explicit linear basis for the coordinate rings of the varieties x and y

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

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

galois representations arising from some compact shimura varieties our aim is to establish some new cases of the global langlands correspondence for glm along the way we obtain a new result on the description of the cohomology of some compact shimura varieties let f be a cm eld with complex conjugation c and

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

regular formal moduli spaces and arithmetic transfer conjectures we define various formal moduli spaces of p divisible groups which are regular and morphisms between them we formulate arithmetic transfer conjectures which are variants of the arithmetic fundamental lemma conjecture of the third author in the presence of ramification these conjectures include the at conjecture of our previous joint work we prove these conjectures in low dimensional cases

a hormander type multiplier theorem for multilinear operators abstract in this paper we prove a hormander type multiplier theorem for multilinear operators as a corollary we can weaken the regularity assumption for multilinear fourier multipliers to assure the boundedness

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

galois representations and torsion in the coherent cohomology of hilbert modular varieties let f be a totally real number eld and let p be a prime unramied in f we prove the existence of galois pseudo representations attached to modp m hecke eigenclasses of paritious weight occurring in the coherent cohomology of hilbert modular varieties forf of level prime to p

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

the langlands kottwitz method and deformation spaces of p divisible groups we extend the results of kottwitz 17 on points of shimura varieties over finite fields to cases of bad reduction the test function whose twisted orbital integrals appear in the final expression is defined geometrically using deformation spaces of p divisible groups

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

special cycles on unitary shimura varieties i unramified local theory the supersingular locus in the fiber at p of a shimura variety attached to a unitary similitude group gu 1 n 1 over is uniformized by a formal scheme mathcal n in the case when p is an inert prime we define special cycles mathcal z bold x in mathcal n associated to collections bold x of m special homomorphisms with fundamental matrix t hermm ok when m n and t is nonsingular we show that the cycle mathcal z bold x is either empty or is a union of components of the ekedahl oort stratification and we give a necessary and sufficient condition in terms of t for mathcal z bold x to be irreducible when mathcal z bold x is zero dimensional in which case it reduces to a single point we determine the length of the corresponding local ring by using a variant of the theory of quasi canonical liftings we show that this length coincides with the derivative of a representation density for hermitian forms

convex polytopes for the central degeneration of the affine grassmannian abstract we study the algebraic geometry and combinatorics of the central degeneration the degeneration that shows up in local models of shimura varieties and gaitsgory s central sheaves in type a more specifically we elucidate the central degeneration of semi infinite orbits and explain its relations with levi restriction also we discuss the central degeneration of mirkovic vilonen cycles in the affine grassmannian and the corresponding transformations of mirkovic vilonen polytopes in addition we shed some light on the geometry of iwahori mv cycles in the affine grassmannian and generalized mv cycles in the affine flag variety which are closely related to demazure modules and affine deligne lusztig varieties respectively

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

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

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

on calderon s conjecture with constants cfi p1 p2 depending only on fi p1 p2 and p p1p2 p1 p2 hold the flrst result of this type is proved in 4 and the purpose of the current paper is to extend the range of exponents p1 and p2 for which 2 is known in particular the case p1 2 p2 1 is solved to the a rmative this was

on the generic part of the cohomology of compact unitary shimura varieties the goal of this paper is to show that the cohomology of compact unitary shimura varieties is concentrated in the middle degree and torsion free after localizing at a maximal ideal of the hecke algebra satisfying a suitable genericity assumption along the way we establish various foundational results on the geometry of the hodge tate period map in particular we compare the fibres of the hodge tate period map with igusa varieties

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

cells in affine weyl groups and tensor categories abstract in this paper we construct a tensor or monoidal category for any two sided cell in a finite or affine weyl group the isomorphism classes of simple objects of this category correspond to the elements of the two sided cell the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold which imitates geometrically the definition of the ring j given in g lusztig j algebra 109 1987 536 548 similarly we construct for any left cell a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse a right cell this contains as a special case the category of equivariant perverse sheaves on an affine grassmannian which as a consequence of g lusztig asterisque 101 102 1983 208 209 is a tensor category for convolution without truncation closely related to the tensor category of representations of the group dual in the sense of langlands to the group which gives rise to the affine grassmannian but our construction can be specialized in other ways and it gives a conjectural realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the langlands dual group

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

polynomials vanishing on grids the elekes ronyai problem revisited in this paper we characterize real bivariate polynomials which have a small range over large cartesian products we show that for every constant degree bivariate real polynomial f either f a b omega n 4 3 for every pair of finite sets a b subset bbb r with a b n where the constant of proportionality depends on deg f or else f must be of one of the special forms f u v h varphi u psi v or f u v h varphi u cdot psi v for some univariate polynomials varphi psi h over bbb r this significantly improves a result of elekes and r onyai 2000 our results are cast in a more general form in which we give an upper bound for the number of zeros of z f x y on a triple cartesian product a times b times c when the sizes a b c need not be the same the upper bound is o n 11 6 when a b c n where the constant of proportionality depends on deg f unless f has one of the aforementioned special forms this result provides a unified tool for improving bounds in various erd h os type problems in geometry and additive combinatorics several applications of our results to problems of these kinds are presented for example we show that the number of distinct distances between n points lying on a constant degree algebraic curve that has a polynomial parameterization and that does not contain a line in any dimension is omega n 4 3 extending the result of pach and de zeeuw 2014 and improving the bound of charalambides 2014 for the special case where the curve under consideration has a polynomial parameterization we also derive improved lower bounds for several variants of the sum product problem in additive combinatorics

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

schubert varieties in twisted affine flag varieties and local models abstract our concern in this paper is the dimension and inclusion relations of schubert varieties in twisted partial affine flag varieties in the end we apply our results to some local models of certain shimura varieties

the hormander multiplier theorem ii the bilinear local l 2 case we use wavelets of tensor product type to obtain the boundedness of bilinear multiplier operators on mathbb r n times mathbb r n associated with hormander multipliers on mathbb r 2n with minimal smoothness we focus on the local l 2 case and we obtain boundedness under the minimal smoothness assumption of n 2 derivatives we also provide counterexamples to obtain necessary conditions for all sets of indices

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

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

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

limit linear series for curves not of compact type we introduce a notion of limit linear series for nodal curves which are not of compact type we give a construction of a moduli space of limit linear series which works also in smoothing families and we prove a corresponding specialization result for a more restricted class of curves which simultaneously generalizes two component curves and curves of compact type we give an equivalent definition of limit linear series which is visibly a generalization of the eisenbud harris definition finally for the same class of curves we prove a smoothing theorem which constitutes an improvement over known results even in the compact type case

some remarks on multilinear maps and interpolation a multilinear version of the boyd interpolation theorem is proved in the context of quasi normed rearrangement invariant spaces a multilinear marcinkiewicz interpolation theorem is obtained as a corollary several applications are given including estimates for bilinear fractional integrals

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

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

the erdős szemeredi problem on sum set and product set summary the basic theme of this paper is the fact that if a is a finite set of integers then the sum and product sets cannot both be small a precise formulation of this fact is conjecture 1 below due to erd os szemeredi e s see also el t and k t for related aspects only much weaker results or very special cases of this conjecture are presently known one approach consists of assuming the sum set a a small and then deriving that the product set aa is large using freiman s structure theorem cf n t na3 we follow the reverse route and prove that if aa c a 2 see theorem 1 a quantitative version of this phenomenon combined with the plunnecke type of inequality due to ruzsa permit us to settle completely a related conjecture in e s on the growth in k i f

a counterexample to weighted estimates for multilinear fourier multipliers with sobolev regularity abstract the problem whether the weighted estimates for multilinear fourier multipliers with sobolev regularity hold under weak condition on weights is considered

on the conormal bundle of the determinantal variety call w such variety if we choose bases b v a a b u v i b for each module we can identify w with the variety of pairs of matrices y x with entries in r x being an m x n matrix y an ii x m matrix such that x y omx y x onxm now let a be the coordinate ring of the affine space a of dimension 2 m x n i e the polynomial ring r x yji i l m j i n and

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

the local langlands correspondence for gln over p adic fields we extend our methods from scholze invent math 2012 doi 10 1007 s00222 012 0419 y to reprove the local langlands correspondence for gl n over p adic fields as well as the existence of l adic galois representations attached to most regular algebraic conjugate self dual cuspidal automorphic representations for which we prove a local global compatibility statement as in the book of harris taylor the geometry and cohomology of some simple shimura varieties 2001 in contrast to the proofs of the local langlands correspondence given by henniart invent math 139 2 439 455 2000 and harris taylor the geometry and cohomology of some simple shimura varieties 2001 our proof completely by passes the numerical local langlands correspondence of henniart ann sci ec norm super 21 4 497 544 1988 instead we make use of a previous result from scholze invent math 2012 doi 10 1007 s00222 012 0419 y describing the inertia invariant nearby cycles in certain regular situations

bounding multiplicative energy by the sumset abstract we prove that the sumset or the productset of any finite set of real numbers a is at least a 4 3 e improving earlier bounds our main tool is a new upper bound on the multiplicative energy e a a

on a grauert riemenschneider vanishing theorem for frobenius split varieties in characteristic p it is known that the grauert riemenschneider vanishing theorem is not valid in characteristic p 1 here we show that it may be restored in the presence of a suitable frobenius splitting the proof uses interchanging two projective limits one involving iterated frobenius maps cf 2 and 4 the other coming from grothendieck s theorem on formal functions that leads to the following general vanishing theorem which we then apply in the situation of the grauert riemenschneider theorem

integral models of hilbert modular varieties in the ramified case deformations of modular galois representations and weight one forms we prove the strong artin conjecture for continuous totally odd two dimensional representations of the absolute galois group of a totally real field f

new sum product type estimates over finite fields abstract let f be a field with positive odd characteristic p we prove a variety of new sum product type estimates over f they are derived from the theorem that the number of incidences between m points and n planes in the projective three space p g 3 f with m n o p 2 is o m n k m where k denotes the maximum number of collinear planes the main result is a significant improvement of the state of the art sum product inequality over fields with positive characteristic namely that 1 a a a a a 1 1 5 for any a such that a p 5 8

the hormander multiplier theorem for multilinear operators in this paper we provide a version of the mikhlin hormander multiplier theorem for multilinear operators in the case where the target space is l p for pe 1 this extends a recent result of tomita 15 who proved an analogous result for p 1

the langlands kottwitz approach for some simple shimura varieties we show how the langlands kottwitz method can be used to determine the semisimple local factors of the hasse weil zeta function of certain shimura varieties this is made possible by a new result describing part of the nearby cycle sheaves in certain situations in combination with a general base change lemma in harmonic analysis we use this to prove a conjecture of haines and kottwitz for these shimura varieties

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