counting points on igusa varieties igusa varieties are smooth varieties over fp which are higher dimensional analogues of igusa curves they were introduced by harris and taylor ht to study the bad reduction of some pel shimura varieties and were generalized by mantovan m1 m2 the present article gives a group theoretic formula for the traces of certain operators on the cohomology of igusa varieties suitable for applications via comparison with the arthur selberg trace formula our formula generalizes the results of ht chap v prop 4 8 to the case of any pel shimura varieties of types a and c and puts it in a more natural framework in the spirit of k7

galois and cartan cohomology of real groups suppose that g is a complex reductive algebraic group a real form of g is an antiholomorphic involutive automorphism so g r g c is a real lie group write h1 g for the galois cohomology pointed set h1 gal c r g a cartan involution for is an involutive holomorphic automorphism of g commuting with so that is a compact real form of g let h1 g be the set h1 z2 g where the action of the nontrivial element of z2 is by by analogy with the galois group we refer to h1 g as the cartan cohomology of g with respect to cartan s classification of real forms of a connected group in terms of their maximal compact subgroups amounts to an isomorphism h1 gad h1 gad where gad is the adjoint group our main result is a generalization of this there is a canonical isomorphism h1 g h1 g we apply this result to give simple proofs of some well known structural results the kostant sekiguchi correspondence of nilpotent orbits matsuki duality of orbits on the flag variety conjugacy classes of cartan subgroups and structure of the weyl group we also use it to compute h1 g for all simple simply connected groups and to give a cohomological interpretation of strong real forms for the applications it is important that we do not assume that g is connected

stable trace formula cuspidal tempered terms consider a connected reductive group g over a number field f for technical reasons we assume that the derived group of g is simply connected see l1 in l3 langlands partially stabilizes the trace formula for g after making certain assumptions he writes the elliptic regular part of the trace formula for g as a linear combination of the elliptic g regular parts of the stable trace formulas for the elliptic endoscopic groups h of g the function f used in the stable trace formula forh is obtained from the function f used in the trace formula for g by transferring orbital integrals

on reversal bounded counter machines and on pushdown automata with a bound on the size of the pushdown store the two main results of the paper are 1 proving a fine hierarchy of reversal bounded counter machine languages and 2 showing that a tape is better than a pushdown store for two way machines in the case where their size is sublinear

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

parabolic transfer for real groups this paper is the second of two articles in real harmonic analysis in the first paper a14 we established asymptotic formulas for some natural distributions on a real group in this paper we shall establish important relationships among the distributions as the group varies the group is the set of real points of a connected reductive group g over r the distributions are weighted orbital integrals jm f on g r and their invariant counterparts im f here m g is a levi subgroup of g while mg reg r is a strongly g regular conjugacy class in m r the relationships are defined by the invariant transfer of functions on g r to functions attached to endoscopic groups of g this necessitates our working with the invariant distributions im f we refer the reader to the introduction of a14 for some general remarks on these objects the distributions im f are the generic archimedean terms in the invariant trace formula we cannot review the trace formula here the reader might consult the introductions to a13 and its two predecessors for a brief summary the purpose of the paper a13 was to stabilize the invariant trace formula subject to a condition on the fundamental lemma that has been established in some special cases the stable trace formula is a milestone of sorts it is expected to lead to reciprocity laws which relate fundamental arithmetic data attached to automorphic representations on different groups the stable trace formula of a13 relies upon the results of this paper as well as a paper a16 in preparation this has been our guiding motivation the relevant identities among the nonarchimedean forms of the distributions im f were actually established in a13 they were a part of the global argument that culminated in the stable trace formula as such they are subject to the condition on the fundamental lemma mentioned above our goal here is to establish the outstanding archimedean identities we shall do so by purely local means which are independent of the fundamental lemma to simplify the introducton we assume that the derived group of g is simply connected the identities then relate the invariant distributions ong r with stable distributions on endoscopic groups g r we recall that a stable distribution on g r depends only on the average values assumed by a test function over strongly regular stable conjugacy classes in greg r which is to say intersections of g reg r

reversal bounded nondeterministic multicounter machines and complementation abstract it is proved that the family of languages recognized by one way real time nondeterministic multicounter machines with constant number of counter reversals is not closed under complementation this solves an open problem of wagner and wechsung 1986

formulas for the unramified brauer group of a principal homogeneous space of a linear algebraic group abstract for a smooth compactification v of a principal homogeneous space e under a connected linear algebraic group g defined over a field k of characteristic zero we present two formulas expressing br v br k in terms of g

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

the pluricomplex poisson kernel for strongly convex domains let d be a bounded strongly convex domain in the complex space of dimension n for a fixed point p epsilon partial derivative d we consider the solution of a homogeneous complex monge ampere equation with a simple pole at p we prove that such a solution enjoys many properties of the classical poisson kernel in the unit disc and thus deserves to be called the pluricomplex poisson kernel of d with pole at p in particular we discuss extremality properties such as a generalization of the classical phragmen lindelof theorem relations with the pluricomplex green function of d uniqueness in terms of the associated foliation and boundary behaviors finally using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions

existence de f cristaux avec structures supplementaires abstract in this article we prove under some hypothesis of non ramification a conjecture of kottwitz and rapoport giving the existence of crystals with additional structures

extended picard complexes and linear algebraic groups for a smooth geometrically integral variety x over a field k of characteristic 0 we introduce and investigate the extended picard complex upic x it is a certain complex of galois modules of length 2 whose zeroth cohomology is k x k and whose first cohomology is pic x where k is a fixed algebraic closure of k and x is obtained from x by extension of scalars to k when x is a k torsor of a connected linear k group g we compute upic x upic g in the derived category in terms of the algebraic fundamental group 1 g as an application we compute the elementary obstruction for such x

local global compatibility and the action of monodromy on nearby cycles we strengthen the local global compatibility of langlands correspondences for gln in the case when n is even and l 6d p let l be a cm field and let be a cuspidal automorphic representation of gln al which is conjugate self dual assume that 1 is cohomological and not slightly regular as defined by shin in this case chenevier and harris constructed an l adic galois representationrl and proved the local global compatibility up to semisimplification at primes v not dividing l we extend this compatibility by showing that the frobenius semisimplification of the restriction of rl to the decomposition group at v corresponds to the image of v via the local langlands correspondence we follow the strategy of taylor and yoshida where it was assumed that is square integrable at a finite place to make the argument work we study the action of the monodromy operator n on the complex of nearby cycles on a scheme which is locally etale over a product of strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case we also prove the ramanujan petersson conjecture for as above

tradeoffs for language recognition on alternating machines the alternating machine having a separate input tape with k two way read only heads and a certain number of internal configurations am k is considered as a parallel computing model for the complexity measure time space parallelism tsp the optimal lower bounds n2 and n3 2 respectively are proved for the recognition of specific languages on am 1 and am k respectively for the complexity measure reversals space parallelism rsp the lower bound n1 2 is established for the recognition of a specific language on am k this result implies a polynomial lower bound on parallel time hardware of parallel ram s lower bounds on the complexity measures time space and reversals space of nondeterministic machines are direct consequences of the result introduced above all lower bounds obtained are substantially improved in the case that space n for 0 1 several strongest lower bounds for two way and one way alternating deterministic nondeterministic multihead finite automata are obtained as direct consequences of these results the hierarchies for the complexity measures tsp rsp ts and rs can be immediately achieved too

on the cohomology of compact unitary group shimura varieties at ramified split places in this article we prove results about the cohomology of compact unitary group shimura varieties at split places in nonendoscopic cases we are able to give a full description of the cohomology after restricting to integral hecke operators at p on the automorphic side we allow arbitrary ramification at p even the pel data may be ramified this gives a description of the semisimple local hasse weil zeta function in these cases we also treat cases of nontrivial endoscopy for this purpose we give a general stabilization of the expression given in 38 following the stabilization given by kottwitz in 25 this introduces endoscopic transfers of the functions h introduced in 38 we state a general conjecture relating these endoscopic transfers with langlands parameters we verify this conjecture in all cases of el type and deduce new results about the endoscopic part of the cohomology of shimura varieties this allows us to simplify the construction of galois representations attached to conjugate self dual regular algebraic cuspidal automorphic representations of gln as previously constructed by one of us 41

on endoscopic transfer of deligne lusztig functions in this paper we prove the fundamental lemma for deligne lusztig functions namely for every deligne lusztig functionon a p adic group g we write down an explicit linear combinationh of deligne lusztig functions on an endoscopic group h such thatandh have matching orbital integrals in particular we prove a conjecture of kottwitz ko4 more precisely we do it under some mild restriction on p

the base change fundamental lemma for central elements in parahoric hecke algebras let g be an unramified group over a p adic field f and let e f be a finite unramified extension field let denote a generator of gal e f this paper concerns the matching at all semi simple elements of orbital integrals on g f with twisted orbital integrals on g e more precisely suppose belongs to the center of a parahoric hecke algebra for g e this paper introduces a base change homomorphism 7 b taking values in the center of the corresponding parahoric hecke algebra for g f it proves that the functions and b are associated in the sense that the stable orbital integrals for semi simple elements of b can be expressed in terms of the stable twisted orbital integrals of in the special case of spherical hecke algebras which are commutative this result becomes precisely the base change fundamental lemma proved previously by clozel cl90 and labesse lab90 as has been explained in h05 the fundamental lemma proved in this paper is a key ingredient for the study of shimura varieties with parahoric level structure at the prime p

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

real homogeneous spaces galois cohomology and reeder puzzles abstract let g be a simply connected absolutely simple algebraic group defined over the field of real numbers r let h be a simply connected semisimple r subgroup of g we consider the homogeneous space x g h we ask how many connected components has x r we give a method of answering this question our method is based on our solutions of generalized reeder puzzles

rigid geometry lefschetz verdier trace formula and deligne s conjecture in abstract algebraic geometry the lefschetz verdier trace formula expresses the global trace of an endomorphism induced on cohomology by a correspondence as a sum of local contributions called local terms near fixed points though the formula holds quite generally the explicit calculation of local terms is very hard to carry out and no good formula is known for general correspondences over the complex numbers a useful formula is known for a good class called weakly hyperbolic g m on the other hand in positive characteristics p deligne conjectured that the local terms would be much simplified if we composed a given correspondence with a high power of frobenius take a proper variety x over an algebraic closure k of a finite field fq a correspondence a y x xk x and a smooth qr k on an open set u of x fixed by a denote the inclusion by j and is invertible in k then

on reversal bounded alternating turing machines abstract it is known that for one tape nondeterministic turing machines s n space and s n reversal bounded machines s n n recognize the same class of languages we present a simulation of s n space bounded alternating turing machines atm by one tape lg s n reversal bounded atms we also show that atms making a constant number of reversals recognize only regular languages this shows that there is a striking difference in computational power between machines making a constant number of reversals and those making an almost constant i e lg n number of reversals

endoscopic character identities for depth zero supercuspidal l packets one of the conjectural properties of a langlands correspondence is its compatibility with endoscopic induction debacker and reeder have recently constructed a partial local langlands correspondence for p adic groups focusing on l packets consisting of depth zero supercuspidal representations in this paper we prove the conjectural endoscopic transfer for these l packets

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

asymptotic behavior of supercuspidal representations and sato tate equidistribution for families abstract we establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes for the tame supercuspidals constructed by j k yu we prove the limit multiplicity property with error terms thereby we obtain a sato tate equidistribution for the hecke eigenvalues of these families the main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p adic group tend to zero uniformly for every noncentral semisimple element

alternation for sublogarithmic space bounded alternating pushdown automata this paper investigates infinite hierarchies on alternation depth and alternation size of alternating pushdown automata apda s with sublogarithmic space we first show that there is an infinite hierarchy on alternation depth for apda s with sublogarithmic space we next investigate a relationship between alternation depth and alternation size and show that for sublogarithmic space bounded apda s alternation depth k is better than alternation size k for each k 3 we finally show that there is an infinite hierarchy on alternation size for apda s with sublogarithmic space

sur certaines identites endoscopiques entre transformees de fourier the main geometric terms of the arthur selberg trace formula are the weighted orbital integrals an important problem is to express these distributions as images by transfer of analogous distributions on endoscopic groups arthur has given a solution but under the hypothesis of the fundamental lemma of langlands in this paper we prove some relations between the fourier transforms of weighted orbital integrals on the lie alge bras of groups which are inner forms of each other these relations hold unconditionnally and they provide some evidence for arthur s solution besides they should enable us to prove new results of transfer in the context of the non invariant trace formula

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

sign changes in harmonic analysis on reductive groups let g be a connected reductive group over a field f in this note the author constructs an element e g of the brauer group of f the square of this element is trivial for a local field e g may be regarded as an element of i and is needed for harmonic analysis on reductive groups over that field for a global field there is a product formula the purpose of this note is to provide a cohomological interpretation of the sign changes in harmonic analysis on reductive groups over local fields that are caused by inner twistings the best understood example occurs in the work of shelstad 8 on inner twistings of real groups theorem 6 3 of that paper gives the following character identity 1 q g q g x here g is a connected reductive group over r g is its quasi split inner form x and x are stabilized characters of matched tempered l packets of g r and g r and y and y are matched regular semisimple elements of g r and g r the sign change in the character identity is q g q g where q g is one half of the dimension of the symmetric space attached to g more precisely attached to the simply connected cover of the derived group of g although q g may be only half integral the difference q g q g is always integral it is reasonable to expect an analogous character identity for groups g over a nonarchimedean local field f we can determine what the sign change must be by considering the steinberg characters of g f and g f since these should be related by the character identity let r g denote the f rank of the derived group of g the steinberg character of g f has value i r g on the elliptic regular set of g f and hence the sign change in the character identity must be l r g r g there are two groups for which the character identity has been proved the multiplicative group of a central division algebra of dimension d2 over f for d 2 see proposition 15 5 on p 484 of 4 and d 3 see theorem 1 of 1 later in this paper we will use a cohomological construction to define a sign e g 1 for any connected reductive group g over a local field let g denote a quasi split inner form of g we will show that e g l r g r g when the base received by the editors may 10 1982 1980 mathematics subject classification primary 22e50 partially supported by the national science foundation under grant mcs 78 02331 1983 american mathematical society 0002 9947 82 0000 11 93 03 25

moduli for pointed convex domains a moduli space for the class of pointed strictly linearly convex domains in n is obtained it is shown that the space of pointed smoothly bounded strictly linearly convex domains with a fixed indicatrix is parameterized by a class of deformations of the cr structure of the boundary of the indicatrix these deformations are constructed by using the circular representation of a domain to pull back its complex structure tensor to the indicatrix a careful study of the pull back structure shows that the allowable deformations are parameterized by a class of complex hamiltonian vector fields the proof of this fact is based on the folland stein estimates for the bar partial b complex of the boundary of the indicatrix

on the cohomology of rapoport zink spaces of el type in this article we are interested in determining the l adic cohomology of rapoport zink spaces associated to rm gl n over an unramified extension of bbb q p called unramified el type in connection with the local langlands correspondence for rm gl n in fact we compute the alternating sum of certain representation theoretic functors defined in terms of their cohomology on the level of grothendieck groups in case rapoport zink spaces parametrize p divisible groups of dimension one the above alternating sum is determined by harris and taylor without the unramifiedness assumption for p divisible groups of rm dim 1 fargues has obtained an answer for the supercuspidal part when the p divisible groups are basic i e their associated newton polygons are straight lines our main result is an identity among the cohomology of rapoport zink spaces that appear in the newton stratification of the same unitary shimura variety our result implies a theorem of fargues providing a second proof and reveals new information about the cohomology of rapoport zink spaces for non basic p divisible groups we propose an inductive procedure to completely determine the above alternating sum assuming a strengthening of our main result along with a conjecture of harris