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

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

langlands classification for l parameters abstract let f be a non archimedean local field and g g f the group of f rational points of a connected reductive f group the langlands classification of complex irreducible admissible representations of g gives a parametrization of the set of all such representations in terms of triples p where p g is a standard f parabolic subgroup is an irreducible tempered representation of the standard levi group m p and r x m p is regular and positive with respect to p in this paper we consider langlands l parameters which conjecturally will serve as a system of parameters for the representations and which are equivalence classes of l homomorphisms of the absolute galois group gal f f with image in langlands l group g l our goal is to establish a classification of l parameters in terms of triples p b such that p is as before and b is a bounded l parameter of m p in an appendix we also deal with the archimedean case f r c this case is easier because the weil group w r has a natural polar decomposition w r r w r 0

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

depth zero supercuspidal l packets for inner forms of gsp4 abstract we show that for any tame regular discrete series parameter of g s p 4 or its inner form g u 2 d the l packet attached by the local langlands conjecture gan and takeda 2011 14 gan and tantono submitted for publication 15 agrees with the l packet of depth zero supercuspidal representations constructed by debacker and reeder 2009 12

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

counting integral points in certain homogeneous spaces the leading term of asymptotic formula of the number of integral points in non compact symmetric homogeneous spaces of semi simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the brauer manin obstruction the similar result is also true for homogeneous spaces of reductive groups with some restriction as application we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and prove the leading term of asymptotic formula of the number of integral matrices with a fixed irreducible characteristic polynomial over z studied by eskin mozes shah is equal to the product of the number of local integral solutions over all primes although the density function defined by borovoi and rudnick is not trivial in general we also answer a question raised by borovoi and rudnick for comparing the number of integral symmetric matrices with the given determinant with the product of local densities c 2015 elsevier all rights reserved

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

fibration de hitchin et endoscopie we propose a geometric interpretation of the theory of elliptic endoscopy due to langlands and kottwitz in terms of the hitchin fibration as applications we prove a global analog of a purity conjecture due to goresky kottwitz and macpherson for unitary groups this global purity statement has been used in a joint work with g laumon to prove the fundamental lemma over a local fields of equal characteristics

global rigid inner forms and multiplicities of discrete automorphic representations we study the cohomology of certain galois gerbes over number fields this cohomology provides a bridge between refined local endoscopy as introduced in kaletha ann math 2 184 2 559 632 2016 and classical global endoscopy as particular applications we express the canonical adelic transfer factor that governs the stabilization of the arthur selberg trace formula as a product of normalized local transfer factors we give an explicit constriction of the pairing between an adelic l packet and the corresponding s group based on the conjectural pairings in the local setting that is the essential ingredient in the description of the discrete automorphic spectrum of a reductive group and we give a proof of some expectations of arthur

on the shimura variety having mumford s fake projective plane as a connected component we give an explicit construction of a unitary shimura surface that has mumford s fake projective plane as one of its connected components moreover as a byproduct of the construction we show that mumford s fake projective place has a model defined over the 7th cyclotomic field

discontinuous parameter estimates with least squares estimators we discuss weighted least squares estimates of ill conditioned linear inverse problems where weights are chosen to be inverse error covariance matrices least squares estimators are the maximum likelihood estimate for normally distributed data and parameters but here we do not assume particular probability distributions weights for the estimator are found by ensuring its minimum follows a g 2 distribution previous work with this approach has shown that it is competitive with regularization methods such as the l curve and generalized cross validation gcv 20 in this work we extend the method to find diagonal weighting matrices rather than a scalar regularization parameter diagonal weighting matrices are advantageous because they give piecewise smooth least squares estimates and hence are a mechanism through which least squares can be used to estimate discontinuous parameters this is explained by viewing least squares estimation as a constrained optimization problem results with diagonal weighting matrices are given for a benchmark discontinuous inverse problem from 13 in addition the method is used to estimate soil moisture from data collected in the dry creek watershed near boise idaho parameter estimates are found that combine two different types of measurements and weighting matrices are found that incorporate uncertainty due to spatial variation so that the parameters can be used over larger scales than those that were measured

linearizing torsion classes in the picard group of algebraic curves over finite fields we address the problem of computing in the group of ell k torsion rational points of the jacobian variety of algebraic curves over finite fields with a view toward computing modular representations

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

on galois cohomology of semisimple groups over local and global fields of positive characteristic ii we show that the recent results of prasad and rapinchuk adv math 207 2 646 660 2006 on the existence and uniqueness of certain global forms of semisimple algebraic groups with given local behaviour in the case of number fields still hold in the case of global function fields

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

distinguished regular supercuspidal representations based on recent work of kaletha we apply hakim murnaghan s result to study distinguished regular supercuspidal representations of tamely ramified reductive p adic groups assuming p is sufficiently large we obtain a necessary and sufficient condition for regular supercuspidal representations to be distinguished we also investigate the relation between the distinction problem and the langlands functoriality and confirm a conjecture of lapid for regular depth zero or epipelagic supercuspidal representations

a rigorous time bound for factoring integers in this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most ln 2 1 o for n oo where l a b exp b logx a loglogx l a many practical factoring algorithms including the quadratic sieve and the elliptic curve method are conjectured to have an expected running time that satisfies the same bound but this is the first algorithm for which the bound can be rigorously proved nevertheless this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms this is due to the advent of a new factoring algorithm the number field sieve which is conjectured to factor any positive integer n in time ln i 0 1 the algorithm analyzed in this paper is a variant of the class group relations method which makes use of class groups of binary quadratic forms of negative discriminant this algorithm was first suggested by seysen and later improved by a k lenstra who showed that the algorithm runs in expected time at most ln 12 1 o 1 if one assumes the generalized riemann hypothesis the main device for removing the use of the generalized riemann hypothesis from the proof is the use of multipliers in addition a character sum estimate for algebraic number fields is used with an explicit dependence on possible exceptional zeros of the corresponding l functions another factoring algorithm using class groups that has been proposed is the random class groups method it is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought department of mathematics university of california berkeley california 94720 e mail address hwl math berkeley edu department of mathematics university of georgia athens georgia 30602 e mail address carl ada math uga edu this content downloaded from 157 55 39 186 on tue 12 apr 2016 08 54 17 utc all use subject to http about jstor org terms

supercuspidal l packets of positive depth and twisted coxeter elements the local langlands correspondence is a conjectural connection between representations of groups g k for connected reductive groups g over a padic field k and certain homomorphisms langlands parameters from the galois or weil deligne group of k into a complex lie group g which is dual in a certain sense to g and which encodes the splitting structure of g over k more introductory remarks on the local langlands correspondence can be found in 21 wheng gl1 this correspondence should reduce to local abelian class field theory for g gln the langlands correspondence is uniquely determined by local factors 24 and was shown to exist in 23 and 25 so far this correspondence is not completely explicit but much progress has been made in this direction see 9 10 for example for groups other than gln or pgln the theory is much less advanced new phenomena appear arising on the arithmetic side from the difference between conjugacy and stable conjugacy and on the dual side from nontrivial monodromy of langlands parameters this means that a single langlands parameter should determine not just one but a finite set of representations these are the l packets of the title however since local factors have not been defined in general there is no precise characterization of an l packet for general groups one can at present only hope to define finite sets of representations attached to langlands parameters and show that they have properties expected or perhaps unexpected of l packets see 14 chap 3 for some of these properties one is thereby proposing a definition of local factors for the representations in the sets cf 4 chap 3 this paper is a sequel to 14 the aim of both papers is to verify in an explicit and natural way the local langlands correspondence for the simplest kinds of non abelian extensions of k and the simplest kinds of

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

modeling water retention curves of sandy soils using neural networks we used neural networks nns to model the drying water retention curve wrc of 204 sandy soil samples from particle size distribution psd soil organic matter content som and bulk density bd neural networks can relate multiple model input data to multiple model output data without the need of an a priori model concept in this way a high performance black box model is created which is very useful in a data exploration effort to assess the maximum obtainable prediction accuracy we used a series of nn models with an increasing parametrization of input and output variables to get a better interpretability of model results in the first two models we used the nine psd fractions bd and som as input while we predicted the nine points of the water retention curve these nns had 12 input and 9 output variables predicting wrcs with an average root mean square residual rmsr water content of 0 020 cm3 cm 3 after a few intermediary models with increasing parametrization of psd and wrc using adapted van genuchten 1980 equations we arrived at a final nn model that used six input variables to predict three van genuchten 1980 parameters resulting in a rmsr of 0 024 cm3 cm 3 we found saturated and residual water contents to be unrelated to the psd bd or som therefore the saturated water content was considered to be an independent input variable while the residual water content was set to zero sensitivity analyses showed that the psd had a major influence on the shape of the wrc while bd and som were less important on the basis of these sensitivity analyses we established more explicit equations that demonstrated similarity relations between psd and wrc and incorporated effects of som and bd in an empirical way despite the fact that we considered a large number of linear and nonlinear variants these equations had a weaker performance rmsr 0 029 cm3 cm 3 than the nn models proving the modeling power of that technique

rigid inner forms of real and p adic groups we dene a new cohomology set h 1 u w z g for an ane algebraic group g and a nite central subgroup both dened over a local eld of characteristic zero which is an enlargement of the usual rst galois cohomology set of g we show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered l packets for arbitrary connected reductive groups that extends the well known conjectural description for quasi split groups in the case of real groups we show that this description is correct using shelstad s work

endoscopy theta liftings and period integrals for the unitary group in three variables l packets for the quasi split unitary group in three variables u 3 we shall give an explicit parametrization of these l packets using theta liftings and describe some relations between the structure of l packets and certain period integrals we also prove that an endoscopic l packet contains a unique generic representation in local and global cases suppose that g is a reductive algebraic group over a local or global field

hat g local systems on smooth projective curves are potentially automorphic dfg grants fg 1920 and spp 1489 erc grant 290766 aamot nsf grant dms 1404769 nsf grant dms 1161671 humboldt research award clay mathematics institute erc grant no 714405 gmlp

twisted relative trace formulae with a view towards unitary groups we introduce a twisted relative trace formula which simultaneously generalizes the twisted trace formula of langlands et al in the quadratic case and the relative trace formula of jacquet and lai certain matching statements relating this twisted relative trace formula to a relative trace formula are also proven including the relevant fundamental lemma in the biquadratic case using recent work of jacquet lapid and their collaborators and the rankin selberg integral representation of the asai l function obtained by flicker using the theory of jacquet piatetskii shapiro and shalika we give the following application let e f be a totally real quadratic extension with gal e f let u be a quasi split unitary group with respect to a cm extension m f and let u res e f u under suitable local hypotheses we show that a cuspidal cohomological automorphic representation of u whose asai l function has a pole at the edge of the critical strip is nearly equivalent to a cuspidal cohomological automorphic representation of u that is u distinguished in the sense that there is a form in the space of admitting a nonzero period over u this provides cohomologically nontrivial cycles of middle dimension on unitary shimura varieties analogous to those on hilbert modular surfaces studied by harder langlands and rapoport

une formule pour le groupe de brauer algebrique dʼun torseur abstract given a homogeneous space x of a connected algebraic group g with connected stabilizers over a field k of characteristic zero we construct a complex of galois modules of length 3 and a canonical isomorphism between a hypercohomology group of this complex and an explicit subgroup of the brauer group of x this result is obtained as a consequence of a formula describing the algebraic brauer group of a torsor and it generalizes recent results by borovoi and van hamel by considering non linear groups g and by taking into account some transcendental elements in the brauer group of x

on multiplicity in restriction of tempered representations of p adic groups we establish an equality between two multiplicities one in the restriction of tempered representations of a p adic group to its closed subgroup with the same derived group and one occurring in their corresponding component groups in langlands dual sides so called mathcal s groups under working hypotheses about the tempered local langlands conjecture and the internal structure of tempered l packets this provides a formula of the multiplicity for p adic groups by means of dimensions of irreducible representations of their mathcal s groups

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

supercuspidal l packets via isocrystals in a recent paper debacker and reeder construct and parameterize l packets on pure inner forms of unramified p adic groups that consist of depth zero supercuspidal representations we generalize their work to non pure inner forms by providing an alternative construction based on the theory of isocrystals with additional structure due to kottwitz furthermore we show the stability and 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

hardy littlewood varieties and semisimple groups we are interested in counting integer and rational points in affine algebraic varieties also under congruence conditions we introduce the notions of a strongly hardy littlewood variety and a relatively hardy littlewood variety in terms of counting rational points satisfying congruence conditions the definition of a strongly hardy littlewood variety is given in such a way that varieties for which the hardy littlewood circle method is applicable are strongly hardy littlewood

la formule des traces stable pour le groupe metaplectique les termes elliptiques soient f un corps de nombres et tilde g le revetement metaplectique de weil de g mathbb a f mathrm sp 2n mathbb a f nous stabilisons les termes elliptiques semi simples de la partie specifique de la formule des trace pour tilde g a l aide du transfert des fonctions test et du lemme fondamental etablis dans un article anterieur nous obtenons ainsi une expression en termes des integrales orbitales stables le long de certaines classes f elliptiques dans mathrm so 2n 1 times mathrm so 2n 1 ce que nous appelons equi singulieres pour les paires n n avec n n n une formule simple des coefficients est aussi obtenue

quantification of uncertainty in pedotransfer function based parameter estimation for unsaturated flow modeling 1 while pedotransfer functions ptfs have long been applied to estimate soil hydraulic parameters for unsaturated flow and solute transport modeling the uncertainty associated with the estimates is often ignored the objective of this study is to evaluate uncertainty of the ptf estimated soil hydraulic parameters and its effect on numerical simulation of moisture flow contributing to the parameter estimation uncertainty are 1 the ptf intrinsic uncertainty caused by limited data used for ptf training and 2 the ptf input uncertainty in pedotransfer variables i e ptf inputs the ptf intrinsic uncertainty is assessed using the bootstrap method by generating multiple bootstrap realizations of the soil hydraulic parameters the realizations follow normal or lognormal distributions the ptf input variables i e bulk density and soil texture are obtained using the cokriging technique the ptf input uncertainty is quantified by assuming that the cokriging estimates follow a normal distribution our results show that the ptf input uncertainty dominates over the ptf intrinsic uncertainty and determines the spatial distribution of the ptf parameter estimation uncertainty when the parameter estimation uncertainty is included the spatial variability of the measured soil hydraulic parameters is better captured this is also the case for the observed moisture contents whose spatial variability is well bracketed by the prediction intervals however this is only possible after the ptf input uncertainty is considered these results suggest that additional sample acquisition for the ptf input variables would have a more favorable impact on reduction of the parameter estimation uncertainty than collecting additional soil hydraulic parameter measurements for ptf development

epipelagic l packets and rectifying characters we provide an explicit construction of the local langlands correspondence for general tamely ramified reductive p adic groups and a class of wildly ramified langlands parameters furthermore we verify that our construction satisfies many expected properties of such a correspondence more precisely we show that each l packet we construct admits a parameterization in terms of the langlands dual group contains a unique generic element for a fixed whittaker datum satisfies the formal degree conjecture is compatible with central and cocentral characters provides a stable virtual character and satisfies the expected endoscopic character identities moreover we show that in the case of mathrm gl n our construction coincides with the established local langlands correspondence our techniques provide a general approach to the construction of the local langlands correspondence for tamely ramified groups and regular supercuspidal parameters

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method

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

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

congruence relations on some shimura varieties the main purpose of this paper is the generalization of the well known eichler shimura congruence relations for modular curves to shimura varieties of peltype whose shimura group g is split over a fixed prime p in particular we prove a conjecture of blasius and rogawski for these shimura varieties further we give a formula for the cycle which is the reduction to positive characteristic of the hecke operator tp in terms of the root data of g and calculate this cycle in two examples msc classification 14k10 14l05 20g25

from laplace to langlands via representations of orthogonal groups in the late 1960s robert langlands proposed a new and far reaching connection between the representation theory of lie groups over real and p adic fields and the structure of the galois groups of these fields 24 even though this local langlands correspondence remains largely conjectural the relation that it predicts between representation theory and number theory has profoundly changed our views of both fields moreover we now know enough about the correspondence to address and sometimes solve traditional problems in representation theory that were previously inaccessible roughly speaking the local langlands correspondence predicts that complex irreducible representations of a reductive group g over a local field k should be

the local theta correspondence and the local gan gross prasad conjecture for the symplectic metaplectic case we prove the local gan gross prasad conjecture for the symplectic metaplectic case under some assumptions this is the last case of the local gan gross prasad conjectures which has not been established we also prove two of prasad s conjectures on the local theta correspondence in the almost equal rank case

arithmetical birational invariants of linear algebraic groups over two dimensional geometric fields let g be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by colliot thelene gille and parimala basing on their results we compute the group of classes of r equivalence g k r the defect of weak approximation a g the first galois cohomology h 1 k g and the tate shafarevich kernel x 1 k g for suitable k in terms of the algebraic fundamental group 1 g we prove that the groups g k r and a g and the set x 1 k g are stably k birational invariants of g

arithmetic invariants of discrete langlands parameters the local langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of p adic groups and their langlands parameters which are homomorphisms from the local weil deligne group to the l group in this article we refine a conjecture of hiraga ichino and ikeda which relates the formal degree of a discrete series representation to the value of the local gamma factor of its parameter we attach a rational function in x with rational coefficients to each discrete parameter which specializes at x q the cardinality of the residue field to the quotient of this local gamma factor by the gamma factor of the steinberg parameter the order of this rational function at x 0 is also an important invariant of the parameter it leads to a conjectural inequality for the swan conductor of a discrete parameter acting on the adjoint representation of the l group we verify this conjecture in many cases when we impose equality we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations both of which are found and described in this article

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