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

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

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

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

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

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

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

tage iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations abstract in the present paper we discuss three point difference method based on nonpolynomial spline basis for the second order ordinary differential equation difference schemes are derived for linear and nonlinear case and are used to solve via two parameter alternating group explicit iterative algorithm the schemes have a fourth and second order of uniform convergence for the choice of the parameters involved in the method computational results are presented comparing the two methods in terms of accuracy and execution times the results indicate the advantage of using parallel implementation of the new method

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

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

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

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

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 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