We prove the local Langlands correspondence for GSp 4 (F ), where F is a non-archimedean local field of positive characteristic with residue characteristic > 2.

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

We establish the existence and uniqueness of twisted exterior and symmetric square γ-factors in positive characteristic by studying the Siegel Levi case of generalized spinor groups. The corresponding theory in characteristic zero is due to Shahidi. In addition, in characteristic p we prove that these twisted local factors are compatible with the local Langlands correspondence. As a consequence, still in characteristic p, we obtain a proof of the stability property of γ-factors under twists by highly ramified characters. Next we use the results on the compatibility of the Langlands-Shahidi local coefficients with the Deligne-Kazhdan theory over close local fields to show that the twisted symmetric and exterior square γ-factors, L-functions and ε-factors are preserved over close local fields. Furthermore, we obtain a formula for Plancherel measures in terms of local factors and we also show that they also preserved over close local fields.

Abstract. We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of SU 3 (Qp). To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.

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Let G be a connected reductive group over a non-archimedean local field F and I be an Iwahori subgroup of G(F ). Let In is the n-th Moy-Prasad filtration subgroup of I. The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with In-level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups G that split over an unramified extension of F .The first main result of this paper is a presentation of the Hecke algebra H(G(F ), In), generalizing the previous work of Iwahori-Matsumoto on the affine Hecke algebras. For split GLn, Howe gave a refined presentation of the Hecke algebra H(G(F ), In). To generalize such a refined presentation to other groups requires the existence of some nice lifting of the Iwahori-Weyl group W to G(F ). The study of a certain nice lifting of W is the second main motivation of this paper, which we discuss below.In 1966, Tits introduced a certain subgroup of G(k), which is an extension of W by an elementary abelian 2-group. This group is called the Tits group and provides a nice lifting of the elements in the finite Weyl group. The "Tits group" T for the Iwahori-Weyl group W is a certain subgroup of G(F ), which is an extension of the Iwahori-Weyl group W by an elementary abelian 2-group. The second main result of this paper is a construction of Tits group T for W when G splits over an unramified extension of F . As a consequence, we generalize Howe's presentation to such groups. We also show that when G is ramified over F , such a group T of W may not exist.are the characteristic functions ½ gIn for g ∈ I I n and ½ Inm(w)In , where w runs over elements 2010 Mathematics Subject Classification. 22E70, 20C08.

This special issue of the Journal of IISc is devoted to the interrelated areas of number theory and representation theory, two evergreen and central subfields of mathematics that continue to see groundbreaking progress with each passing decade. This volume contains a collection of articles by experts, each surveying an important and active domain in the two subfields. We now briefly describe each of these works, starting with three articles on distinct themes in number theory that have seen extensive activity in recent times.The work of Wiles and Taylor on the Taniyama-Shimura conjecture catalysed a large amount of activity in the Langlands program. Roughly speaking, the Langlands program postulates a one-to-one correspondence between Galois representations on the one hand and certain automorphic forms on the other. Counterparts of some operations that are easily carried out for Galois representations turn out to be extremely difficult for automorphic forms (and vice versa). These form a part of Langlands's functoriality conjecture. One such operation is taking symmetric powers. The article "Modularity of Galois representations and Langlands functoriality" by James Newton gives a survey of recent results on modularity with a focus on his joint work with Thorne on proving Langlands functoriality of symmetric powers of a cuspidal Hecke eigenform. The article also provides an extensive list of references to expositions on various aspects and developments of modularity results that can be very helpful for someone to get initiated in the subject.Another very important theme in algebraic number theory is the relationship between analytic and arithmetic invariants. The subject has its origin in the works of Dirichlet and Kummer in the nineteenth century. Although there is a very satisfying conjectural picture due to the insights of many eminent mathematicians including Stark, Tate, Deligne, Beilinson, Bloch, and Kato, very few cases of these conjectures are proved. Perhaps the most studied case is the Birch and Swinnerton-Dyer conjecture (BSD).