This article grew out of my talk in ‘The Legacy of Srinivasa Ramanujan’ conference where I spoke about some techniques to prove algebraicity results for the special values of symmetric cube L-functions attached to the Ramanujan ∆-function. If one wishes to compare these different techniques, then one needs to compare various automorphic periods attached to the symmetric cube transfer of ∆. Moti...

We report briefly on an endoscopic classification of representations by focusing on one aspect of the problem, the question of embedded Hecke eigenvalues. 1. The problem for G By “eigenvalue”, we mean the family of unramified Hecke eigenvalues of an automorphic representation. The question is whether there are any eigenvalues for the discrete spectrum that are also eigenvalues for the continuou...

is then a unitary representation of G(F) X G(F) on L2(G(F)). Kazhdan has suggested that there should be a local trace formula attached to R which is analogous to the global trace formula for automorphic forms. The purpose of this note is to discuss how one might go about proving such an identity, and to describe the ultimate form the identity is likely to take. To see the analogy with automorph...

Rough exposition of the goal of this course. Hecke actions on Shimura varieties and their cohomology groups. The Matsushima formula in the classical setting; relative Lie algebra cohomology, spectral decomposition on L(Γ\G). The Matsushima formula in the adelic setting — automorphic representations appear in the Betti cohomology of Shimura varieties. Admissible (g, U)-modules and cohomological ...

This paper is concerned with a compatible family of 4-dimensional -adic representations ρ of GQ := Gal(Q/Q) attached to the space of weight-3 cuspforms S3(Γ) on a noncongruence subgroup Γ ⊂ SL2(Z). For this representation we prove that: 1. It is automorphic: the L-function L(s, ρ∨ ) agrees with the L-function for an automorphic form for GL4(AQ), where ρ ∨ is the dual of ρ . 2. For each prime p ...

An important but seldom-heard remark is that there is essentially just one irreducible representation of a p-adic group. Thus, rather than opening a Pandora’s box of disastrous complication, introduction of representation-theoretic methods in study of automorphic forms and L-functions is exactly the right thing: to the extent that there can be simplicity here, it is in the context of representa...

For us it is imperative not to consider functions on the upper half plane but rather to consider functions on GL(2,Q)\GL(2,A(Q)) where A(Q) is the adéle ring of Q. We also replace Q by an arbitrary number field or function field (in one variable over a finite field) F . One can introduce [3] a space of functions, called automorphic forms, and the notion that an irreducible representation π ofGL...

‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...

We prove a geometric version of a classical result on the characterization of an irreducible cuspidal automorphic representation of GLn(AE ) being the base change of a stable cuspidal representation of the quasi-split unitary group associated to the quadratic extension E/F , via the nonvanishing of certain period integrals, called being distinguished. We show that certain cohomology of an autom...

Conjectures concerning the relations between motives and automorphic representations of GL(n) over number elds have been made by Clozel C1]. They make precise part of the programme indicated by Langlands in La]. Once a motive is attached to an automorphic representation, one can derive a compatible series of-adic representations of Gal(Q=Q) attached to it in the usual way. In C2], Clozel has gi...