Our goal is to formulate a theorem that is part of a recent classification of automorphic representations of orthogonal and symplectic groups. To place it in perspective, we devote much of the paper to a historical introduction to the Langlands program. In our attempt to make the article accessible to a general mathematical audience, we have centred it around the theory of L-functions, and its ...

We study the L-functions associated to Siegel modular forms — equivalently, automorphic representations of GSp(4, ‫ށ‬ ‫ޑ‬) — both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform representation theoretic calculations to cast the Langlands L-function in classical terms. We develop a precise notion of what it means to test a conjectured functional equation f...

This will appear as an appendix to the paper Functoriality for the exterior square of GL4 and the symmetric fourth of GL2 ([K]) by Henry Kim. The object here is to prove the following extension (from cuspidal) to isobaric automorphic representations of Proposition 3.6.1 of [Ra], which was itself an extension to GL(n) of the Proposition 4.2 (for GL(2)) in [BR]. The argument is essentially the sa...

A result of Chai–Faltings on Satake parameters of Siegel cusp forms together with the classification of unitary, unramified, irreducible, admissible representations of GSp4 over a p-adic field, imply that the local components of the automorphic representation of GSp4 attached to a cuspidal Siegel eigenform of degree 2 must lie in certain families. Applications include estimates on Hecke eigenva...

The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the framework of quantum mechanics, geometric quantization and optics. In this paper, we study the Weil representations of the Jacobi group and their properties. We also provide their applications to the theory of automorphic forms on the Jacobi group and repr...

In the late 1960s Robert Langlands launched what has become known as the Langlands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis [L]. In particular, Langlands conjectured that Galois representations and motives can be described in terms of the more tangible data of automorphic representations. A striking application of this general principle is...

Given a pair of distinct unitary cuspidal automorphic representations for GL([Formula: see text]) over number field, let [Formula: text] denote the set finite places at which are unramified and their associated Hecke eigenvalues differ. In this paper, we demonstrate how conjectures on automorphy possible cuspidality adjoint lifts Rankin–Selberg products imply lower bounds size text]. We also ob...

We extend the results of [CHT] by removing the ‘minimal ramification’ condition on the lifts. That is we establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), l-adic lifts of certain automorphic mod l Galois representations of any dimension. The main innovation is a new approach to the automorphy of non-minimal lifts which is closer in spi...

We consider restrictions along closed geodesics and geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain a non-trivial bound on the L-norm of such restrictions as the eigenvalue tends to infinity. We use methods from the theory of automorphic functions and in particular the uniqueness of invariant functionals on irreducible unit...

We completely determine the residual automorphic representations coming from the torus of odd orthogonal groups. Under certain technical assumption on disconnected groups, our results are valid for symplectic groups and even orthogonal groups. Moeglin has determined the residual spectrum attached to the trivial character of the torus for split classical groups. However, non-trivial characters p...