Conjecture 1. Let F be a number field, let p be some prime number, and fix an isomorphism ι : C ∼= Qp. Then there is a bijection between the set of algebraic cuspidal automorphic representations of GLn(AF ), and the set of isomorphism classes of irreducible continuous representations of the absolute Galois group of F on n-dimensional Qp-vector spaces which are almost everywhere unramified, and ...

Let F be a global function field with constant $\mathbb{F}_q$. G reductive group over We establish variant of Arthur's truncated kernel for and its Lie algebra which generalizes original construction. coarse geometric expansion our truncation. As applications, we consider some existence uniqueness problems cuspidal automorphic representations the functions projective line $\mathbb{P}^1_{\mathbb...

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 ...

1.1 Whittaker and Bessel models are of importance in the theory of automorphic representations of GSp4. This paper is the first in a series of two, where we study some phenomena corresponding to these models in the geometric Langlands program. The theory of Whittaker functors for GLn is an essential technical tool in Gaitsgory’s proof of the Vanishing Conjecture appearing in the geometric Langl...

1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in Harish-Chandra’s investigations of the representations of semi-sim...

A well-known formula of Whipple relates certain hypergeometric values $$_7F_6(1)$$ and $$_4F_3(1)$$ . In this paper, we revisit relation from the viewpoint underlying data HD, to which there are also associated character sums Galois representations. We explain a special structure behind Whipple’s when HD primitive self-dual. If defined over $$\mathbb Q$$ , by work Katz, Beukers, Cohen, Mellit, ...

Let F be a totally real number field, n a prime integer, and G a unitary group of rank n defined over F that is compact at infinity. We prove an asymptotic formula for the number of automorphic representations of G whose factors at finitely many places are prescribed up to inertia. These factors are specified by local inertial types, and the proof relies crucially on a bound on the traces of th...

The aim of the present note is to develop a study on the feasibility of a unified theory of mean values of automorphic L-functions, a desideratum in the field. This is an outcome of the investigation commenced with the part XII ([13]), where a framework was laid on the basis of the theory of automorphic representations, and a general approach to the mean values was envisaged. Specifically, it i...

Let X be a smooth projective curve. Write BunSO2n for the moduli stack of SO2n-torsors on X. We give a geometric interpretation of the automorphic function f on BunSO2n corresponding to the minimal representation. Namely, we construct a perverse sheaf KH on BunSO2n such that f should be equal to the trace of the Frobenius of KH plus some constant function. The construction is based on some expl...

We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a p-adic distribution interpolating the special values of the twisted Rankin-Selberg L-function attached to cuspidal automorphic representations π and σ of GLn and GLn−1 over a number field k. If π and σ are ordinary at p, our distribution is bounded and gives rise to a p-adic ...