Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations

Abstract

In this paper we show a local Jacquet-Langlands correspondence for all unitary irreducible representations. We prove the global Jacquet-Langlands correspondence in characteristic zero. As consequences we obtain the multiplicity one and strong multiplicity one theorems for inner forms of GL(n) as well as a classification of the residual spectrum and automorphic representations in analogy with results proved by Moeglin-Waldspurger and Jacquet-Shalika for GL(n).

Local root numbers, Bessel models, and a conjecture of Guo and Jacquet

“…First, the case n = 1 was already established by Waldspurger [33]. Waldspurger further proved that, when n = 1, there is a unique such D in part (2), and by work of Tunnell [31] and Saito [30], this D can be determined uniquely in terms of local root numbers. For n > 1 odd, it is also reasonable to expect that the D in part (2) is unique and is determined by root numbers.…”

“…Let G and H be algebraic groups defined over F with common center Z, and suppose H is a closed subgroup of G. In this paper, a (cuspidal) automorphic representation means an irreducible unitary (cuspidal) automorphic representation. We say a cuspidal representation π of G(A) with trivial central character is H-distinguished if the period integral Let E/F be a quadratic extension of number fields and X(E : F ) denote the set of isomorphism classes of quaternion algebras over F which split over E. For D ∈ X(E : F ), let JL = JL D denote the Jacquet-Langlands correspondence of representations from an inner form GL(n, D) to GL(2n) defined by Badulescu [2] and Badulescu-Renard [3], and LJ D denote its inverse. For a cuspidal representation π of GL(2n, A), π E denotes the base change of π to GL(2n, A E ), and X(E : F : π) denotes the set of D ∈ X(E : F ) for which π D = LJ D (π) exists as a (necessarily cuspidal) representation of GL(n, D)(A).…”

Shalika periods on GL2(D) and GL4

“…It is conjectured that there is a similar Jacquet-Langlands correspondence between representations π of G(‫)ށ‬ and π of G ‫)ށ(‬ (or more generally between any inner forms of GL n ) such that π v π v when G v G v . A consequence of this conjecture is that multiplicity one and strong multiplicity one theorems should hold for G. Such a correspondence has been established when D is split at each infinite place [Badulescu 2007].…”

Refined Limit Multiplicity for Varying Conductor

“…Our decision to count representations π where π S is generic is natural; these are the automorphic D(A) 1 -representations whose image under the global Jacquet-Langlands functor of [Bad07] is cuspidal. Moreover, the stipulation that S contain a place v 0 at which D splits is a technical condition used to show that the proportion of non-generic representations in the limit multiplicity formula vanishes.…”