Hervé Jacquet

Distinguished representations and quadratic base change for (3)

Let E/F be a quadratic extension of number fields. Suppose that every real place of F splits in E and let H be the unitary group in 3 variables. Suppose that Π is an automorphic cuspidal representation of GL(3, EA). We prove that there is a form φ in the space of Π such that the integral of φ over H(F )\H(FA) is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.

The continuous spectrum of the relative trace formula for GL(3) over a quadratic extension

AbstractLetE/F be a quadratic extension of number fields,G the group GL(3,E) regarded as an algebraic group overF andU a quasi-split unitary group in three variables. Let alsoϑ be a generic character of a maximal unipotent subgroupN ofG. We derive an explicit expression for the integral

Generic representations for the unitary group in three variables

\int {\int {K_{cont} (u, n)du\theta (n)dn} }

Germs of Kloosterman Integrals for (3)

whereKcont is the continuous part of the kernel attached to a smooth function of compact support onG(A). In particular, we prove that this expression is absolutely convergent. The result can be used to show that a cuspidal representation ofG contains a vectorφ such thatεφ(u)du≠0 if and only if it is a base change from a representation of GL(3,F).

Représentations génériques du groupe unitaire à trois variables

The non-vanishing, at the centre of symmetry, of theL-function attached to an automorphic representation of GL(2) or its twists by quadratic characters has been extensively investigated, in particular by Waldspurger. The purpose of this paper is to outline a new proof of Waldspurger’s results. The automorphic representations of GL(2) and its metaplectic cover are compared in two different ways; one way is by means of a “relative trace formula”; the relative trace formula presented here is actually a generalization of the work of Iwaniec.

Rankin-Selberg Convolutions

We show that for the quasi-split unitary group in three variables every tempered packet of cuspidal automorphic representations contains a globally generic representation.

Automorphic forms and automorphic representations

We establish the existence of smooth transfer between absolute Kloosterman integrals and Kloosterman integrals relative to a quadratic extension.

A relation between automorphic representations of

In an earlier paper we introduced the concept of Shalika germs for certain Kloosterman integrals. We compute explicitly the germs in the case of the group GL(3).