Corps de nombres peu ramifies et formes automorphes autoduales

TL;DR

This paper investigates the structure of certain Galois groups and constructs selfdual automorphic representations of GL(2n) with specific local properties, using advanced trace formula techniques.

Contribution

It establishes injectivity of Galois group maps and develops methods to construct selfdual automorphic cuspidal representations with prescribed local behaviors.

Findings

Injectivity of Galois group maps for specified primes.

Construction of selfdual automorphic representations with controlled local properties.

Insights into the orthogonal/symplectic dichotomy for selfdual representations.

Abstract

Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n).