Corps de nombres peu ramifiés et formes automorphes autoduales

Abstract:

Let $S$ be a finite set of primes, $p$ in $S$, and $\mathbb {Q}_S$ a maximal algebraic extension of $\mathbb {Q}$ unramified outside $S$ and $\infty$. Assume that $|S|\geq 2$. We show that the natural maps \[ \operatorname {Gal}(\overline {\mathbb {Q}_p}/\mathbb {Q}_p) \rightarrow \operatorname {Gal}(\mathbb {Q}_S/\mathbb {Q})\] are injective. Much of the paper is devoted to the problem of constructing self-dual automorphic cuspidal representations of $\operatorname {GL}(2n,\mathbb {A}_{\mathbb {Q}})$ with prescribed properties at all places, which we study via Arthur’s twisted trace formula. The techniques we develop also shed some light on the orthogonal/symplectic alternative for self-dual representations of $\operatorname {GL}(2n)$.