Level raising for automorphic representations of GL(2n)
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of Cambridge
Department of Pure Mathematics and Mathematical Statistics
Trinity College
Department of Pure Mathematics and Mathematical Statistics
Trinity College
Abstract
Description
To each regular algebraic, conjugate self-dual, cuspidal automorphic representation $\Pi$ of $\mathrm{GL}(N)$ over a CM number field $E$ (or, more generally, to a regular algebraic isobaric sum of conjugate self-dual, cuspidal representations), we can attach a continuous $\ell$-adic Galois representation $r(\Pi)$ of the absolute Galois group of $E$. The residual Galois representation $\overline{r}(\Pi):\mathrm{Gal}(\overline{E}/E)\to\mathrm{GL}_N(\overline{\mathbb{F}}_\ell)$ of $\pi$ is defined to be the semisimplification of the reduction of $r(\Pi)$ (modulo the maximal ideal of $\overline{\mathbb{Z}}_\ell$), with respect to any invariant $\overline{\mathbb{Z}}_\ell$-lattice. The aim of this thesis is to prove a level raising theorem for automorphic representations of $\mathrm{GL}(2n)$. More precisely, given a regular algebraic automorphic representation $\Pi$ of $\mathrm{GL}(2n)$ over $E$, which is either unitary, conjugate self-dual and cuspidal or an isobaric sum $\Pi_1 \boxplus \Pi_2$ of two unitary, conjugate self-dual cuspidal representations of $\mathrm{GL}(n)$, we want to construct a unitary, conjugate self-dual cuspidal representation $\Pi'$ of $\mathrm{GL}(2n)$ that has the same residual Galois representation as $\Pi$ and whose component at a finite place $w$ of $E$ is an unramified twist of the Steinberg representation.
We prove that this is possible, after replacing $\Pi$ with its base change along a CM biquadratic extension, under certain assumptions on $\Pi$ (including a local obstruction at the place $w$). Our proof uses the results of Kaletha, Minguez, Shin and White on the endoscopic classification of representations of (inner forms of) unitary groups to descend $\Pi$ to an automorphic representation of a totally definite unitary group $G$ over the maximal totally real subfield of $E$. We then prove a level raising theorem for the group $G$; we do this by proving an analogue of “Ihara's lemma” for $G$, using the strong approximation theorem for the derived subgroup of $G$.