Level raising for automorphic representations of GL(2n)

Abstract

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 $G$