Level raising for automorphic representations of GL(2n)

To each regular algebraic, conjugate self-dual, cuspidal automorphic representation $\Pi$ of 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):Gal(\overline{E}/E)\toGL_N(\overline{\mathbb{F}}_\ell) of π 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 GL(2n). More precisely, given a regular algebraic automorphic representation $\Pi$ of GL(2n) over $E$, which is either unitary, conjugate self-dual and cuspidal or an isobaric sum \Pi₁ \boxplus \Pi₂ of two unitary, conjugate self-dual cuspidal representations of GL(n), we want to construct a unitary, conjugate self-dual cuspidal representation $\Pi'$ of 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 $G$.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2019

Author dc:creator

• Anastassiades, Christos

Advisor dc:contributor.advisor

• Thorne, Jack

• Number Theory
• Automorphic Representations
• Galois Representations
• Level Raising

Licence

• https://www.repository.cam.ac.uk/bitstreams/af93c49d-6f58-4767-a7b0-35c8d4ed9441/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.39690

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/292530