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Level raising for automorphic representations of GL(2n)


Type

Thesis

Change log

Authors

Anastassiades, Christos 

Abstract

To each regular algebraic, conjugate self-dual, cuspidal automorphic representation Π 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 -adic Galois representation r(Π) of the absolute Galois group of E. The residual Galois representation r―(Π):Gal(E/E)→GLN(F) of π is defined to be the semisimplification of the reduction of r(Π) (modulo the maximal ideal of Z), with respect to any invariant Z-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 Π of GL(2n) over E, which is either unitary, conjugate self-dual and cuspidal or an isobaric sum Π1⊞Π2 of two unitary, conjugate self-dual cuspidal representations of GL(n), we want to construct a unitary, conjugate self-dual cuspidal representation Π of GL(2n) that has the same residual Galois representation as Π 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 Π with its base change along a CM biquadratic extension, under certain assumptions on Π (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 Π 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.

Description

Date

2019-02-04

Advisors

Thorne, Jack

Keywords

Number Theory, Automorphic Representations, Galois Representations, Level Raising

Qualification

Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge