Title:

Level raising for automorphic representations of GL(2n)

To each regular algebraic, conjugate selfdual, cuspidal automorphic representation Π of GL(N) over a CM number field E (or, more generally, to a regular algebraic isobaric sum of conjugate selfdual, cuspidal representations), we can attach a continuous ℓadic Galois representation r(Π) of the absolute Galois group of E. The residual Galois representation r̅(Π):GalE̅/E)→GL_N(Fℓ) of π is defined to be the semisimplification of the reduction of r(Π) (modulo the maximal ideal of Z̅ℓ), with respect to any invariant Z̅ℓlattice. 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 selfdual and cuspidal or an isobaric sum Π₁ ⊞Π₂ of two unitary, conjugate selfdual cuspidal representations of GL(n), we want to construct a unitary, conjugate selfdual 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.
