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Level raising for automorphic representations of GL(2n) 10.17863/CAM.39690 Anastassiades, Christos Thorne, Jack 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 $G$. University of Cambridge 2019-05-18 Thesis Doctoral Doctor of Philosophy (PhD) en https://www.repository.cam.ac.uk/handle/1810/292530 https://www.repository.cam.ac.uk/bitstreams/3c0c1e4c-2a2f-4235-a09d-f0d58e00de4a/download b991a04fcdfbfb1f5bc806de3a6d789b https://www.repository.cam.ac.uk/bitstreams/af93c49d-6f58-4767-a7b0-35c8d4ed9441/download 87eda9de84448d1f82354d60eee3eb5f https://www.rioxx.net/licenses/all-rights-reserved/ Number Theory Automorphic Representations Galois Representations Level Raising