{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/292530"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/292530","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"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 $G$.","abstract_html":"To each regular algebraic, conjugate self-dual, cuspidal automorphic representation $\\Pi$ of <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">GL</span>(N)</span> 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 <span class=\"etd-inline-math\">\\overline{r}(\\Pi):<span class=\"etd-inline-math-roman\">Gal</span>(\\overline{E}/E)\\to<span class=\"etd-inline-math-roman\">GL</span><sub>N</sub>(\\overline{\\mathbb{F}}<sub>\\</sub>ell)</span> of <span class=\"etd-inline-math\">&pi;</span> is defined to be the semisimplification of the reduction of $r(\\Pi)$ (modulo the maximal ideal of <span class=\"etd-inline-math\">\\overline{\\mathbb{Z}}<sub>\\</sub>ell</span>), with respect to any invariant <span class=\"etd-inline-math\">\\overline{\\mathbb{Z}}<sub>\\</sub>ell</span>-lattice. The aim of this thesis is to prove a level raising theorem for automorphic representations of <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">GL</span>(2n)</span>. More precisely, given a regular algebraic automorphic representation $\\Pi$ of <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">GL</span>(2n)</span> over $E$, which is either unitary, conjugate self-dual and cuspidal or an isobaric sum <span class=\"etd-inline-math\">\\Pi<sub>1</sub> \\boxplus \\Pi<sub>2</sub></span> of two unitary, conjugate self-dual cuspidal representations of <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">GL</span>(n)</span>, we want to construct a unitary, conjugate self-dual cuspidal representation $\\Pi&#x27;$ of <span class=\"etd-inline-math\"><span class=\"etd-inline-math-roman\">GL</span>(2n)</span> 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&#x27;s lemma” for $G$, using the strong approximation theorem for the derived subgroup of $G$.","abstract_has_math":true,"creators":["Anastassiades, Christos"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Thorne, Jack"],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-05-18","date_published":"2019-05-18","updated_at":"2026-07-24T01:33:33Z","subjects":["Number Theory","Automorphic Representations","Galois Representations","Level Raising"],"languages":["en"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/af93c49d-6f58-4767-a7b0-35c8d4ed9441/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.39690","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Thorne, Jack"]},{"key":"dc:creator","label":"Author","values":["Anastassiades, Christos"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2019-05-18"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/292530"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Number Theory","Automorphic Representations","Galois Representations","Level Raising"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/af93c49d-6f58-4767-a7b0-35c8d4ed9441/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.39690"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/3c0c1e4c-2a2f-4235-a09d-f0d58e00de4a/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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 $G$."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["b991a04fcdfbfb1f5bc806de3a6d789b","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Level raising for automorphic representations of GL(2n)"]}]}],"canonical_facts":{"dc:contributor.advisor":["Thorne, Jack"],"dc:creator":["Anastassiades, Christos"],"dc:date.issued":["2019-05-18"],"dc:description.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 $G$."],"dc:format.checksum.md5":["b991a04fcdfbfb1f5bc806de3a6d789b","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.39690"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/3c0c1e4c-2a2f-4235-a09d-f0d58e00de4a/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/292530"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/af93c49d-6f58-4767-a7b0-35c8d4ed9441/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Number Theory","Automorphic Representations","Galois Representations","Level Raising"],"dc:title":["Level raising for automorphic representations of GL(2n)"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:33:33Z"}