{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/392933"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/392933","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Realising The Smooth Representations Of GL2(O_F) Using p-adic Geometric Methods","abstract":"Let p be an odd prime number and F be a finite extension of the p-adic numbers Q_p with valuation ring O_F and residue field k. In this thesis, we study the smooth representation theory of the profinite group GL_2(OO_F) using the theory of dagger analytic geometry developed by Grosse-Kl\\\"onne [GK00]. For a locally profinite group G that acts continuously on a smooth dagger space X we adapt the techniques of Ardakov and Wadsley from [AW24] in order to develop a theory of G-equivariant vector bundles with a flat connection on X. In the case that X is affinoid and connected, the de Rham cohomology groups of these vector bundles on X are a source of smooth representations of G. By applying this theory to GL_2(O_F)-stable (dagger) affinoid subdomains of the Drinfeld upper half plane, and related spaces, we are able to provide p-adic geometric realisations of several families of smooth representations of GL_2(O_F). For example, we reconstruct the cuspidal representations of GL_2(k) and provide a complete classification of the so-called principal split representations of GL_2(O_F).","abstract_html":"Let p be an odd prime number and F be a finite extension of the p-adic numbers Q_p with valuation ring O_F and residue field k. In this thesis, we study the smooth representation theory of the profinite group GL_2(OO_F) using the theory of dagger analytic geometry developed by Grosse-Kl\\"onne [GK00]. For a locally profinite group G that acts continuously on a smooth dagger space X we adapt the techniques of Ardakov and Wadsley from [AW24] in order to develop a theory of G-equivariant vector bundles with a flat connection on X. In the case that X is affinoid and connected, the de Rham cohomology groups of these vector bundles on X are a source of smooth representations of G. By applying this theory to GL_2(O_F)-stable (dagger) affinoid subdomains of the Drinfeld upper half plane, and related spaces, we are able to provide p-adic geometric realisations of several families of smooth representations of GL_2(O_F). For example, we reconstruct the cuspidal representations of GL_2(k) and provide a complete classification of the so-called principal split representations of GL_2(O_F).","abstract_has_math":false,"creators":["Adams, Thomas"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Wadsley, Simon"],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-04-04","date_published":"2025-04-04","updated_at":"2026-07-22T22:23:54Z","subjects":["p-adic","Rigid Analytic Geometry","de Rham Cohomology","Smooth Representations"],"languages":["eng"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/7ffaa14d-2af7-4398-83cb-04f1fa187c64/download","http://purl.org/NET/rdflicense/allrightsreserved"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.123464","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Wadsley, Simon"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["DPMMS"]},{"key":"dc:creator","label":"Author","values":["Adams, Thomas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2025-04-04"]},{"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/1810/392933"]},{"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":["p-adic","Rigid Analytic Geometry","de Rham Cohomology","Smooth Representations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/7ffaa14d-2af7-4398-83cb-04f1fa187c64/download","http://purl.org/NET/rdflicense/allrightsreserved"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.17863/CAM.123464"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/dd672c13-1343-4723-9935-02a1edbb4faf/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let p be an odd prime number and F be a finite extension of the p-adic numbers Q_p with valuation ring O_F and residue field k. In this thesis, we study the smooth representation theory of the profinite group GL_2(OO_F) using the theory of dagger analytic geometry developed by Grosse-Kl\\\"onne [GK00]. For a locally profinite group G that acts continuously on a smooth dagger space X we adapt the techniques of Ardakov and Wadsley from [AW24] in order to develop a theory of G-equivariant vector bundles with a flat connection on X. In the case that X is affinoid and connected, the de Rham cohomology groups of these vector bundles on X are a source of smooth representations of G. By applying this theory to GL_2(O_F)-stable (dagger) affinoid subdomains of the Drinfeld upper half plane, and related spaces, we are able to provide p-adic geometric realisations of several families of smooth representations of GL_2(O_F). For example, we reconstruct the cuspidal representations of GL_2(k) and provide a complete classification of the so-called principal split representations of GL_2(O_F)."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["b2ed2e97a26bf2249c29268e99eb09dc","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Realising The Smooth Representations Of GL2(O_F) Using p-adic Geometric Methods"]}]}],"canonical_facts":{"dc:contributor.advisor":["Wadsley, Simon"],"dc:contributor.sponsor":["DPMMS"],"dc:creator":["Adams, Thomas"],"dc:date.issued":["2025-04-04"],"dc:description.abstract":["Let p be an odd prime number and F be a finite extension of the p-adic numbers Q_p with valuation ring O_F and residue field k. In this thesis, we study the smooth representation theory of the profinite group GL_2(OO_F) using the theory of dagger analytic geometry developed by Grosse-Kl\\\"onne [GK00]. For a locally profinite group G that acts continuously on a smooth dagger space X we adapt the techniques of Ardakov and Wadsley from [AW24] in order to develop a theory of G-equivariant vector bundles with a flat connection on X. In the case that X is affinoid and connected, the de Rham cohomology groups of these vector bundles on X are a source of smooth representations of G. By applying this theory to GL_2(O_F)-stable (dagger) affinoid subdomains of the Drinfeld upper half plane, and related spaces, we are able to provide p-adic geometric realisations of several families of smooth representations of GL_2(O_F). For example, we reconstruct the cuspidal representations of GL_2(k) and provide a complete classification of the so-called principal split representations of GL_2(O_F)."],"dc:format.checksum.md5":["b2ed2e97a26bf2249c29268e99eb09dc","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["https://doi.org/10.17863/CAM.123464"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/dd672c13-1343-4723-9935-02a1edbb4faf/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/392933"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/7ffaa14d-2af7-4398-83cb-04f1fa187c64/download","http://purl.org/NET/rdflicense/allrightsreserved"],"dc:subject":["p-adic","Rigid Analytic Geometry","de Rham Cohomology","Smooth Representations"],"dc:title":["Realising The Smooth Representations Of GL2(O_F) Using p-adic Geometric Methods"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:23:54Z"}