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Realising The Smooth Representations Of GL2(O_F) Using p-adic Geometric Methods https://doi.org/10.17863/CAM.123464 Adams, Thomas Wadsley, Simon 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). University of Cambridge 2025-04-04 Thesis Doctoral Doctor of Philosophy (PhD) eng DPMMS https://www.repository.cam.ac.uk/handle/1810/392933 https://www.repository.cam.ac.uk/bitstreams/dd672c13-1343-4723-9935-02a1edbb4faf/download b2ed2e97a26bf2249c29268e99eb09dc https://www.repository.cam.ac.uk/bitstreams/7ffaa14d-2af7-4398-83cb-04f1fa187c64/download 87eda9de84448d1f82354d60eee3eb5f http://purl.org/NET/rdflicense/allrightsreserved p-adic Rigid Analytic Geometry de Rham Cohomology Smooth Representations