Realising The Smooth Representations Of GL2(O_F) Using p-adic Geometric Methods

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).

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Adams, Thomas

Advisor dc:contributor.advisor

• Wadsley, Simon

• p-adic
• Rigid Analytic Geometry
• de Rham Cohomology
• Smooth Representations

Licence

• https://www.repository.cam.ac.uk/bitstreams/7ffaa14d-2af7-4398-83cb-04f1fa187c64/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.123464

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/392933