Fundamental domains for quaternionic S-arithmetic groups

Abstract

Let F be a totally real number field, fix a prime p of F, and let F_p be the completion of F at p. In this thesis we develop an algorithm to compute fundamental domains for the action of certain discrete subgroups of SL_2(F_p) on the Bruhat-Tits tree associated to GL_2(F_p). The discrete groups that we consider arise from some Eichler orders on definite quaternion algebras defined over F. For Shimura curves that have bad reduction at p, the structure of the bad special fiber is encoded by these fundamental domains. We computed an extensive list of examples of fundamental domains related to p-adic uniformizations of Shimura curves. These fundamental domains can be used to integrate numerically rigid-analytic modular forms. As an application, we use these integrals to compute p-adic Heegner points on elliptic curves over totally real number fields.

Date of Award	9 Dec 2025
Original language	English
Awarding Institution	Universitat Autònoma de Barcelona (UAB)
Supervisor	Marc Masdeu Sabate (Director)