Computing fundamental domains for the Bruhat-Tits tree for GL2(Qp), p-adic automorphic forms, and the canonical embedding of Shimura curves

TL;DR

This paper presents algorithms for computing quaternionic quotients of the Bruhat-Tits tree for GL2(Qp) and for deriving equations of Shimura curves' canonical embeddings, advancing computational methods in p-adic automorphic forms.

Contribution

It introduces new algorithms for quaternionic quotients and for obtaining equations of Shimura curves' embeddings, linking p-adic automorphic forms with explicit geometric models.

Findings

Algorithms successfully compute quaternionic quotients of the Bruhat-Tits tree.

Method derives conjectural equations for Shimura curves' canonical embeddings.

Enhances computational tools for p-adic automorphic forms and Shimura varieties.

Abstract

We describe an algorithm for computing certain quaternionic quotients of the Bruhat-Tits tree for GL2(Qp). As an application, we describe an algorithm to obtain (conjectural) equations for the canonical embedding of Shimura curves.