# On the genus of congruence surfaces from maximal orders

## TL;DR

This paper examines the possible genera of closed congruence surfaces derived from maximal orders in quaternion algebras, demonstrating that genus 212 cannot be realized within this specific class, thus answering a restricted version of Breuillard and Reid's question.

## Contribution

It provides a negative answer to the question of which genera can be obtained from such surfaces, focusing on those constructed from maximal orders in quaternion algebras.

## Key findings

- No surface of genus 212 exists in this class
- Certain genera are impossible for these surfaces
- The restricted question has a negative answer

## Abstract

In this paper, we investigate a question of Breuillard and Reid concerning which genera can be obtained by closed congruence surfaces. Specifically, we study a smaller set of objects, namely the closed congruence surfaces which can be constructed by a maximal order in a quaternion algebra, and show that there is no surface of genus 212 in this class. In particular, we show that Breuillard and Reid's question restricted to such surfaces has a negative answer.