Arithmetic Aspects of Bianchi Groups

TL;DR

This paper explores the arithmetic properties of Bianchi groups, focusing on computational methods for homology, automorphic forms, Galois representations, and torsion, while highlighting open problems and providing numerical data.

Contribution

It provides a comprehensive survey of computational techniques and open problems related to Bianchi groups, connecting homology, automorphic forms, and Galois representations.

Findings

Computed homology of Bianchi groups with Hecke action

Identified connections with automorphic forms and Galois representations

Presented numerical data and open conjectures

Abstract

We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms, abelian varieties, Galois representations and the torsion in the homology of Bianchi groups. Along the way, we list several open problems and conjectures, survey the related literature, presenting concrete examples and numerical data.