The Hilbert Modular Group and Orthogonal Groups

TL;DR

This paper establishes an explicit isomorphism between the Hilbert modular group and specific subgroups of the orthogonal group SO(2, 2), using linear algebra techniques tailored for number theory.

Contribution

It provides a novel explicit isomorphism linking Hilbert modular groups with orthogonal groups, enhancing understanding of their algebraic structures.

Findings

Explicit isomorphism between Hilbert modular group and SO(2, 2) subgroups

Application of linear algebra methods to number theory

New insights into the structure of congruence subgroups

Abstract

We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group $SO(2, 2)$ on the other hand. The proof is based on an application of linear algebra adapted to number theoretical needs.