Congruence Subgroups and Orthogonal Groups

TL;DR

This paper establishes explicit isomorphisms between various congruence subgroups of modular and orthogonal groups, enhancing understanding of their algebraic structures over different number fields.

Contribution

It provides new explicit isomorphisms connecting congruence subgroups of Siegel, Hermitian, and quaternionic modular groups with orthogonal groups over specific number fields.

Findings

Explicit isomorphisms between congruence subgroups and orthogonal groups.

Application of linear algebra techniques to number theory.

Connections across modular groups over different algebraic structures.

Abstract

We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2 and the discriminant kernels of special orthogonal groups SO 0 (2, n), n = 3, 4, 6. The proof is based on an application of linear algebra adapted to the number theoretical needs.