The maximal discrete extension of the Hermitian modular group

TL;DR

This paper determines the maximal discrete extension of the Hermitian modular group over an imaginary quadratic field, describing its structure via class groups, torsion subgroups, and involutions, with explicit cases for n=2.

Contribution

It provides a complete description of the normalizer of the Hermitian modular group, linking it to ideal class groups and involutions, and characterizes it within SO(2,4).

Findings

Maximal discrete extension coincides with the normalizer in SU(n,n).

Explicit description for n=2 involves generalized Atkin-Lehner involutions.

The group is characterized within SO(2,4).

Abstract

Let $\Gamma_n(\mathcal{\scriptstyle{O}}_\mathbb{K})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $SU(n,n;\mathbb{C})$, which coincides with the normalizer of $\Gamma_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$. The description involves the $n$-torsion subgroup of the ideal class group of $\mathbb{K}$. This group is defined over a particular number field $\widehat{\mathbb{K}}_n$ and we can describe the ramified primes in it. In the case $n=2$ we give an explicit description, which involves generalized Atkin-Lehner involutions. Moreover we find a natural characterization of this group in $SO(2,4)$.