The maximal discrete extension of the Hermitian modular group.

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 $SO(2,4)$.