The maximal discrete extension of the Hermitian modular group

Abstract

Let \Gamma_n(\mathcal{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 \operatorname{SU}(n, n; \mathbb{C}) , which coincides with the normalizer of \Gamma_n(\mathcal{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 \operatorname{SO}(2,4) .