The maximal discrete extension of the Hermitian modular group

Documenta Mathematica, vol. 26, pp. 1871–1888

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) .

Authors 3

1. Aloys Krieg Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl A für Mathematik, RWTH Aachen University, D-52056 Aachen, Germany

2. Martin Raum

   University of Gothenburg · Chalmers University of Technology

   Affiliation as printed

   Chalmers Tekniska Högskola och Göteborgs Universitet, Institutionen för Matematiska Vetenskaper, SE-412 96 Göteborg, Sweden

3. Annalena Wernz Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl A für Mathematik, RWTH Aachen University, D-52056 Aachen, Germany