On hermitian modular groups and modular forms

• Zu Hermiteschen Modulgruppen und Modulformen

Wernz, Annalena Barbara Margit; Krieg, Aloys (Thesis advisor); Pumplün, Susanne (Thesis advisor)

Aachen (2019)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2019

Abstract

In this doctoral thesis, we find that for $n=2$, the Hermitian modular group and other subgroups of the unitary group are isomorphic to suitable subgroups of the orthogonal group $O(2,4)$. The proof is completely explicit and we find a closed formula for the image under the isomorphism. Furthermore, we compute the normalizer of the Hermitian modular group and its image on the orthogonal side, respectively. Finally, we consider Hermitian and orthogonal modular forms. We are able to identify Hermitian with orthogonal modular forms and derive some conditions for the weights of modular forms with trivial character. Lastly, we consider two prominent examples of Hermitian modular forms, namely Maaß forms and theta series.