The Hermitian modular group of degree $n$ over an imaginary quadratic field

$K=\mathbb{Q}(\sqrt{-m})$ was introduced by Hel Braun in the 1940s

as an analogue for the well known Siegel modular group. It acts on the

Hermitian half space and the accociated Hermitian modular forms have

been studied thoroughly in the past. However, this talk does not concentrate

on the modular forms but on the modular group itself. For $n=2$ and $m \neq 1,3$,

$m$ squarefree, we will prove that the Hermitian modular group $U(2,2;\mathcal{O}_K)$,

where $\mathcal{O}_K$ is the ring of integers in $K$, is isomorphic to the discriminant

kernel of the orthogonal group O(2,4) and we will provide an explicit homomorphism.

Furthermore, we compute the normalizer of the Hermitian modular group in the symplectic

group and show that it is isomorphic to the integral orthogonal group which is the normalizer

of the discriminant kernel in the real orthogonal group.

© MPI f. Mathematik, Bonn | Impressum & Datenschutz |