Geodesic
The maximal discrete extension of the Hermitian modular group
Documenta mathematica, Tome 26 (2021), pp. 1871-1888.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $\operatorname{SO}(2,4)$.
Classification : 11F06, 11F55
Keywords: Hermitian modular group, normalizer, maximal discrete extension, Atkin-Lehner involution, orthogonal group 
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     title = {The maximal discrete extension of the {Hermitian} modular group},
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Krieg, Aloys; Raum, Martin; Wernz, Annalena. The maximal discrete extension of the Hermitian modular group. Documenta mathematica, Tome 26 (2021), pp. 1871-1888. http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a5/