Hermitian Theta Series and Maaß Spaces Under the Action of the Maximal Discrete Extension of the Hermitian Modular Group

Abstract

AbstractLet $$\Gamma _n(\mathcal {\scriptstyle {O}}_{\mathbb {K}})$$ Γ n ( O K ) denote the Hermitian modular group of degree n over an imaginary quadratic number field $$\mathbb {K}$$ K and $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ its maximal discrete extension in the special unitary group $$SU(n,n;\mathbb {C})$$ S U ( n , n ; C ) . In this paper we study the action of $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ on Hermitian theta series and Maaß spaces. For $$n=2$$ n = 2 we will find theta lattices such that the corresponding theta series are modular forms with respect to $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ as well as examples where this is not the case. Our second focus lies on studying two different Maaß spaces. We will see that the new found group $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ consolidates the different definitions of the spaces.