Adelic Maass spaces on U(2,2)

TL;DR

This paper constructs an adelic Maass space for hermitian modular forms on U(2,2), demonstrating its invariance under Hecke operators and linking it to elliptic modular forms, extending prior work to a broader adelic context.

Contribution

It introduces an adelic version of the Maass space for U(2,2) and proves its invariance under local Hecke algebras, establishing a Hecke-equivariant embedding into elliptic modular forms.

Findings

Maass space is invariant under local Hecke algebras for odd class number.

Established a Hecke-equivariant injective map to elliptic modular forms.

Extended classical results to an adelic setting for hermitian modular forms.

Abstract

Generalizing the results of Kojima, Gritsenko and Krieg, we define an adelic version of the Maass space for hermitian modular forms of weight k regarded as functions on adelic points of the quasi-split unitary group U(2,2) associated with an imaginary quadratic extension F/Q of discriminant D_F. When the class number h_F of F is odd, we show that the Maass space is invariant under the action of the local Hecke algebras of U(2,2)(Q_p) for all p not dividing D_F. As a consequence we obtain a Hecke-equivariant injective map from the Maass space to the h_F-fold direct product of the space of elliptic modular forms of weight k-1 and level D_F.