Hyper-Algebras of Vector-Valued Modular Forms

TL;DR

This paper introduces hyper-algebras for vector-valued Siegel modular forms, enabling new insights into their tensor products and Hecke operators, and demonstrates how large-weight cusp forms can be generated from fixed Eisenstein series.

Contribution

It defines graded hyper-algebras and vector-valued Hecke operators for Siegel modular forms, bridging classical and representation theoretic approaches, and shows how large-weight cusp forms derive from simple products.

Findings

Large-weight cusp forms are generated from fixed Eisenstein series.

Inclusions of cuspidal automorphic representations into tensor products of principal series.

New algebraic structures facilitate the study of tensor products and Hecke actions.

Abstract

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.