### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 108, 17 pages      arXiv:1810.02048      https://doi.org/10.3842/SIGMA.2018.108
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

### Hyper-Algebras of Vector-Valued Modular Forms

Martin Raum
Chalmers tekniska högskola och Göteborgs Universitet, Institutionen för Matematiska vetenskaper, SE-412 96 Göteborg, Sweden

Received May 07, 2018, in final form September 30, 2018; Published online October 04, 2018

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.

Key words: Siegel modular forms; vector-valued Hecke operators; automorphic representations.

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