Products of Vector Valued Eisenstein Series

TL;DR

This paper demonstrates that products of up to two vector valued Eisenstein series at level 1 generate all cusp forms for congruence subgroups, using vector valued Hecke operators to connect classical constructions.

Contribution

It establishes a level aspect analogue of classical weight aspect results, showing the spanning property of products of Eisenstein series for cusp forms.

Findings

Products of two Eisenstein series span cusp form spaces

Vector valued Hecke operators recover classical operators

A vanishing condition for modular forms is derived

Abstract

We prove that products of at most two vector valued Eisenstein series that originate in level 1 span all spaces of cusp forms for congruence subgroups. This can be viewed as an analogue in the level aspect to a result that goes back to Rankin, and Kohnen and Zagier, which focuses on the weight aspect. The main feature of the proof are vector valued Hecke operators. We recover several classical constructions from them, including classical Hecke operators, Atkin-Lehner involutions, and oldforms. As a corollary to our main theorem, we obtain a vanishing condition for modular forms reminiscent of period relations deduced by Kohnen and Zagier in the context of their previously mentioned result.