The basis problem for modular forms for the Weil representation

TL;DR

This paper proves that the space of cusp forms for the Weil representation is generated by vector valued theta series, solving Eichler's basis problem and providing new proofs for related modular form results.

Contribution

It establishes that vector valued theta series generate the cusp form space for the Weil representation, confirming Eichler's basis problem and offering new proofs for existing theorems.

Findings

Cusp form space is generated by vector valued theta series

Provides a positive answer to Eichler's basis problem

Offers a new proof of Borcherds lift surjectivity

Abstract

The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such functions. This gives a positive answer to Eichler's basis problem in this case. As applications we derive Waldspurger's result on the basis problem for scalar valued modular forms and give a new proof of the surjectivity of the Borcherds lift based on the analysis of local Picard groups.