Toric modular forms and nonvanishing of L-functions

Abstract

In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) 6= 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.