Abstract: In a previous paper we defined the toric forms $\T_*(\ell)$, and showed that they constitute a finitely-generated subring of the holomorphic modular forms of integral weight on the congruence group $\Gamma_1(\ell)$. 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 forms $f$ such that $L(f,1) \not = 0$. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.
@misc{math.NT/9910141,
title = {{Toric modular forms and nonvanishing of L-functions}},
author = {Lev A. Borisov and Paul E. Gunnells},
eprint = {arxiv:math.NT/9910141}}