Toric modular forms and nonvanishing of L-functions, by Lev A. Borisov and Paul E. Gunnells

In a previous paper, we defined the space of toric forms, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Gamma_1(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) is nonzero. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

Lev A. Borisov and Paul E. Gunnells <,>