Algebraic Geometry Commons^™

Toric Modular Forms And Nonvanishing Of L-Functions, Lev A. Borisov, Paul E. Gunnells

Paul Gunnells

In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, 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)≠0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.