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Toric modular forms and nonvanishing of L-functions

Lev A. Borisov·Paul E. Gunnells
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Abstract

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 $Γ_1(l)$. 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 $L(f,1) \not = 0$. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.

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Lev A. Borisov, Paul E. Gunnells. 2000-10-03. Toric modular forms and nonvanishing of L-functions. https://arxiv.org/abs/math/9910141

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