Toric modular forms and nonvanishing of L-functions

Abstract

In a previous paper [1], we de®ned the space of toric forms Tl, and showed that it is a ®nitely generated subring of the holomorphic modular forms of integral weight on the congruence group q 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 Y 1 Q 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols. First, in O2we recall results about the Manin symbols. We discuss various homology groups associated to the modular curve in terms of modular symbols, and

Elliptic functions and equations of modular curves

“…The following proposition follows easily from Lemma 4.8 in [2] for N = 2, and Propositions 4.7 and 4.8 in [3]: Proposition 2.4. The weight two modular forms…”

“…Notice that each s a (τ ) is an Eisenstein series. Our notation here differs slightly from that of [2], [3] and [4]. What we denote by s a here was called s a/p in those papers.…”

“…2) Furthermore, the embedding is not quadratically normal in general, due to the existence of Hecke eigenforms of positive analytic rank, see [3]. The first occurrence is at p = 37.…”

“…In Section 2 we recall first results from [2] and [3] concerning modular forms for Γ 1 (p), then show that weight one Eisenstein series define an embedding of the modular curve and compute equations vanishing on the image of this embedding (Proposition 2.4). In Section 3 we introduce a system of differential equations (2) that mimics the system satisfied by elliptic functions with poles of order one along a subgroup of order p. We construct Laurent series solutions to this system, and show that they satisfy the defining quadratic equations of a (possibly degenerate) elliptic normal curve in P p−1 .…”

Regulators of Siegel units and applications

“…It would be interesting to study further the properties of these modular forms and to understand their possible relations with the toric modular forms introduced by L. Borisov and P. Gunnells [1].…”