On Toric Varieties and Modular Forms

Let M∗(l) = M∗(Γ1(l),C) be the ring of holomorphic modular forms on Γ1(l). In this talk we use the combinatorics of complete toric varieties to construct a subring T∗(l) ⊂ M∗(l), the subring of toric modular forms (§2). This is a natural subring, in the sense that it behaves nicely with respect to natural operations on M∗(l) (namely, Hecke operators, Fricke involution, and the theory of oldforms and newforms). Moreover, an explicit structure theorem for T∗(l) together with the theory of Manin symbols allows us to describe the cuspidal part of T∗(l) in terms of nonvanishing of special values of L-functions (§3). Finally, we discuss an explicit scheme-theoretic embedding of the modular curve X1(l) = Γ1(l)\H ∗ in a weighted projective space that was inspired by the structure of T∗(l) (§4). The results of §§2–3 are joint work with Lev Borisov, and can be found in the papers [1, 2, 3]; the embedding of the modular curve in §4 is joint work with with Lev Borisov and Sorin Popescu and appears in [4]. It is a pleasure to thank them for their stimulating and interesting collaboration.