Toric varieties and modular forms

Let W ⊂ ℝ^r be a lattice, and let deg : N → ℂ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N, deg) determines a function f : Heng hooktop sign → ℂ that is a holomorphic modular form of weight r for the congruence subgroup Γ₁(l). Moreover, by considering all possible pairs (N, deg), we obtain a natural subring Script T sign (l) of modular forms with respect to Γ₁(l). We construct an explicit set of generators for Script T sign (l), and show that Script T sign (l) is stable under the action of the Hecke operators. Finally, we relate Script T sign (l) to the Hirzebruch elliptic genera that are modular with respect to Γ₁(l).