Toric varieties and modular forms

Abstract.

Let N⊂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:ℌ→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 ? (l) of modular forms with respect to Γ₁ (l). We construct an explicit set of generators for ? (l), and show that ? (l) is stable under the action of the Hecke operators. Finally, we relate ? (l) to the Hirzebruch elliptic genera that are modular with respect to Γ₁ (l).