Toric varieties and modular forms

Abstract

Let $N\subset \RR^{r}$ be a lattice, and let $°\colon N \to \CC$ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on $°$, the data $(N,°)$ determines a function $f\colon {\HHH}\to \CC$ that is a holomorphic modular form of weight $r$ for the congruence subgroup $Γ_{1} (l) $. Moreover, by considering all possible pairs $(N ,°)$, we obtain a natural subring ${\TTT} (l)$ of modular forms with respect to $Γ_{1} (l) $. We construct an explicit set of generators for $\TTT (l)$, and show that ${\TTT} (l)$ is stable under the action of the Hecke operators. Finally, we relate ${\TTT} (l)$ to the Hirzebruch elliptic genera that are modular with respect to $Γ_{1} (l) $.