Abstract: Let $N\subset \RR^{r}$ be a lattice, and let $\deg\colon N \rightarrow \CC$ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Then the data $(N,\deg)$ determines a function $f\colon {\HHH}\rightarrow \CC$ that is, under certain conditions on $\deg$, a modular form of weight $r$ for the congruence subgroup $\Gamma_{1} (l) $.

Moreover, by considering all possible pairs $(N ,\deg)$, we obtain a natural subring ${\TTT} (l)$ of modular forms with respect to $\Gamma_{1} (l) $. We show that ${\TTT} (l)$ is stable under the action of the Hecke operators, and relate forms in ${\TTT} (l)$ to the Hirzebruch elliptic genera that are modular with respect to $\Gamma_{1} (l) $.

@misc{math.NT/9908138,
    title = {{Toric varieties and modular forms}},
    author = {Lev A. Borisov and Paul E. Gunnells},
    eprint = {arxiv:math.NT/9908138}}