Toric varieties and modular forms, by Lev A. Borisov and Paul E. Gunnells

Let N be a lattice, and let deg be a complex-valued 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 holomorphic modular form of weight r on congruence subgroup Gamma_1(l).

Moreover, by considering all possible pairs (N, deg), we obtain a natural subring T(l) of modular forms with respect to Gamma_1(l). We construct an explicit set of generators for T(l), and show that T(l) is stable under the action of the Hecke operators. Finally, we relate T(l) to the Hirzebruch elliptic genera that are modular with respect to Gamma_1(l).


Lev A. Borisov and Paul E. Gunnells <gunnells@math.columbia.edu>