Let N⊂ℝr 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).
KeywordsModular Form Toric Variety Elliptic Genus Congruence Subgroup Holomorphic Modular Form
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