Toric varieties and modular forms

Lev A. Borisov, Paul E. Gunnells

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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

Original languageEnglish (US)
Pages (from-to)297-325
Number of pages29
JournalInventiones Mathematicae
Issue number2
StatePublished - 2001
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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