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Inventiones mathematicae

, Volume 144, Issue 2, pp 297–325 | Cite as

Toric varieties and modular forms

  • Lev A. Borisov
  • Paul E. Gunnells

Abstract.

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).

Keywords

Modular Form Toric Variety Elliptic Genus Congruence Subgroup Holomorphic Modular Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Lev A. Borisov
    • 1
  • Paul E. Gunnells
    • 1
  1. 1.Department of Mathematics, Columbia University, New York, NY 10027, USA (e-mail: {lborisov,gunnells}@math.columbia.edu)US

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