Toric varieties and modular forms

1999/08/26 by Lev Borisov, Lev A. Borisov, Borisov, Lev A. +2
Mathematics · #11F11 #11F25 #14M25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT #msc:11F11 #msc:11F25 #msc:14M25

paper · pdf · doi:10.48550/arxiv.math/9908138

27 pp., 1 figure, AMS-LaTeX

arxiv created 1999/08/26 · openalex publication_date 1999/08/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let N⊂ \RR^r be a lattice, and let °\colon N → \CC be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on °, the data (N,°) determines a function f\colon \HHH→ \CC that is a holomorphic modular form of weight r for the congruence subgroup Γ₁ (l) . Moreover, by considering all possible pairs (N ,°), we obtain a natural subring \TTT (l) of modular forms with respect to Γ₁ (l) . We construct an explicit set of generators for \TTT (l), and show that \TTT (l) is stable under the action of the Hecke operators. Finally, we relate \TTT (l) to the Hirzebruch elliptic genera that are modular with respect to Γ₁ (l) .

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