2018/09/20 by Philip Engel, Engel, Philip · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1809.07434
openalex publication_date 2018/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An elliptic orbifold is the quotient of an elliptic curve by a finite group. In 2001, Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for SL2(ℤ). In 2006, they generalized this theorem to the enumeration of branched covers of the quotient of an elliptic curve by ± 1, proving quasi-modularity for Γ1(2). In 2017, the author generalized their work to the quotient of an elliptic curve by ⟨ ζN⟩ for N=3, 4, 6, proving quasimodularity for Γ1(N). In these works, both Eskin-Okounkov and the author had to assume that there was at least one orbifold point of order N over which there was no ramification. Here we remove that assumption, with the caveat that the generating functions are only quasimodular for Γ(N). We deduce the following corollary: Let h6(κ,q) be the generating function whose qn coefficient is the number of surface triangulations with 2n triangles, such that the set of non-zero curvatures is κi. Here the curvature of a vertex is six minus its valence. Then under the substitution q=e2πi τ, the function h6(κ,q) is a quasimodular form for Γ1(6) with weight bounded in terms of κ. This statement in turn implies that the Masur-Veech volume of any stratum of sextic differentials is polynomial in π.