Hurwitz Theory of Elliptic Orbifolds, II

Abstract

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 $SL_2(\mathbb{Z}).$ In 2006, they generalized this theorem to the enumeration of branched covers of the quotient of an elliptic curve by $\pm 1$, proving quasi-modularity for $\Gamma_1(2)$. In 2017, the author generalized their work to the quotient of an elliptic curve by $\langle \zeta_N\rangle$ for $N=3, 4, 6$, proving quasimodularity for $\Gamma_1(N)$. 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 $\Gamma(N)$. We deduce the following corollary: Let $h_6(\vec{\kappa},q)$ be the generating function whose $q^n$ coefficient is the number of surface triangulations with $2n$ triangles, such that the set of non-zero curvatures is $\kappa_i$. Here the curvature of a vertex is six minus its valence. Then under the substitution $q=e^{2\pi i \tau}$, the function $h_6(\vec{\kappa},q)$ is a quasimodular form for $\Gamma_1(6)$ with weight bounded in terms of $\vec{\kappa}$. This statement in turn implies that the Masur-Veech volume of any stratum of sextic differentials is polynomial in $\pi$.