# Hurwitz Theory of Elliptic Orbifolds, I

## TL;DR

This paper extends Hurwitz theory for elliptic orbifolds, proving quasi-modularity for cyclic quotients and connecting it to tiling enumeration and moduli space volume calculations.

## Contribution

It generalizes previous quasi-modularity results to cyclic quotients of elliptic curves by orders 3, 4, and 6, and links these to tiling enumeration and volume computations.

## Key findings

- Proves quasi-modularity for cyclic quotients of elliptic curves by orders 3, 4, 6.
- Establishes a method to compute Masur-Veech volumes of moduli spaces.
- Shows volume is polynomial in π.

## Abstract

An elliptic orbifold is the quotient of an elliptic curve by a finite group. Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for the full modular group $SL_2(\mathbb{Z})$. They later generalized this theorem to the enumeration of branched covers of a pillowcase, i.e. the quotient of an elliptic curve by the elliptic involution, proving quasi-modularity for $\Gamma_1(2)$. We generalize their work to the quotient of an elliptic curve by cyclic groups of orders $N=3$, $4$, $6$, proving quasi-modularity for level $\Gamma_1(N)$. One corollary is that certain generating functions of hexagon, square, and triangle tilings of compact surfaces are quasi-modular. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotic behavior as the number of tiles goes to infinity, theoretically giving an algorithm to compute the Masur-Veech volumes of moduli spaces of cubic, quartic, and sextic differentials. We also deduce that the volume is polynomial in $\pi$.