#### Volume 25, issue 1 (2021)

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Hurwitz theory of elliptic orbifolds, I

### Philip Engel

Geometry & Topology 25 (2021) 229–274
##### 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 ${SL}_{2}\left(ℤ\right)$. In 2006, they generalized this theorem to branched covers of the quotient of an elliptic curve by $±1$, proving quasimodularity for ${\Gamma }_{\phantom{\rule{-0.17em}{0ex}}0}\left(2\right)$. We generalize their work to the quotient of an elliptic curve by $⟨{\zeta }_{N}⟩$ for $N=3$, $4$, $6$, proving quasimodularity for $\Gamma \left(N\right)$, and extend their work in the case $N=2$.

It follows that certain generating functions of hexagon, square and triangle tilings of compact surfaces are quasimodular forms. These tilings enumerate lattice points in moduli spaces of flat surfaces. We analyze the asymptotics as the number of tiles goes to infinity, providing an algorithm to compute the Masur–Veech volumes of strata of cubic, quartic, and sextic differentials. We conclude a generalization of the Kontsevich–Zorich conjecture: these volumes are polynomial in $\pi$.

##### Keywords
Hurwitz theory, elliptic orbifold, higher differentials, Masur–Veech volume, tilings, enumeration
##### Mathematical Subject Classification 2010
Primary: 05B45, 14H30, 14H52, 14K25, 17B69