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
. In 2006,
they generalized this theorem to branched covers of the quotient of an elliptic curve by
, proving
quasimodularity for
.
We generalize their work to the quotient of an elliptic curve by
for
,
,
, proving quasimodularity
for
, and extend their
work in the case
.
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
.
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