Volume 25, issue 1 (2021)

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Hurwitz theory of elliptic orbifolds, I

Philip Engel

Geometry & Topology 25 (2021) 229–274

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(). In 2006, they generalized this theorem to branched covers of the quotient of an elliptic curve by ± 1, proving quasimodularity for Γ0(2). We generalize their work to the quotient of an elliptic curve by ζN for N = 3, 4, 6, proving quasimodularity for Γ(N), 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 π.

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Hurwitz theory, elliptic orbifold, higher differentials, Masur–Veech volume, tilings, enumeration
Mathematical Subject Classification 2010
Primary: 05B45, 14H30, 14H52, 14K25, 17B69
Received: 25 October 2018
Revised: 17 January 2020
Accepted: 21 February 2020
Published: 2 March 2021
Proposed: Jim Bryan
Seconded: Anna Wienhard, Ian Agol
Philip Engel
Department of Mathematics
Harvard University
Cambridge, MA
United States
Department of Mathematics
University of Georgia
Athens, GA
United States