Hurwitz theory of elliptic orbifolds, I

Engel, Philip

doi:10.2140/gt.2021.25.229

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author Index

To Appear

Other MSP Journals

This article is available for purchase or by subscription. See below.

Hurwitz theory of elliptic orbifolds, I

Philip Engel

Geometry & Topology 25 (2021) 229–274

DOI: 10.2140/gt.2021.25.229

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 ${SL}_{2} (ℤ)$ . In 2006, they generalized this theorem to branched covers of the quotient of an elliptic curve by $\pm 1$ , proving quasimodularity for $Γ_{0} (2)$ . We generalize their work to the quotient of an elliptic curve by $⟨ ζ_{N} ⟩$ 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 $π$ .

PDF Access Denied

We have not been able to recognize your IP address 52.14.126.74 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords

Hurwitz theory, elliptic orbifold, higher differentials, Masur–Veech volume, tilings, enumeration

Mathematical Subject Classification 2010

Primary: 05B45, 14H30, 14H52, 14K25, 17B69

References

Publication

Received: 25 October 2018

Revised: 17 January 2020

Accepted: 21 February 2020

Published: 2 March 2021

Proposed: Jim Bryan

Seconded: Anna Wienhard, Ian Agol

Authors

Philip Engel
Department of Mathematics Harvard University Cambridge, MA United States	Department of Mathematics University of Georgia Athens, GA United States

Volume 25, issue 1 (2021)