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Quasimodularity and large genus limits of Siegel-Veech constants

Authors: Dawei Chen, Martin Möller and Don Zagier
Journal: J. Amer. Math. Soc.
MSC (2010): Primary 32G15; Secondary 05A17, 11F23, 37A25, 57M12
Published electronically: April 30, 2018
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Abstract: Quasimodular forms were first studied systematically in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veech constants.

In Part I we connect the geometric definition of Siegel-Veech constants both with a combinatorial counting problem and with intersection numbers on Hurwitz spaces. We also introduce certain modified Siegel-Veech weights whose generating functions will later be shown to be quasimodular.

Parts II and III are devoted to the study of the (quasi) modular properties of the generating functions arising from weighted counting of torus coverings. These two parts contain little geometry and can be read independently of the rest of the paper. The starting point is the theorem of Bloch and Okounkov saying that certain weighted averages, called $ q$-brackets, of shifted symmetric functions on partitions are quasimodular forms. In Part II we give an expression for the growth polynomials (a certain polynomial invariant of quasimodular forms) of these $ q$-brackets in terms of Gaussian integrals, and we use this to obtain a closed formula for the generating series of cumulants that is the basis for studying large genus asymptotics. In Part III we show that the even hook-length moments of partitions are shifted symmetric polynomials, and we prove a surprising formula for the $ q$-bracket of the product of such a hook-length moment with an arbitrary shifted symmetric polynomial as a linear combination of derivatives of Eisenstein series. This formula gives a quasimodularity statement also for the $ (-2)$-nd hook-length moments by an appropriate extrapolation, and this in turn implies the quasimodularity of the Siegel-Veech weighted counting functions.

Finally, in Part IV these results are used to give explicit generating functions for the volumes and Siegel-Veech constants in the case of the principal stratum of abelian differentials. The generating functions have an amusing form in terms of the inversion of a power series (with multiples of Bernoulli numbers as coefficients) that gives the asymptotic expansion of a Hurwitz zeta function. To apply these exact formulas to the Eskin-Zorich conjectures on large genus asymptotics (both for the volume and the Siegel-Veech constant) we provide in a separate appendix a general framework for computing the asymptotics of rapidly divergent power series.

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Additional Information

Dawei Chen
Affiliation: Department of Mathematics, Maloney 549, Boston College, Chestnut Hill, Massachusetts 02467

Martin Möller
Affiliation: Goethe-Universitat Frankfurt, FB 12 Mathematik, Robert-Mayer-Str 6-8, 53757 Frankfurt, Germany

Don Zagier
Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany

Received by editor(s): September 26, 2016
Received by editor(s) in revised form: February 7, 2018
Published electronically: April 30, 2018
Additional Notes: The first author was partially supported by NSF under the grant 1200329 and the CAREER award 1350396.
The second author was partially supported by the ERC starting grant 257137 “Flat surfaces” and the DFG-project MO 1884/1-1. He would also like to thank the Max Planck Institute for Mathematics in Bonn, where much of this work was done.
Article copyright: © Copyright 2018 American Mathematical Society

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