[Paper Review] Quasimodularity and large genus limits of Siegel-Veech constants

Dawei Chen, Martin Moeller+1 more|arXiv (Cornell University)|Jun 13, 2016

Mathematical Dynamics and Fractals34 references8 citations

TL;DR

This paper establishes the quasimodularity of Siegel-Veech constants in the principal stratum of abelian differentials by connecting weighted torus coverings to quasimodular forms via Bloch-Okounkov q-brackets. It proves the Eskin-Zorich conjecture on large genus asymptotics of Masur-Veech volumes and Siegel-Veech constants using generating functions built from Eisenstein series and cumulant expansions.

ABSTRACT

Quasimodular forms were first studied 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 introduce 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 quasimodularity of the generating functions arising from weighted counting of torus coverings. The starting point is the theorem of Bloch and Okounkov saying that q-brackets of shifted symmetric functions are quasimodular forms. In Part II we give an expression for their growth polynomials in terms of Gaussian integrals and 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 prove a formula for the q-bracket of the product of such a hook-length moment with an arbitrary shifted symmetric polynomial. This formula proves quasimodularity also for the (-2)-nd hook-length moments by extrapolation, and 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. To apply these exact formulas to the Eskin-Zorich conjectures we provide a general framework for computing the asymptotics of rapidly divergent power series.

Motivation & Objective

• To establish the quasimodularity of Siegel-Veech constants in the principal stratum of abelian differentials through weighted counting of torus coverings.
• To resolve the Eskin-Zorich conjecture on the large genus asymptotics of Masur-Veech volumes and Siegel-Veech constants.
• To provide explicit generating functions for Siegel-Veech constants and volumes using intersection theory on Hurwitz spaces and modular forms.
• To develop a framework for asymptotics of rapidly divergent power series to analyze large genus limits.
• To demonstrate that q-brackets of shifted symmetric polynomials and hook-length moments yield quasimodular forms, extending to negative indices via extrapolation.

Proposed method

• Uses the Bloch-Okounkov theorem to express q-brackets of shifted symmetric functions as quasimodular forms.
• Derives closed formulas for growth polynomials of q-brackets using Gaussian integrals and cumulant generating series.
• Proves that even hook-length moments of partitions are shifted symmetric polynomials and computes their q-brackets as linear combinations of derivatives of Eisenstein series.
• Applies extrapolation to define the (−2)-nd hook-length moment, yielding quasimodularity for Siegel-Veech weighted counting functions.
• Constructs generating functions for Siegel-Veech constants and volumes in the principal stratum via inversion of power series with Bernoulli number coefficients.
• Applies a general asymptotic framework for rapidly divergent series to extract large genus expansions from the generating functions.

Experimental results

Research questions

• RQ1 Are Siegel-Veech constants in the principal stratum of abelian differentials quasimodular forms?
• RQ2 What is the large genus asymptotic behavior of Masur-Veech volumes and Siegel-Veech constants in the principal stratum?
• RQ3 Can the q-bracket of a product of a shifted symmetric polynomial and a hook-length moment be expressed as a linear combination of derivatives of Eisenstein series?
• RQ4 How can the generating functions for Siegel-Veech constants be expressed in terms of modular forms and power series inversion?
• RQ5 What is the asymptotic expansion of rapidly divergent series arising from modular forms in this context?

Key findings

• The generating series for Siegel-Veech constants in the principal stratum are quasimodular forms, proven via q-bracket identities and extrapolation to negative indices.
• The leading asymptotic term of the Siegel-Veech constant $ c_p^0({ m Tr}^n) $ for $ p > 0 $ is given by a sum over combinatorial terms involving Bernoulli numbers and rational intersection numbers.
• For $ p = -1 $, the leading term arises from an additional contribution to the $ X^{k+1} $-term, with a distinct asymptotic structure due to the $ G_2^{(k-1)} $-term in the generating function.
• The asymptotic expansion of the Masur-Veech volume and Siegel-Veech constant ratio is derived from the leading coefficients of modular forms via the $ { m Ev} $-map, confirming the Eskin-Zorich conjecture.
• The generating function for the Siegel-Veech constant $ C_p^0(u) $ is expressed as a double sum involving $ G_{p+i+1}^{(i+k)} $ and cumulant generating functions of $ p_2 $-operators.
• The asymptotics of the series are computed using a general framework for rapidly divergent series, with the leading term determined by the highest-degree contribution in the expansion of the inverse power series with Bernoulli number coefficients.