## Quasimodularity and large genus limits of Siegel-Veech constants

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- by Dawei Chen, Martin Möller and Don Zagier
- J. Amer. Math. Soc.
**31**(2018), 1059-1163 - DOI: https://doi.org/10.1090/jams/900
- 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.

## References

- Enrico Arbarello and Maurizio Cornalba,
*Calculating cohomology groups of moduli spaces of curves via algebraic geometry*, Inst. Hautes Études Sci. Publ. Math.**88**(1998), 97–127 (1999). MR**1733327**, DOI 10.1007/BF02701767 - Jayadev S. Athreya, Alex Eskin, and Anton Zorich,
*Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\Bbb C\rm P^1$*, Ann. Sci. Éc. Norm. Supér. (4)**49**(2016), no. 6, 1311–1386 (English, with English and French summaries). With an appendix by Jon Chaika. MR**3592359**, DOI 10.24033/asens.2310 - Artur Avila, Carlos Matheus, and Jean-Christophe Yoccoz,
*$SL(2,\Bbb {R})$-invariant probability measures on the moduli spaces of translation surfaces are regular*, Geom. Funct. Anal.**23**(2013), no. 6, 1705–1729. MR**3132901**, DOI 10.1007/s00039-013-0244-5 - Roland Bacher and Laurent Manivel,
*Hooks and powers of parts in partitions*, Sém. Lothar. Combin.**47**(2001/02), Art. B47d, 11. MR**1894024** - Matt Bainbridge,
*Billiards in L-shaped tables with barriers*, Geom. Funct. Anal.**20**(2010), no. 2, 299–356. MR**2671280**, DOI 10.1007/s00039-010-0065-8 - Max Bauer and Elise Goujard,
*Geometry of periodic regions on flat surfaces and associated Siegel-Veech constants*, Geom. Dedicata**174**(2015), 203–233. MR**3303050**, DOI 10.1007/s10711-014-0014-z - Spencer Bloch and Andrei Okounkov,
*The character of the infinite wedge representation*, Adv. Math.**149**(2000), no. 1, 1–60. MR**1742353**, DOI 10.1006/aima.1999.1845 - Raoul Bott and Loring W. Tu,
*Differential forms in algebraic topology*, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR**658304**, DOI 10.1007/978-1-4757-3951-0 - Dawei Chen,
*Square-tiled surfaces and rigid curves on moduli spaces*, Adv. Math.**228**(2011), no. 2, 1135–1162. MR**2822219**, DOI 10.1016/j.aim.2011.06.002 - Dawei Chen and Martin Möller,
*Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus*, Geom. Topol.**16**(2012), no. 4, 2427–2479. MR**3033521**, DOI 10.2140/gt.2012.16.2427 - Dawei Chen and Martin Möller,
*Quadratic differentials in low genus: exceptional and non-varying strata*, Ann. Sci. Éc. Norm. Supér. (4)**47**(2014), no. 2, 309–369 (English, with English and French summaries). MR**3215925**, DOI 10.24033/asens.2216 - Robbert Dijkgraaf,
*Mirror symmetry and elliptic curves*, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 149–163. MR**1363055**, DOI 10.1007/978-1-4612-4264-2_{5} - Alex Eskin, Maxim Kontsevich, and Anton Zorich,
*Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow*, Publ. Math. Inst. Hautes Études Sci.**120**(2014), 207–333. MR**3270590**, DOI 10.1007/s10240-013-0060-3 - Alex Eskin and Howard Masur,
*Asymptotic formulas on flat surfaces*, Ergodic Theory Dynam. Systems**21**(2001), no. 2, 443–478. MR**1827113**, DOI 10.1017/S0143385701001225 - Alex Eskin, Howard Masur, and Anton Zorich,
*Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants*, Publ. Math. Inst. Hautes Études Sci.**97**(2003), 61–179. MR**2010740**, DOI 10.1007/s10240-003-0015-1 - A. Eskin and M. Mirzakhani,
*Invariant and stationary measures for the SL(2,R) action on Moduli space*, 2013. Preprint, arXiv: 1302.3320. - Alex Eskin, Maryam Mirzakhani, and Amir Mohammadi,
*Isolation, equidistribution, and orbit closures for the $\textrm {SL}(2,\Bbb R)$ action on moduli space*, Ann. of Math. (2)**182**(2015), no. 2, 673–721. MR**3418528**, DOI 10.4007/annals.2015.182.2.7 - Alex Eskin and Andrei Okounkov,
*Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials*, Invent. Math.**145**(2001), no. 1, 59–103. MR**1839286**, DOI 10.1007/s002220100142 - Alex Eskin, Andrei Okounkov, and Rahul Pandharipande,
*The theta characteristic of a branched covering*, Adv. Math.**217**(2008), no. 3, 873–888. MR**2383889**, DOI 10.1016/j.aim.2006.08.001 - Alex Eskin and Anton Zorich,
*Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera*, Arnold Math. J.**1**(2015), no. 4, 481–488. MR**3434506**, DOI 10.1007/s40598-015-0028-0 - Gavril Farkas and Mihnea Popa,
*Effective divisors on $\overline {\scr M}_g$, curves on $K3$ surfaces, and the slope conjecture*, J. Algebraic Geom.**14**(2005), no. 2, 241–267. MR**2123229**, DOI 10.1090/S1056-3911-04-00392-3 - Simion Filip,
*Splitting mixed Hodge structures over affine invariant manifolds*, Ann. of Math. (2)**183**(2016), no. 2, 681–713. MR**3450485**, DOI 10.4007/annals.2016.183.2.5 - James Lee Hafner and Aleksandar Ivić,
*On sums of Fourier coefficients of cusp forms*, Enseign. Math. (2)**35**(1989), no. 3-4, 375–382. MR**1039952** - J. Harris and I. Morrison,
*Slopes of effective divisors on the moduli space of stable curves*, Invent. Math.**99**(1990), no. 2, 321–355. MR**1031904**, DOI 10.1007/BF01234422 - Joe Harris and Ian Morrison,
*Moduli of curves*, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR**1631825** - Joe Harris and David Mumford,
*On the Kodaira dimension of the moduli space of curves*, Invent. Math.**67**(1982), no. 1, 23–88. With an appendix by William Fulton. MR**664324**, DOI 10.1007/BF01393371 - Masanobu Kaneko and Don Zagier,
*A generalized Jacobi theta function and quasimodular forms*, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 165–172. MR**1363056**, DOI 10.1007/978-1-4612-4264-2_{6} - R. Kaufmann, Yu. Manin, and D. Zagier,
*Higher Weil-Petersson volumes of moduli spaces of stable $n$-pointed curves*, Comm. Math. Phys.**181**(1996), no. 3, 763–787. MR**1414310**, DOI 10.1007/BF02101297 - Serguei Kerov and Grigori Olshanski,
*Polynomial functions on the set of Young diagrams*, C. R. Acad. Sci. Paris Sér. I Math.**319**(1994), no. 2, 121–126 (English, with English and French summaries). MR**1288389** - M. Kontsevich,
*Lyapunov exponents and Hodge theory*, The mathematical beauty of physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, World Sci. Publ., River Edge, NJ, 1997, pp. 318–332. MR**1490861** - Maxim Kontsevich and Anton Zorich,
*Connected components of the moduli spaces of Abelian differentials with prescribed singularities*, Invent. Math.**153**(2003), no. 3, 631–678. MR**2000471**, DOI 10.1007/s00222-003-0303-x - Sergei K. Lando and Alexander K. Zvonkin,
*Graphs on surfaces and their applications*, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004. With an appendix by Don B. Zagier; Low-Dimensional Topology, II. MR**2036721**, DOI 10.1007/978-3-540-38361-1 - Michel Lassalle,
*An explicit formula for the characters of the symmetric group*, Math. Ann.**340**(2008), no. 2, 383–405. MR**2368985**, DOI 10.1007/s00208-007-0156-5 - Adam Logan,
*The Kodaira dimension of moduli spaces of curves with marked points*, Amer. J. Math.**125**(2003), no. 1, 105–138. MR**1953519**, DOI 10.1353/ajm.2003.0005 - Howard Masur,
*Interval exchange transformations and measured foliations*, Ann. of Math. (2)**115**(1982), no. 1, 169–200. MR**644018**, DOI 10.2307/1971341 - Howard Masur,
*The growth rate of trajectories of a quadratic differential*, Ergodic Theory Dynam. Systems**10**(1990), no. 1, 151–176. MR**1053805**, DOI 10.1017/S0143385700005459 - Peter McCullagh,
*Cumulants and partition lattices*, Selected works of Terry Speed, Sel. Works Probab. Stat., Springer, New York, 2012, pp. 277–282. MR**2933865**, DOI 10.1007/978-1-4614-1347-9_{6} - Maryam Mirzakhani and Peter Zograf,
*Towards large genus asymptotics of intersection numbers on moduli spaces of curves*, Geom. Funct. Anal.**25**(2015), no. 4, 1258–1289. MR**3385633**, DOI 10.1007/s00039-015-0336-5 - Martin Möller,
*Teichmüller curves, mainly from the viewpoint of algebraic geometry*, Moduli spaces of Riemann surfaces, IAS/Park City Math. Ser., vol. 20, Amer. Math. Soc., Providence, RI, 2013, pp. 267–318. MR**3114688**, DOI 10.1090/pcms/020/09 - A. Okun′kov and G. Ol′shanskiĭ,
*Shifted Schur functions*, Algebra i Analiz**9**(1997), no. 2, 73–146 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**9**(1998), no. 2, 239–300. MR**1468548** - Gian-Carlo Rota,
*On the foundations of combinatorial theory. I. Theory of Möbius functions*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**2**(1964), 340–368 (1964). MR**174487**, DOI 10.1007/BF00531932 - T. P. Speed,
*Cumulants and partition lattices*, Austral. J. Statist.**25**(1983), no. 2, 378–388. MR**725217**, DOI 10.1111/j.1467-842X.1983.tb00391.x - William A. Veech,
*Gauss measures for transformations on the space of interval exchange maps*, Ann. of Math. (2)**115**(1982), no. 1, 201–242. MR**644019**, DOI 10.2307/1971391 - William A. Veech,
*Siegel measures*, Ann. of Math. (2)**148**(1998), no. 3, 895–944. MR**1670061**, DOI 10.2307/121033 - Ya. B. Vorobets,
*Ergodicity of billiards in polygons*, Mat. Sb.**188**(1997), no. 3, 65–112 (Russian, with Russian summary); English transl., Sb. Math.**188**(1997), no. 3, 389–434. MR**1462024**, DOI 10.1070/SM1997v188n03ABEH000211 - Edward Witten,
*Two-dimensional gravity and intersection theory on moduli space*, Surveys in differential geometry (Cambridge, MA, 1990) Lehigh Univ., Bethlehem, PA, 1991, pp. 243–310. MR**1144529** - Fei Yu and Kang Zuo,
*Weierstrass filtration on Teichmüller curves and Lyapunov exponents*, J. Mod. Dyn.**7**(2013), no. 2, 209–237. MR**3106711**, DOI 10.3934/jmd.2013.7.209 - Don Zagier,
*Periods of modular forms and Jacobi theta functions*, Invent. Math.**104**(1991), no. 3, 449–465. MR**1106744**, DOI 10.1007/BF01245085 - Don Zagier,
*Elliptic modular forms and their applications*, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 1–103. MR**2409678**, DOI 10.1007/978-3-540-74119-0_{1} - Don Zagier,
*Partitions, quasimodular forms, and the Bloch-Okounkov theorem*, Ramanujan J.**41**(2016), no. 1-3, 345–368. MR**3574638**, DOI 10.1007/s11139-015-9730-8 - Anton Zorich,
*Square tiled surfaces and Teichmüller volumes of the moduli spaces of abelian differentials*, Rigidity in dynamics and geometry (Cambridge, 2000) Springer, Berlin, 2002, pp. 459–471. MR**1919417** - A. Zorich,
*Flat surfaces*, Frontiers in number theory, physics and geometry. volume 1: On random matrices, zeta functions and dynamical systems, 2006, pp. 439–586.

## Bibliographic Information

**Dawei Chen**- Affiliation: Department of Mathematics, Maloney 549, Boston College, Chestnut Hill, Massachusetts 02467
- MR Author ID: 848983
- Email: dawei.chen@bc.edu
**Martin Möller**- Affiliation: Goethe-Universitat Frankfurt, FB 12 Mathematik, Robert-Mayer-Str 6-8, 53757 Frankfurt, Germany
- Email: moeller@math.uni-frankfurt.de
**Don Zagier**- Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 186205
- Email: don.zagier@mpim-bonn.mpg.de
- 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. - © Copyright 2018 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**31**(2018), 1059-1163 - MSC (2010): Primary 32G15; Secondary 05A17, 11F23, 37A25, 57M12
- DOI: https://doi.org/10.1090/jams/900
- MathSciNet review: 3836563