Euler characteristics of Teichmüller curves in genus two. (English) Zbl 1131.32007
The author obtains a formula for the Euler characteristics of all Teichmüller curves in the muduli space \(\mathcal{M}_2\) of genus two Riemann surfaces that are generated by holomorphic one-forms with a single double zero. Such Teichmüller curves are naturally embedded in Hilbert modular surfaces, and the formula given shows the Euler characteristic of all Teichmüller curve as a constant times the Euler characteristic of the Hilbert curve on which it lies. The result was conjectured by Zagier after numerical evidence. The author obtains several related results, in particular generalizing formulas that were obtained by Eskin, Masur and Schmoll. He also relates his results to some formulas obtained by McMullen, and by Lelièvre-Royer.
The techniques used come from algebraic geometry and are based on a calculatation of the fundamental classes of the Teichmüller curves in certain compactifications of the Hilbert modular surfaces obtained by expressing these fundamental classes as zero loci of certain meromorphic sections of line bundles over Hilbert modular surfaces.
There is a well-known correspondence between closed billard paths on rational-angled Euclidean polygons and closed geodesics on some singular flat surface defined by an Abelian differential on a Riemann surface. The author applies his results to calculate the Siegel-Veech constants for the number of closed billiard paths in certain \(L\)-shaped polygons. Siegel-Veech constants are the coefficients that appear in the formula for the (quadratic) asymptotics of the counting function.
The author also calculates the Lyapunov exponents of the Konstevich-Zorich cocycle for any ergodic \(\mathrm{SL}(2,\mathbb{R})\)-invariant measure on the moduli space \(\mathcal{M}_2\), giving a proof of an unpublished result by Kontsevich.
The paper contains very interesting material and is carefully written. The work is based on the author’s Ph.D. thesis written under McMullen.
The techniques used come from algebraic geometry and are based on a calculatation of the fundamental classes of the Teichmüller curves in certain compactifications of the Hilbert modular surfaces obtained by expressing these fundamental classes as zero loci of certain meromorphic sections of line bundles over Hilbert modular surfaces.
There is a well-known correspondence between closed billard paths on rational-angled Euclidean polygons and closed geodesics on some singular flat surface defined by an Abelian differential on a Riemann surface. The author applies his results to calculate the Siegel-Veech constants for the number of closed billiard paths in certain \(L\)-shaped polygons. Siegel-Veech constants are the coefficients that appear in the formula for the (quadratic) asymptotics of the counting function.
The author also calculates the Lyapunov exponents of the Konstevich-Zorich cocycle for any ergodic \(\mathrm{SL}(2,\mathbb{R})\)-invariant measure on the moduli space \(\mathcal{M}_2\), giving a proof of an unpublished result by Kontsevich.
The paper contains very interesting material and is carefully written. The work is based on the author’s Ph.D. thesis written under McMullen.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 30F30 | Differentials on Riemann surfaces |
| 30F60 | Teichmüller theory for Riemann surfaces |
Keywords:
Teichmüller curve; Teichmüller space; moduli space; Abelian differential; Siegel-Veech constant; Hilbert modular surface; billiards; Lyapunov exponentsReferences:
| [1] | W Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. \((2)\) 105 (1977) 29 · Zbl 0347.32010 · doi:10.2307/1971024 |
| [2] | L V Ahlfors, The complex analytic structure of the space of closed Riemann surfaces., Princeton Univ. Press (1960) 45 · Zbl 0100.28903 |
| [3] | J S Athreya, Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata 119 (2006) 121 · Zbl 1108.32007 · doi:10.1007/s10711-006-9058-z |
| [4] | W L Baily Jr., Satake’s compactification of \(V_n\), Amer. J. Math. 80 (1958) 348 · Zbl 0087.08301 · doi:10.2307/2372789 |
| [5] | W L Baily Jr., On the moduli of Jacobian varieties, Ann. of Math. \((2)\) 71 (1960) 303 · Zbl 0178.55001 · doi:10.2307/1970084 |
| [6] | W L Baily Jr., On the theory of \(\theta \)-functions, the moduli of abelian varieties, and the moduli of curves, Ann. of Math. \((2)\) 75 (1962) 342 · Zbl 0147.39702 · doi:10.2307/1970178 |
| [7] | M Bainbridge, Billiards in L-shaped tables with barriers · Zbl 1213.37062 · doi:10.1007/s00039-010-0065-8 |
| [8] | L Bers, Correction to “Spaces of Riemann surfaces as bounded domains”, Bull. Amer. Math. Soc. 67 (1961) 465 |
| [9] | L Bers, Holomorphic differentials as functions of moduli, Bull. Amer. Math. Soc. 67 (1961) 206 · Zbl 0102.06702 · doi:10.1090/S0002-9904-1961-10569-7 |
| [10] | L Bers, Fiber spaces over Teichmüller spaces, Acta. Math. 130 (1973) 89 · Zbl 0249.32014 · doi:10.1007/BF02392263 |
| [11] | L Bers, On spaces of Riemann surfaces with nodes, Bull. Amer. Math. Soc. 80 (1974) 1219 · Zbl 0294.32017 · doi:10.1090/S0002-9904-1974-13686-4 |
| [12] | L Bers, Finite-dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. \((\)N.S.\()\) 5 (1981) 131 · Zbl 0485.30002 · doi:10.1090/S0273-0979-1981-14933-8 |
| [13] | C Birkenhake, H Lange, Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer (2004) · Zbl 1056.14063 |
| [14] | A I Borevich, I R Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press (1966) · Zbl 0145.04902 |
| [15] | N Bourbaki, Éléments de mathématique. Fasc. XXXV. Livre VI: Intégration. Chapitre IX: Intégration sur les espaces topologiques séparés, Actualités Scientifiques et Industrielles, No. 1343, Hermann (1969) 133 · Zbl 0189.14201 |
| [16] | I Bouw, M Möller, Teichmüller curves, triangle groups, and Lyapunov exponents · Zbl 1203.37049 · doi:10.4007/annals.2010.172.139 |
| [17] | K Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004) 871 · Zbl 1073.37032 · doi:10.1090/S0894-0347-04-00461-8 |
| [18] | W Chen, Y Ruan, Orbifold Gromov-Witten theory, Contemp. Math. 310, Amer. Math. Soc. (2002) 25 · Zbl 1091.53058 |
| [19] | H Cohen, Sums involving the values at negative integers of \(L\)-functions of quadratic characters, Math. Ann. 217 (1975) 271 · Zbl 0316.10015 · doi:10.1007/BF01436180 |
| [20] | P Deligne, D Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969) 75 · Zbl 0181.48803 · doi:10.1007/BF02684599 |
| [21] | A Eskin, H Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems 21 (2001) 443 · Zbl 1096.37501 · doi:10.1017/S0143385701001225 |
| [22] | A Eskin, H Masur, M Schmoll, Billiards in rectangles with barriers, Duke Math. J. 118 (2003) 427 · Zbl 1038.37031 · doi:10.1215/S0012-7094-03-11832-3 |
| [23] | A Eskin, H Masur, A Zorich, Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci. (2003) 61 · Zbl 1037.32013 · doi:10.1007/s10240-003-0015-1 |
| [24] | A Eskin, A Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math. 145 (2001) 59 · Zbl 1019.32014 · doi:10.1007/s002220100142 |
| [25] | A Eskin, A Okounkov, R Pandharipande, The theta characteristic of a branched covering · Zbl 1157.14014 · doi:10.1016/j.aim.2006.08.001 |
| [26] | H Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962) 331 · Zbl 0173.33004 · doi:10.1007/BF01441136 |
| [27] | P Griffiths, J Harris, Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons (1978) · Zbl 0408.14001 |
| [28] | R C Gunning, Lectures on complex analytic varieties: The local parametrization theorem, Mathematical Notes, Princeton University Press (1970) · Zbl 0213.35904 |
| [29] | R C Gunning, H Rossi, Analytic functions of several complex variables, Prentice-Hall (1965) · Zbl 0141.08601 |
| [30] | E Gutkin, C Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000) 191 · Zbl 0965.30019 · doi:10.1215/S0012-7094-00-10321-3 |
| [31] | J Harris, I Morrison, Moduli of curves, Graduate Texts in Mathematics 187, Springer (1998) · Zbl 0913.14005 |
| [32] | R Hartshorne, Residues and duality, Lecture Notes in Mathematics 20, Springer (1966) |
| [33] | R Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer (1977) · Zbl 0367.14001 |
| [34] | F E P Hirzebruch, Hilbert modular surfaces, Enseignement Math. \((2)\) 19 (1973) 183 · Zbl 0285.14007 |
| [35] | P Hubert, S Lelièvre, Prime arithmetic Teichmüller discs in \(\mathcal H(2)\), Israel J. Math. 151 (2006) 281 · Zbl 1138.37016 · doi:10.1007/BF02777365 |
| [36] | K Ireland, M Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics 84, Springer (1990) · Zbl 0712.11001 |
| [37] | E Kani, The number of curves of genus two with elliptic differentials, J. Reine Angew. Math. 485 (1997) 93 · Zbl 0867.11045 · doi:10.1515/crll.1997.485.93 |
| [38] | E Kani, Hurwitz spaces of genus 2 covers of an elliptic curve, Collect. Math. 54 (2003) 1 · Zbl 1077.14529 |
| [39] | R Kenyon, J Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000) 65 · Zbl 0967.37019 · doi:10.1007/s000140050113 |
| [40] | S Kerckhoff, H Masur, J Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. \((2)\) 124 (1986) 293 · Zbl 0637.58010 · doi:10.2307/1971280 |
| [41] | F F Knudsen, D Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”, Math. Scand. 39 (1976) 19 · Zbl 0343.14008 |
| [42] | M Kontsevich, Lyapunov exponents and Hodge theory, Adv. Ser. Math. Phys. 24, World Sci. Publ., River Edge, NJ (1997) 318 · Zbl 1058.37508 |
| [43] | M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631 · Zbl 1087.32010 · doi:10.1007/s00222-003-0303-x |
| [44] | I Kra, Horocyclic coordinates for Riemann surfaces and moduli spaces. I. Teichmüller and Riemann spaces of Kleinian groups, J. Amer. Math. Soc. 3 (1990) 499 · Zbl 0714.30040 · doi:10.2307/1990927 |
| [45] | H B Laufer, Normal two-dimensional singularities, Annals of Mathematics Studies 71, Princeton University Press (1971) · Zbl 0245.32005 |
| [46] | S Lelièvre, Siegel-Veech constants in \(\mathcal H(2)\), Geom. Topol. 10 (2006) 1157 · Zbl 1131.30017 · doi:10.2140/gt.2006.10.1157 |
| [47] | S Lelièvre, E Royer, Orbitwise countings in \(\mathcal H(2)\) and quasimodular forms, Int. Math. Res. Not. (2006) 30 · Zbl 1142.11025 · doi:10.1155/IMRN/2006/42151 |
| [48] | A Marden, Geometric complex coordinates for Teichmüller space, Adv. Ser. Math. Phys. 1, World Sci. Publishing (1987) 341 · Zbl 0678.32014 |
| [49] | H Masur, Interval exchange transformations and measured foliations, Ann. of Math. \((2)\) 115 (1982) 169 · Zbl 0497.28012 · doi:10.2307/1971341 |
| [50] | H Masur, The growth rate of trajectories of a quadratic differential, Ergodic Theory Dynam. Systems 10 (1990) 151 · Zbl 0706.30035 · doi:10.1017/S0143385700005459 |
| [51] | H Masur, S Tabachnikov, Rational billiards and flat structures, North-Holland (2002) 1015 · Zbl 1057.37034 |
| [52] | C T McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003) 857 · Zbl 1030.32012 · doi:10.1090/S0894-0347-03-00432-6 |
| [53] | C T McMullen, Teichmüller geodesics of infinite complexity, Acta Math. 191 (2003) 191 · Zbl 1131.37052 · doi:10.1007/BF02392964 |
| [54] | C T McMullen, Teichmüller curves in genus two: discriminant and spin, Math. Ann. 333 (2005) 87 · Zbl 1086.14024 · doi:10.1007/s00208-005-0666-y |
| [55] | C T McMullen, Teichmüller curves in genus two: the decagon and beyond, J. Reine Angew. Math. 582 (2005) 173 · Zbl 1073.32004 · doi:10.1515/crll.2005.2005.582.173 |
| [56] | C T McMullen, Teichmüller curves in genus two: torsion divisors and ratios of sines, Invent. Math. 165 (2006) 651 · Zbl 1103.14014 · doi:10.1007/s00222-006-0511-2 |
| [57] | C T McMullen, Dynamics of \(\mathrm{SL}_2(\mathbb R)\) over moduli space in genus two, Ann. of Math. \((2)\) 165 (2007) 397 · Zbl 1131.14027 · doi:10.4007/annals.2007.165.397 |
| [58] | C T McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math. 129 (2007) 183 · Zbl 1154.14020 · doi:10.1353/ajm.2007.0002 |
| [59] | T Miyake, Modular forms, Springer (1989) |
| [60] | M Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math. 165 (2006) 633 · Zbl 1111.14019 · doi:10.1007/s00222-006-0510-3 |
| [61] | D Mumford, The topology of normal singularities of an algebraic surface and a criterion for simplicity, Inst. Hautes Études Sci. Publ. Math. (1961) 5 · Zbl 0108.16801 · doi:10.1007/BF02698717 |
| [62] | D Mumford, The structure of the moduli spaces of curves and Abelian varieties, Gauthier-Villars (1971) 457 · Zbl 0222.14023 |
| [63] | D Mumford, Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math. 42 (1977) 239 · Zbl 0365.14012 · doi:10.1007/BF01389790 |
| [64] | D Mumford, Towards an enumerative geometry of the moduli space of curves, Progr. Math. 36, Birkhäuser (1983) 271 · Zbl 0554.14008 |
| [65] | D Mumford, The red book of varieties and schemes, Lecture Notes in Mathematics 1358, Springer (1999) · Zbl 0945.14001 · doi:10.1007/b62130 |
| [66] | Y Namikawa, On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform, Nagoya Math. J. 52 (1973) 197 · Zbl 0271.14014 |
| [67] | M Ratner, Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4 (1994) 236 · Zbl 0801.22008 · doi:10.1007/BF01895839 |
| [68] | I Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956) 359 · Zbl 0074.18103 · doi:10.1073/pnas.42.6.359 |
| [69] | I Satake, On the compactification of the Siegel space, J. Indian Math. Soc. \((\)N.S.\()\) 20 (1956) 259 · Zbl 0072.30002 |
| [70] | M Schmoll, Spaces of elliptic differentials, Contemp. Math. 385, Amer. Math. Soc. (2005) 303 · Zbl 1082.14034 |
| [71] | J P Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955-1956) 1 · Zbl 0075.30401 · doi:10.5802/aif.59 |
| [72] | J P Serre, Groupes algébriques et corps de classes, Publications de l’institut de mathématique de l’université de Nancago, VII. Hermann, Paris (1959) 202 · Zbl 0097.35604 |
| [73] | C L Siegel, The volume of the fundamental domain for some infinite groups, Trans. Amer. Math. Soc. 39 (1936) 209 · Zbl 0013.24901 · doi:10.2307/1989745 |
| [74] | C L Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969) 87 · Zbl 0186.08804 |
| [75] | G van der Geer, Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer (1988) · Zbl 0634.14022 |
| [76] | W A Veech, The Teichmüller geodesic flow, Ann. of Math. \((2)\) 124 (1986) 441 · Zbl 0658.32016 · doi:10.2307/2007091 |
| [77] | W A Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989) 553 · Zbl 0676.32006 · doi:10.1007/BF01388890 |
| [78] | W A Veech, Moduli spaces of quadratic differentials, J. Analyse Math. 55 (1990) 117 · Zbl 0722.30032 · doi:10.1007/BF02789200 |
| [79] | W A Veech, The billiard in a regular polygon, Geom. Funct. Anal. 2 (1992) 341 · Zbl 0760.58036 · doi:10.1007/BF01896876 |
| [80] | W A Veech, Siegel measures, Ann. of Math. \((2)\) 148 (1998) 895 · Zbl 0922.22003 · doi:10.2307/121033 |
| [81] | H Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont. (1972) · Zbl 0265.32008 |
| [82] | S A Wolpert, On obtaining a positive line bundle from the Weil-Petersson class, Amer. J. Math. 107 (1985) · Zbl 0581.14022 · doi:10.2307/2374413 |
| [83] | O Zariski, P Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Princeton, NJ-Toronto-London-New York (1960) · Zbl 0121.27801 |
| [84] | A Zorich, How do the leaves of a closed \(1\)-form wind around a surface?, Amer. Math. Soc. Transl. Ser. 2 197, Amer. Math. Soc. (1999) 135 · Zbl 0976.37012 |
| [85] | A Zorich, Flat surfaces, Springer (2006) 437 · Zbl 1129.32012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
