2011, 5(1): 1-32. doi: 10.3934/jmd.2011.5.1

Shimura and Teichmüller curves

1. 

Institut für Mathematik, Johann Wolfgang Goethe-Universität Frankfurt, Robert-Mayer-Str. 6–8, 60325 Frankfurt am Main, Germany

Received  July 2009 Revised  January 2011 Published  April 2011

We classify curves in the moduli space of curves $M_g$ that are both Shimura and Teichmüller curves: for both $g=3$ and $g=4$ there exists precisely one such curve, for $g=2$ and $g \geq 6$ there are no such curves.
   We start with a Hodge-theoretic description of Shimura curves and of Teichmüller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces.
   Finally, we translate our main result into a classification of Teichmüller curves with totally degenerate Lyapunov spectrum.
Citation: Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1
References:
[1]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1.

[2]

A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant 4 fibres singulières,, C. R. Acad. Sc. Paris 294, (1982), 657.

[3]

C. Birkenhake and H. Lange, "Complex Abelian Varieties," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, 302. Springer-Verlag, (2004).

[4]

A. Beauville, L'inégalité $p_g \geq 2q-4$ pour les surfaces de type général,, (French) [Numerical inequalities for surfaces of general type] With an appendix by A. Beauville, 110 (1982), 343.

[5]

F. Beukers and G. Heckman, Monodromy for the hypergeometric function $_nF_{n-1}$,, Invent.\ Math., 95 (1989), 325. doi: 10.1007/BF01393900.

[6]

S. Bosch, W. Lütkebohmert and M. Raynaud, "Néron Models,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1990).

[7]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, Ann. Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139.

[8]

B. Conrad and W. Stein, Component groups of purely toric quotients,, Math. Res. Letters, 8 (2001), 745.

[9]

P. Deligne, Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques,, (French) Automorphic forms, (1977), 247.

[10]

T. Fischbacher, Introducing LambdaTensor1.0,, available on \arXiv{hep-th/0208218}, (2002).

[11]

G. Forni, Deviation of ergodic measures for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150.

[12]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, Handbook of Dynamical systems, 1B (2006), 549.

[13]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with totally degenerate Kontsevich-Zorich spectrum,, preprint \arXiv{0810.0023}, (2008).

[14]

J. Guardia, A family of arithmetic surfaces of genus 3,, Pacific J. of Math., 212 (2003), 71. doi: 10.2140/pjm.2003.212.71.

[15]

E. Gutkin and C. Judge, Affine mappings of translation surfaces,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3.

[16]

J. Harris and I. Morrison, "Moduli of Curves,", Graduate Texts in Mathematics, (1998).

[17]

F. Herrlich, Teichmüller curves defined by characteristic origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 133.

[18]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve,, Math. Nachr., 1 (2008), 219. doi: 10.1002/mana.200510597.

[19]

J. de Jong and R. Noot, Jacobians with complex multiplication,, Arithmetic algebraic geometry (Texel, (1989), 177.

[20]

K. Kodaira, On compact analytic surfaces III,, Annals of Math., 78 (1963), 1. doi: 10.2307/1970500.

[21]

J. Kollár, Subadditivity of the Kodaira dimension: Fibers of general type,, Algebraic geometry, (1985), 361.

[22]

I. Kra, The Carathéodory metric on abelian Teichmüller disks,, J. Analyse Math., 40 (1981), 129.

[23]

A. Kuribayashi and K. Komiya, On Weierstrass points and automorphisms of curves of genus three,, Algebraic geometry (Proc. Summer Meeting, (1978), 253.

[24]

H. Masur, On a class of geodesics in Teichmüller space,, Annals of Math. (2), 102 (1975), 205. doi: 10.2307/1971031.

[25]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in: Handbook of dynamical systems, 1A (2002), 1015.

[26]

C. McMullen, Billiards and Teichmüller curves on Hilbert modular sufaces,, J. Amer. Math. Soc., 16 (2003), 857. doi: 10.1090/S0894-0347-03-00432-6.

[27]

M. Möller, Maximally irregularly fibred surfaces of general type,, Manusc. Math., 116 (2005), 71. doi: 10.1007/s00229-004-0517-2.

[28]

M. Möller, Variations of Hodge structures of Teichmüller curves,, J. Amer. Math. Soc., 19 (2006), 327. doi: 10.1090/S0894-0347-05-00512-6.

[29]

M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teich-müller curve,, Invent. Math., 165 (2006), 633. doi: 10.1007/s00222-006-0510-3.

[30]

B. Moonen, Linearity properties of Shimura varieties I.,, J. Alg. Geom., 7 (1998), 539.

[31]

D. Mumford, A note of Shimura's paper: Discontinuous groups and Abelian varieties,, Math. Ann., 181 (1969), 345. doi: 10.1007/BF01350672.

[32]

F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves,, Journées de Géometrie Algébrique d'Angers, (1979), 157.

[33]

I. Satake, "Algebraic Structures of Symmetric Domains,", Kanô Memorial Lectures, (1980).

[34]

G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functios,", Kanô Memorial Lectures, (1971).

[35]

S.-L. Tan, Y. Tu and A. Zamora, On complex surfaces with $5$ or $6$ semistable singular fibres over $\mathbbP^1$,, Math. Z., 249 (2005), 427. doi: 10.1007/s00209-004-0706-4.

[36]

W. Veech, The Teichmüller geodesic flow,, Ann. Math. (2), 124 (1986), 441. doi: 10.2307/2007091.

[37]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 533. doi: 10.1007/BF01388890.

[38]

E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties,, J. of Diff. Geometry, 66 (2004), 233.

[39]

E. Viehweg and K. Zuo, Numerical bounds for families of curves or of certain higher-dimensional manifolds,, J. Alg. Geom., 15 (2006), 771.

[40]

J. Wolfart, Werte hypergeometrischer Funktionen,, (German) [Values of hypergeometric functions], 92 (1988), 187. doi: 10.1007/BF01393999.

[41]

G. Xiao, "Surfaces Fibrées en Courbes de Genre Deux,", (French) [Surfaces fibered by curves of genus two] Lecture Notes in Mathematics, (1137).

[42]

A. Zorich, Flat surfaces,, Frontiers in number theory, (2006), 437.

show all references

References:
[1]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1.

[2]

A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant 4 fibres singulières,, C. R. Acad. Sc. Paris 294, (1982), 657.

[3]

C. Birkenhake and H. Lange, "Complex Abelian Varieties," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],, 302. Springer-Verlag, (2004).

[4]

A. Beauville, L'inégalité $p_g \geq 2q-4$ pour les surfaces de type général,, (French) [Numerical inequalities for surfaces of general type] With an appendix by A. Beauville, 110 (1982), 343.

[5]

F. Beukers and G. Heckman, Monodromy for the hypergeometric function $_nF_{n-1}$,, Invent.\ Math., 95 (1989), 325. doi: 10.1007/BF01393900.

[6]

S. Bosch, W. Lütkebohmert and M. Raynaud, "Néron Models,", Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1990).

[7]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents,, Ann. Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139.

[8]

B. Conrad and W. Stein, Component groups of purely toric quotients,, Math. Res. Letters, 8 (2001), 745.

[9]

P. Deligne, Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques,, (French) Automorphic forms, (1977), 247.

[10]

T. Fischbacher, Introducing LambdaTensor1.0,, available on \arXiv{hep-th/0208218}, (2002).

[11]

G. Forni, Deviation of ergodic measures for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150.

[12]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, Handbook of Dynamical systems, 1B (2006), 549.

[13]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with totally degenerate Kontsevich-Zorich spectrum,, preprint \arXiv{0810.0023}, (2008).

[14]

J. Guardia, A family of arithmetic surfaces of genus 3,, Pacific J. of Math., 212 (2003), 71. doi: 10.2140/pjm.2003.212.71.

[15]

E. Gutkin and C. Judge, Affine mappings of translation surfaces,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3.

[16]

J. Harris and I. Morrison, "Moduli of Curves,", Graduate Texts in Mathematics, (1998).

[17]

F. Herrlich, Teichmüller curves defined by characteristic origamis,, The geometry of Riemann surfaces and abelian varieties, (2006), 133.

[18]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve,, Math. Nachr., 1 (2008), 219. doi: 10.1002/mana.200510597.

[19]

J. de Jong and R. Noot, Jacobians with complex multiplication,, Arithmetic algebraic geometry (Texel, (1989), 177.

[20]

K. Kodaira, On compact analytic surfaces III,, Annals of Math., 78 (1963), 1. doi: 10.2307/1970500.

[21]

J. Kollár, Subadditivity of the Kodaira dimension: Fibers of general type,, Algebraic geometry, (1985), 361.

[22]

I. Kra, The Carathéodory metric on abelian Teichmüller disks,, J. Analyse Math., 40 (1981), 129.

[23]

A. Kuribayashi and K. Komiya, On Weierstrass points and automorphisms of curves of genus three,, Algebraic geometry (Proc. Summer Meeting, (1978), 253.

[24]

H. Masur, On a class of geodesics in Teichmüller space,, Annals of Math. (2), 102 (1975), 205. doi: 10.2307/1971031.

[25]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in: Handbook of dynamical systems, 1A (2002), 1015.

[26]

C. McMullen, Billiards and Teichmüller curves on Hilbert modular sufaces,, J. Amer. Math. Soc., 16 (2003), 857. doi: 10.1090/S0894-0347-03-00432-6.

[27]

M. Möller, Maximally irregularly fibred surfaces of general type,, Manusc. Math., 116 (2005), 71. doi: 10.1007/s00229-004-0517-2.

[28]

M. Möller, Variations of Hodge structures of Teichmüller curves,, J. Amer. Math. Soc., 19 (2006), 327. doi: 10.1090/S0894-0347-05-00512-6.

[29]

M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teich-müller curve,, Invent. Math., 165 (2006), 633. doi: 10.1007/s00222-006-0510-3.

[30]

B. Moonen, Linearity properties of Shimura varieties I.,, J. Alg. Geom., 7 (1998), 539.

[31]

D. Mumford, A note of Shimura's paper: Discontinuous groups and Abelian varieties,, Math. Ann., 181 (1969), 345. doi: 10.1007/BF01350672.

[32]

F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves,, Journées de Géometrie Algébrique d'Angers, (1979), 157.

[33]

I. Satake, "Algebraic Structures of Symmetric Domains,", Kanô Memorial Lectures, (1980).

[34]

G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functios,", Kanô Memorial Lectures, (1971).

[35]

S.-L. Tan, Y. Tu and A. Zamora, On complex surfaces with $5$ or $6$ semistable singular fibres over $\mathbbP^1$,, Math. Z., 249 (2005), 427. doi: 10.1007/s00209-004-0706-4.

[36]

W. Veech, The Teichmüller geodesic flow,, Ann. Math. (2), 124 (1986), 441. doi: 10.2307/2007091.

[37]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 533. doi: 10.1007/BF01388890.

[38]

E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties,, J. of Diff. Geometry, 66 (2004), 233.

[39]

E. Viehweg and K. Zuo, Numerical bounds for families of curves or of certain higher-dimensional manifolds,, J. Alg. Geom., 15 (2006), 771.

[40]

J. Wolfart, Werte hypergeometrischer Funktionen,, (German) [Values of hypergeometric functions], 92 (1988), 187. doi: 10.1007/BF01393999.

[41]

G. Xiao, "Surfaces Fibrées en Courbes de Genre Deux,", (French) [Surfaces fibered by curves of genus two] Lecture Notes in Mathematics, (1137).

[42]

A. Zorich, Flat surfaces,, Frontiers in number theory, (2006), 437.

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