January  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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

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

[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). Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[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. Google Scholar

[12]

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

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

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. Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[5]

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

[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). Google Scholar

[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. Google Scholar

[8]

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

[9]

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

[10]

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

[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. Google Scholar

[12]

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

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[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. Google Scholar

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

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