Duke Mathematical Journal

Arithmetic Veech sublattices of SL(2,Z)

Jordan S. Ellenberg and D. B. McReynolds

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We prove that every algebraic curve X/Q¯ is birational over C to a Teichmüller curve. This result is a corollary of our main theorem, which asserts that most finite-index subgroups of SL(2,Z) are Veech groups.

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Duke Math. J., Volume 161, Number 3 (2012), 415-429.

First available in Project Euclid: 1 February 2012

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Zentralblatt MATH identifier

Primary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)


Ellenberg, Jordan S.; McReynolds, D. B. Arithmetic Veech sublattices of $\operatorname{SL}(2,\mathbf{Z})$. Duke Math. J. 161 (2012), no. 3, 415--429. doi:10.1215/00127094-1507412. https://projecteuclid.org/euclid.dmj/1328105285

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  • [1] M. Asada, The faithfulness of the monodromy representations associated with certain families of algebraic curves, J. Pure Appl. Algebra 159 (2001), 123–147.
  • [2] J. S. Birman and H. M. Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. (2) 97 (1973), 424–439.
  • [3] M. Boggi, The congruence subgroup property for the hyperelliptic modular group: The open surface case, Hiroshima Math. J. 39 (2009), 351–362.
  • [4] I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172 (2010), 139–185.
  • [5] K.-U. Bux, M. Ershov, and A. S. Rapinchuk, The congruence subgroup property for Aut(F2): A group-theoretic proof of Asada’s theorem, Groups Geom. Dyn. 5 (2011), 327–353.
  • [6] S. Diaz, R. Donagi, and D. Harbater, Every curve is a Hurwitz space, Duke Math. J. 59 (1989), 737–746.
  • [7] B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Univ. Press, Princeton, 2011. Zbl pre05960418
  • [8] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J. 103 (2000), 191–213.
  • [9] P. Hubert and S. Lelievre, Noncongruence subgroups in $\mathcal{H}(2)$, Int. Math. Res. Not. IMRN 2005, no. 1, 47–64.
  • [10] P. Hubert, H. Masur, T. Schmidt, and A. Zorich, “Problems on billiards, flat surfaces and translation surfaces” in Problems on Mapping Class Groups and Related Topics, Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, 2006.
  • [11] P. Lochak, On arithmetic curves in the moduli spaces of curves, J. Inst. Math. Jussieu 4 (2005), 443–508.
  • [12] H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J. 53 (1986), 307–314.
  • [13] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), 857–885.
  • [14] C. T. McMullen, Rigidity of Teichmüller curves, Math. Res. Lett. 16 (2009), 647–649.
  • [15] D. B. McReynolds, The congruence subgroup problem for braid groups: Thurston’s proof, preprint, arXiv:0901.4663v2 [math.GT]
  • [16] M. Möller, Teichmüller curves, Galois actions and GT-relations, Math. Nachr. 278 (2005), 1061–1077.
  • [17] M. Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), 327–344.
  • [18] G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experiment. Math. 13 (2004), 459–472.
  • [19] G. Schmithüsen, Veech groups of origamis, Ph.D. dissertation, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2005.
  • [20] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 553–583; Correction, Invent. Math. 103 (1991), 447.