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  2017, 11: 17-41. doi: 10.3934/jmd.2017002

The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve

Matteo Costantini and  André Kappes

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

Received  February 22, 2016 Revised  August 15, 2016 Published  December 2016

We compute the algebraic equation of the universal family over the Kenyon-Smillie (2, 3, 4)-Teichmüller curve, and we prove that the equation is correct in two different ways. Firstly, we prove it in a constructive way via linear conditions imposed by three special points of the Teichmüller curve. Secondly, we verify that the equation is correct by computing its associated Picard-Fuchs equation. We also notice that each point of the Teichmüller curve has a hyperflex and we see that the torsion map is a central projection from this point.

Keywords: Teichmüller curve, Veech group is triangle group, universal family of curves, Picard-Fuchs equation.
Mathematics Subject Classification: Primary: 14H10, 32G15, 14H30.
Citation: Matteo Costantini, André Kappes. The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve. Journal of Modern Dynamics, 2017, 11: 17-41. doi: 10.3934/jmd.2017002
References:
[1]

I. V. Artamkin, Canonical mappings of punctured curves with the simplest singularities, Mat. Sb., 195 (2004), 3–32; translation in Sb. Math., 195 (2004), 615–642. doi: 10.1070/SM2004v195n05ABEH000818.  Google Scholar

[2]

M. Bainbridge, P. Habegger and M. Möller, Teichmüller curves in genus three and just likely intersections in $ G_{m}^{n}\times G_{a}^{n} $, arXiv: 1410.6835, 2014. Google Scholar

[3]

I. Bouw and M. Möller, Differential equations associated with nonarithmetic Fuchsian groups, J. Lond. Math. Soc.(2), 81 (2010), 65-90.  doi: 10.1112/jlms/jdp059.  Google Scholar

[4]

______, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185.  doi: 10.4007/annals.2010.172.139.  Google Scholar

[5]

M. Bainbridge and M. Möller, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3, Acta Math., 208 (2012), 1-92.  doi: 10.1007/s11511-012-0074-6.  Google Scholar

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I. Bouw, The p-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322.  doi: 10.1023/A:1017513122376.  Google Scholar

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K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908.  doi: 10.1090/S0894-0347-04-00461-8.  Google Scholar

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F. Catanese, M. Franciosi, K. Hulek and M. Reid, Embeddings of curves and surfaces, Nagoya Math. J., 154 (1999), 185-220.  doi: 10.1017/S0027763000025381.  Google Scholar

[9]

D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/surv/068.  Google Scholar

[10]

F. Catanese and R. Pignatelli, Pignatelli R., Fibrations of low genus. I, Ann. Sci. école Norm. Sup. (4), 39 (2006), 1011-1049.  doi: 10.1016/j.ansens.2006.10.001.  Google Scholar

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A. Kuribayashi and K. Komiya, On Weierstrass points of non-hyperelliptic compact Riemann surfaces of genus three, Hiroshima Math. J., 7 (1977), 743-768.   Google Scholar

[12]

A. Kumar and R. Mukamel, Algebraic models and arithmetic geometry of Teichmüller curves in genus two, Int. Math. Res. Notices, 2016. doi: 10.1093/imrn/rnw193.  Google Scholar

[13]

R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108.  doi: 10.1007/s000140050113.  Google Scholar

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C. J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number, Geom. Topol., 8 (2004), 1301-1359.  doi: 10.2140/gt.2004.8.1301.  Google Scholar

[15]

C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885.  doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar

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C. T. McMullen, R. E. Mukamel and A. Wright, Cubic curves and totally geodesic subvarieties of moduli space, preprint, 2016. Available from: http://math.harvard.edu/~ctm/papers/home/text/papers/gothic/gothic.pdf. Google Scholar

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M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., 165 (2006), 633-649.  doi: 10.1007/s00222-006-0510-3.  Google Scholar

[19]

______, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc., 19 (2006), 327-344.  doi: 10.1090/S0894-0347-05-00512-6.  Google Scholar

[20]

______, Teichmüller curves, mainly from the viewpoint of algebraic geometry, in Moduli Spaces of Riemann Surfaces, IAS/Park City Math. Ser., 20, Amer. Math. Soc., Providence, RI, (2013), 267-318.   Google Scholar

[21]

PARI Group, Bordeaux, PARI/GP version 2. 3. 5. Available from: http://pari.math.u-bordeaux.fr/. Google Scholar

[22]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.  Google Scholar

[23]

Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Uspekhi Mat. Nauk, 51 (1996), 3-42.  doi: 10.1070/RM1996v051n05ABEH002993.  Google Scholar

[24]

C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems, 18 (1998), 1019-1042.  doi: 10.1017/S0143385798117479.  Google Scholar

[25]

A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves, Geom. Funct. Anal., 23 (2013), 776-809.  doi: 10.1007/s00039-013-0221-z.  Google Scholar

[26]

F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents, J. Mod. Dyn., 7 (2013), 209-237.  doi: 10.3934/jmd.2013.7.209.  Google Scholar

show all references

References:
[1]

I. V. Artamkin, Canonical mappings of punctured curves with the simplest singularities, Mat. Sb., 195 (2004), 3–32; translation in Sb. Math., 195 (2004), 615–642. doi: 10.1070/SM2004v195n05ABEH000818.  Google Scholar

[2]

M. Bainbridge, P. Habegger and M. Möller, Teichmüller curves in genus three and just likely intersections in $ G_{m}^{n}\times G_{a}^{n} $, arXiv: 1410.6835, 2014. Google Scholar

[3]

I. Bouw and M. Möller, Differential equations associated with nonarithmetic Fuchsian groups, J. Lond. Math. Soc.(2), 81 (2010), 65-90.  doi: 10.1112/jlms/jdp059.  Google Scholar

[4]

______, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185.  doi: 10.4007/annals.2010.172.139.  Google Scholar

[5]

M. Bainbridge and M. Möller, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3, Acta Math., 208 (2012), 1-92.  doi: 10.1007/s11511-012-0074-6.  Google Scholar

[6]

I. Bouw, The p-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322.  doi: 10.1023/A:1017513122376.  Google Scholar

[7]

K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908.  doi: 10.1090/S0894-0347-04-00461-8.  Google Scholar

[8]

F. Catanese, M. Franciosi, K. Hulek and M. Reid, Embeddings of curves and surfaces, Nagoya Math. J., 154 (1999), 185-220.  doi: 10.1017/S0027763000025381.  Google Scholar

[9]

D. A. Cox and S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/surv/068.  Google Scholar

[10]

F. Catanese and R. Pignatelli, Pignatelli R., Fibrations of low genus. I, Ann. Sci. école Norm. Sup. (4), 39 (2006), 1011-1049.  doi: 10.1016/j.ansens.2006.10.001.  Google Scholar

[11]

A. Kuribayashi and K. Komiya, On Weierstrass points of non-hyperelliptic compact Riemann surfaces of genus three, Hiroshima Math. J., 7 (1977), 743-768.   Google Scholar

[12]

A. Kumar and R. Mukamel, Algebraic models and arithmetic geometry of Teichmüller curves in genus two, Int. Math. Res. Notices, 2016. doi: 10.1093/imrn/rnw193.  Google Scholar

[13]

R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108.  doi: 10.1007/s000140050113.  Google Scholar

[14]

C. J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number, Geom. Topol., 8 (2004), 1301-1359.  doi: 10.2140/gt.2004.8.1301.  Google Scholar

[15]

C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885.  doi: 10.1090/S0894-0347-03-00432-6.  Google Scholar

[16]

______, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590.  doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar

[17]

C. T. McMullen, R. E. Mukamel and A. Wright, Cubic curves and totally geodesic subvarieties of moduli space, preprint, 2016. Available from: http://math.harvard.edu/~ctm/papers/home/text/papers/gothic/gothic.pdf. Google Scholar

[18]

M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., 165 (2006), 633-649.  doi: 10.1007/s00222-006-0510-3.  Google Scholar

[19]

______, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc., 19 (2006), 327-344.  doi: 10.1090/S0894-0347-05-00512-6.  Google Scholar

[20]

______, Teichmüller curves, mainly from the viewpoint of algebraic geometry, in Moduli Spaces of Riemann Surfaces, IAS/Park City Math. Ser., 20, Amer. Math. Soc., Providence, RI, (2013), 267-318.   Google Scholar

[21]

PARI Group, Bordeaux, PARI/GP version 2. 3. 5. Available from: http://pari.math.u-bordeaux.fr/. Google Scholar

[22]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583.  doi: 10.1007/BF01388890.  Google Scholar

[23]

Ya. B. Vorobets, Plane structures and billiards in rational polygons: The Veech alternative, Uspekhi Mat. Nauk, 51 (1996), 3-42.  doi: 10.1070/RM1996v051n05ABEH002993.  Google Scholar

[24]

C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems, 18 (1998), 1019-1042.  doi: 10.1017/S0143385798117479.  Google Scholar

[25]

A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves, Geom. Funct. Anal., 23 (2013), 776-809.  doi: 10.1007/s00039-013-0221-z.  Google Scholar

[26]

F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents, J. Mod. Dyn., 7 (2013), 209-237.  doi: 10.3934/jmd.2013.7.209.  Google Scholar

Figure 1.  Real points of the curve $S_t$ for $t=3$ near $P_t=(0, 0)$ and the hyperflex $Q_t=(0, -1)$.
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Figure 2.  The Kenyon-Smillie $(2, 3, 4)$-lattice surface resulting from unfolding the triangle $\Delta$. Sides are labeled by powers of $\zeta_9 = \exp(2\pi i/9)$, and sides with the same label are identified. The triple (simple) zero is marked by a white (black) dot.
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Figure 3.  Dual graphs of the two cusps of the Teichmüller curve. The vertices represent the connected components and the edges correspond to the nodes of the stable curves associated with the cusps.
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Figure 4.  Vertical cylinder decomposition of $S_{\infty}$ with cylinders $A$, $B$, $C$, $D$ (from light to dark).
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Figure 5.  Horizontal cylinder decomposition of $S_0$ with cylinders $C_1$, $C_2$, $C_3$ (from light to dark).
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Table 1.  Degree of $a_{i, j, k}(s_1, s_2)$.
i 4 3 3 2 2 2 1 1 1 1 0 0 0 0 0
j 0 1 0 2 1 0 3 2 1 0 4 3 2 1 0
k 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4
4i+2j+k-7 9 7 6 5 4 3 3 2 1 0 1 0 -1 -2 -3
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i 4 3 3 2 2 2 1 1 1 1 0 0 0 0 0
j 0 1 0 2 1 0 3 2 1 0 4 3 2 1 0
k 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4
4i+2j+k-7 9 7 6 5 4 3 3 2 1 0 1 0 -1 -2 -3
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