# American Institute of Mathematical Sciences

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

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

 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.

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. [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. [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. [4] ______, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139. [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. [6] I. Bouw, The p-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322. doi: 10.1023/A:1017513122376. [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. [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. [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. [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. [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. [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. [13] R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113. [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. [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. [16] ______, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5. [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. [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. [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. [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. [21] PARI Group, Bordeaux, PARI/GP version 2. 3. 5. Available from: http://pari.math.u-bordeaux.fr/. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [4] ______, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139. [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. [6] I. Bouw, The p-rank of ramified covers of curves, Compositio Math., 126 (2001), 295-322. doi: 10.1023/A:1017513122376. [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. [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. [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. [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. [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. [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. [13] R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113. [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. [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. [16] ______, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5. [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. [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. [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. [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. [21] PARI Group, Bordeaux, PARI/GP version 2. 3. 5. Available from: http://pari.math.u-bordeaux.fr/. [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. [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. [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. [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. [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.
Real points of the curve $S_t$ for $t=3$ near $P_t=(0, 0)$ and the hyperflex $Q_t=(0, -1)$.
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.
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.
Vertical cylinder decomposition of $S_{\infty}$ with cylinders $A$, $B$, $C$, $D$ (from light to dark).
Horizontal cylinder decomposition of $S_0$ with cylinders $C_1$, $C_2$, $C_3$ (from light to dark).
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
 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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