American Institute of Mathematical Sciences

Advanced
  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

[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

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 $Q_t=(0, -1)$.
Figure Options

Download full-size image

Download as PowerPoint slide

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.
Figure Options

Download full-size image

Download as PowerPoint slide

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.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Vertical cylinder decomposition of $S_{\infty}$ with cylinders $A$, $B$, $C$, $D$ (from light to dark).
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Horizontal cylinder decomposition of $S_0$ with cylinders $C_1$, $C_2$, $C_3$ (from light to dark).
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
[1]

Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1
[2]

Alex Wright. Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces. Journal of Modern Dynamics, 2012, 6 (3) : 405-426. doi: 10.3934/jmd.2012.6.405
[3]

Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209
[4]

David Aulicino, Chaya Norton. Shimura–Teichmüller curves in genus 5. Journal of Modern Dynamics, 2020, 16: 255-288. doi: 10.3934/jmd.2020009
[5]

Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003
[6]

Vincenzo Michael Isaia. Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3459-3481. doi: 10.3934/dcdsb.2017175
[7]

Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135
[8]

Ursula Hamenstädt. Bowen's construction for the Teichmüller flow. Journal of Modern Dynamics, 2013, 7 (4) : 489-526. doi: 10.3934/jmd.2013.7.489
[9]

Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393
[10]

Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535
[11]

Chi Po Choi, Xianfeng Gu, Lok Ming Lui. Subdivision connectivity remeshing via Teichmüller extremal map. Inverse Problems & Imaging, 2017, 11 (5) : 825-855. doi: 10.3934/ipi.2017039
[12]

Inês Cruz, Helena Mena-Matos, Esmeralda Sousa-Dias. The group of symplectic birational maps of the plane and the dynamics of a family of 4D maps. Journal of Geometric Mechanics, 2020, 12 (3) : 363-375. doi: 10.3934/jgm.2020010
[13]

Rodrigo Abarzúa, Nicolas Thériault, Roberto Avanzi, Ismael Soto, Miguel Alfaro. Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates. Advances in Mathematics of Communications, 2010, 4 (2) : 115-139. doi: 10.3934/amc.2010.4.115
[14]

Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215
[15]

I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191
[16]

Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure & Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028
[17]

Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123
[18]

Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002
[19]

Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83
[20]

Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (42)
  • HTML views (113)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences