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Logarithm laws for unipotent flows, Ⅱ
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 |
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.
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. 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. |
[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. 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. |
[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/. 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. |
[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. |
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. |
[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. |
[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. 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. |
[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/. 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. |
[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. |





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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