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  2020, 16: 255-288. doi: 10.3934/jmd.2020009

Shimura–Teichmüller curves in genus 5

David Aulicino 1, and  Chaya Norton 2,

1. 

Department of Mathematics, Brooklyn College and CUNY Graduate Center, 2900 Bedford Avenue, Brooklyn, NY 11210-2889, USA
2. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

Received  November 18, 2019 Revised  April 29, 2020 Published  September 2020

Fund Project: Partially supported by the National Science Foundation under Award Nos. DMS - 1738381, DMS - 1600360, and PSC-CUNY Grant # 61639-00 49

We prove that there are no Shimura–Teichmüller curves generated by genus five translation surfaces, thereby completing the classification of Shimura–Teichmüller curves in general. This was conjectured by Möller in his original work introducing Shimura–Teichmüller curves. Moreover, the property of being a Shimura–Teichmüller curve is equivalent to having completely degenerate Kontsevich–Zorich spectrum.

The main new ingredient comes from the work of Hu and the second named author, which facilitates calculations of higher order terms in the period matrix with respect to plumbing coordinates. A large computer search is implemented to exclude the remaining cases, which must be performed in a very specific way to be computationally feasible.

Keywords: Teichmüller geodesic flow, Kontsevich–Zorich cocycle, Abelian differentials, translation surfaces, Lyapunov exponents, period matrices, variational formulas.
Mathematics Subject Classification: 32G20, 37D40; Secondary: 14H15, 14G35, 37D25.
Citation: David Aulicino, Chaya Norton. Shimura–Teichmüller curves in genus 5. Journal of Modern Dynamics, 2020, 16: 255-288. doi: 10.3934/jmd.2020009
References:
[1]

D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, Comment. Math. Helv., 90 (2015), 573-643.  doi: 10.4171/CMH/365.

[2]

D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, Ergodic Theory Dynam. Systems, 38 (2018), 10-33.  doi: 10.1017/etds.2016.26.

[3]

David Aulicino and Chaya Norton, Shimura–Teichmüller curves in genus 5, Sage Notebooks, https://github.com/davidaulicino/ST5.

[4]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.

[5]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes \'Etudes Sci., 120 (2014), 207-333.  doi: 10.1007/s10240-013-0060-3.

[6]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL $(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.

[7]

J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973.

[8]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv: 0810.0023v1 (2008).

[9]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.  doi: 10.3934/jmd.2011.5.285.

[10]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.

[11]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,549–580. doi: 10.1016/S1874-575X(06)80033-0.

[12]

S. Grushevsky, I. Krichever and C. Norton, Real-normalized differentials: Limits on stable curves, Russian Math. Surveys, 74 (2019), 265-324.  doi: 10.4213/rm9877.

[13]

X. Hu and C. Norton, General variational formulas for Abelian differentials, Int. Math. Res. Not. IMRN (2020), no. 12, 3540–3581. doi: 10.1093/imrn/rny106.

[14]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237.  doi: 10.1002/mana.200510597.

[15]

H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J., 43 (1976), 623-635. 

[16]

H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314.  doi: 10.1215/S0012-7094-86-05319-6.

[17]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.  doi: 10.3934/jmd.2011.5.1.

[18]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486.  doi: 10.3934/jmd.2010.4.453.

[19]

Yu. L. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Mathematics and its Applications (Soviet Series), vol. 16, D. Reidel Publishing Co., Dordrecht, 1988. doi: 10.1007/978-94-009-2885-5.

[20]

J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557.  doi: 10.1007/s00222-010-0236-0.

[21]

W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.  doi: 10.2307/2007091.

[22]

W. A. 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]

A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143. 

show all references

References:
[1]

D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, Comment. Math. Helv., 90 (2015), 573-643.  doi: 10.4171/CMH/365.

[2]

D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, Ergodic Theory Dynam. Systems, 38 (2018), 10-33.  doi: 10.1017/etds.2016.26.

[3]

David Aulicino and Chaya Norton, Shimura–Teichmüller curves in genus 5, Sage Notebooks, https://github.com/davidaulicino/ST5.

[4]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.  doi: 10.1007/s11511-007-0012-1.

[5]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes \'Etudes Sci., 120 (2014), 207-333.  doi: 10.1007/s10240-013-0060-3.

[6]

A. Eskin, M. Mirzakhani and A. Mohammadi, Isolation, equidistribution, and orbit closures for the SL $(2, \mathbb{R})$ action on moduli space, Ann. of Math. (2), 182 (2015), 673-721.  doi: 10.4007/annals.2015.182.2.7.

[7]

J. D. Fay, Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973.

[8]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum, preprint, arXiv: 0810.0023v1 (2008).

[9]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318.  doi: 10.3934/jmd.2011.5.285.

[10]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  doi: 10.2307/3062150.

[11]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006,549–580. doi: 10.1016/S1874-575X(06)80033-0.

[12]

S. Grushevsky, I. Krichever and C. Norton, Real-normalized differentials: Limits on stable curves, Russian Math. Surveys, 74 (2019), 265-324.  doi: 10.4213/rm9877.

[13]

X. Hu and C. Norton, General variational formulas for Abelian differentials, Int. Math. Res. Not. IMRN (2020), no. 12, 3540–3581. doi: 10.1093/imrn/rny106.

[14]

F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237.  doi: 10.1002/mana.200510597.

[15]

H. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J., 43 (1976), 623-635. 

[16]

H. Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J., 53 (1986), 307-314.  doi: 10.1215/S0012-7094-86-05319-6.

[17]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.  doi: 10.3934/jmd.2011.5.1.

[18]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486.  doi: 10.3934/jmd.2010.4.453.

[19]

Yu. L. Rodin, The Riemann Boundary Problem on Riemann Surfaces, Mathematics and its Applications (Soviet Series), vol. 16, D. Reidel Publishing Co., Dordrecht, 1988. doi: 10.1007/978-94-009-2885-5.

[20]

J. Smillie and B. Weiss, Characterizations of lattice surfaces, Invent. Math., 180 (2010), 535-557.  doi: 10.1007/s00222-010-0236-0.

[21]

W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.  doi: 10.2307/2007091.

[22]

W. A. 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]

A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143. 

Figure 1.  Schematic of the surface coordinates used in the algorithm: Circle vertices represent zeros and square vertices could represent zeros or regular points
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Figure 2.  Proof of Corollary 5.10: The shaded region on the top of $ C_2 $ represents the admissible locations of $ \tau_0 $
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Table 1.  List of Strata Possibly Containing ST-Curves and Values of $ d_{opt} $: Dashed lines separate genus
Stratum $ d_{opt} $
$ \mathscr{H}(1, 1, 1, 1) $ $ 2 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1) $ $ 4 $
$ \mathscr{H}(2, 2, 2) $ $ 3 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1,2) $ $ 36 $
$ \mathscr{H}(1, 1, 1, 1, 2,2) $ $ 18 $
$ \mathscr{H}(1, 1, 2, 2,2) $ $ 12 $
$ \mathscr{H}(2, 2, 2,2) $ $ 9 $
$ \mathscr{H}(1, 1, 1, 1, 1,3) $ $ 16 $
$ \mathscr{H}(1, 1, 3,3) $ $ 8 $
$ \mathscr{H}(1, 1, 1, 1,4) $ $ 10 $
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Stratum $ d_{opt} $
$ \mathscr{H}(1, 1, 1, 1) $ $ 2 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1) $ $ 4 $
$ \mathscr{H}(2, 2, 2) $ $ 3 $
$ \mathscr{H}(1, 1, 1, 1, 1, 1,2) $ $ 36 $
$ \mathscr{H}(1, 1, 1, 1, 2,2) $ $ 18 $
$ \mathscr{H}(1, 1, 2, 2,2) $ $ 12 $
$ \mathscr{H}(2, 2, 2,2) $ $ 9 $
$ \mathscr{H}(1, 1, 1, 1, 1,3) $ $ 16 $
$ \mathscr{H}(1, 1, 3,3) $ $ 8 $
$ \mathscr{H}(1, 1, 1, 1,4) $ $ 10 $
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