Geodesic
Billiards and Teichmüller curves on Hilbert modular surfaces
McMullen, Curtis1
1 Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138-2901
Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 857-885

Voir la notice de l'article provenant de la source American Mathematical Society

This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from L-shaped polygons, give billiard tables with optimal dynamical properties.
DOI : 10.1090/S0894-0347-03-00432-6

McMullen, Curtis 1

1 Mathematics Department, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 02138-2901 
@article{10_1090_S0894_0347_03_00432_6,
     author = {McMullen, Curtis},
     title = {Billiards and {Teichm\~A{\textonequarter}ller} curves on {Hilbert} modular surfaces},
     journal = {Journal of the American Mathematical Society},
     pages = {857--885},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2003},
     doi = {10.1090/S0894-0347-03-00432-6},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00432-6/}
}
TY  - JOUR
AU  - McMullen, Curtis
TI  - Billiards and Teichmüller curves on Hilbert modular surfaces
JO  - Journal of the American Mathematical Society
PY  - 2003
SP  - 857
EP  - 885
VL  - 16
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00432-6/
DO  - 10.1090/S0894-0347-03-00432-6
ID  - 10_1090_S0894_0347_03_00432_6
ER  - 
%0 Journal Article
%A McMullen, Curtis
%T Billiards and Teichmüller curves on Hilbert modular surfaces
%J Journal of the American Mathematical Society
%D 2003
%P 857-885
%V 16
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-03-00432-6/
%R 10.1090/S0894-0347-03-00432-6
%F 10_1090_S0894_0347_03_00432_6
McMullen, Curtis. Billiards and Teichmüller curves on Hilbert modular surfaces. Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 857-885. doi: 10.1090/S0894-0347-03-00432-6

[1] Ahlfors, Lars V. The complex analytic structure of the space of closed Riemann surfaces 1960 45 66

[2] Ciliberto, Ciro, Van Der Geer, Gerard Subvarieties of the moduli space of curves parametrizing Jacobians with nontrivial endomorphisms Amer. J. Math. 1992 551 570

[3] Ciliberto, C., Van Der Geer, G., Teixidor I Bigas, M. On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms J. Algebraic Geom. 1992 215 229

[4] Cohen, Paula, Wolfart, Jã¼Rgen Modular embeddings for some nonarithmetic Fuchsian groups Acta Arith. 1990 93 110

[5] Cornfeld, I. P., Fomin, S. V., Sinaä­, Ya. G. Ergodic theory 1982

[6] Eskin, Alex, Okounkov, Andrei Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials Invent. Math. 2001 59 103

[7] Gross, Benedict H., Rohrlich, David E. Some results on the Mordell-Weil group of the Jacobian of the Fermat curve Invent. Math. 1978 201 224

[8] Gutkin, Eugene, Judge, Chris Affine mappings of translation surfaces: geometry and arithmetic Duke Math. J. 2000 191 213

[9] Hirzebruch, Friedrich, Van Der Geer, Gerard Lectures on Hilbert modular surfaces 1981 193

[10] Kenyon, Richard, Smillie, John Billiards on rational-angled triangles Comment. Math. Helv. 2000 65 108

[11] Kerckhoff, Steven, Masur, Howard, Smillie, John Ergodicity of billiard flows and quadratic differentials Ann. of Math. (2) 1986 293 311

[12] Kontsevich, M. Lyapunov exponents and Hodge theory 1997 318 332

[13] Kra, Irwin The Carathéodory metric on abelian Teichmüller disks J. Analyse Math. 1981

[14] Lind, D. A. The entropies of topological Markov shifts and a related class of algebraic integers Ergodic Theory Dynam. Systems 1984 283 300

[15] Masur, Howard Transitivity properties of the horocyclic and geodesic flows on moduli space J. Analyse Math. 1981 1 10

[16] Masur, Howard Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential 1988 215 228

[17] Masur, Howard Hausdorff dimension of the set of nonergodic foliations of a quadratic differential Duke Math. J. 1992 387 442

[18] Penner, R. C. Bounds on least dilatations Proc. Amer. Math. Soc. 1991 443 450

[19] Puchta, Jan-Christoph On triangular billiards Comment. Math. Helv. 2001 501 505

[20] Rapoport, M. Compactifications de l’espace de modules de Hilbert-Blumenthal Compositio Math. 1978 255 335

[21] Royden, H. L. Invariant metrics on Teichmüller space 1974 393 399

[22] Schmutz Schaller, Paul, Wolfart, Jã¼Rgen Semi-arithmetic Fuchsian groups and modular embeddings J. London Math. Soc. (2) 2000 13 24

[23] Schmidt, Thomas A. Klein’s cubic surface and a “non-arithmetic” curve Math. Ann. 1997 533 539

[24] Van Der Geer, Gerard Hilbert modular surfaces 1988

[25] Veech, W. A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards Invent. Math. 1989 553 583

[26] Veech, William A. Moduli spaces of quadratic differentials J. Analyse Math. 1990 117 171

[27] Veech, William A. The billiard in a regular polygon Geom. Funct. Anal. 1992 341 379

[28] Veech, William A. Geometric realizations of hyperelliptic curves 1995 217 226

[29] Vorobets, Ya. B. Plane structures and billiards in rational polygons: the Veech alternative Uspekhi Mat. Nauk 1996 3 42

[30] Ward, Clayton C. Calculation of Fuchsian groups associated to billiards in a rational triangle Ergodic Theory Dynam. Systems 1998 1019 1042

Cité par Sources :