logo AFST
Dynamical moduli spaces and elliptic curves
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 2, pp. 389-420.

Dans ces notes, nous donnons un lien entre la dynamique complexe d’une famille de fractions rationnelles f t : 1 1 , paramétrée par une surface de Riemann X, et la dynamique arithmétique de f t sur les points rationnels de 1 (k), où k=(X). Une relation explicite entre stabilité et hauteur canonique est établie, avec une preuve qui contient une partie du théorème de Mordell–Weil pour les courbes elliptiques sur un corps de fonctions. Notre but principal est de poser quelques questions et conjectures, guidés par le principe des « unlikely intersections » en géométrie arithmétique (cf. [53]). Nous incluons aussi une preuve du fait que les applications hyperboliques postcritiquement-finies sont Zariski denses dans l’espace des modules 𝕄 d des applications rationnelles de degré donné d>1. Ces notes sont basées sur un cours de 4 séances données à KAWA 2015 à Pise, Italie, destinées à une audience spécialisée en analyse complexe, et développent les principaux résultats de [6, 17, 14].

In these notes, we present a connection between the complex dynamics of a family of rational functions f t : 1 1 , parameterized by t in a Riemann surface X, and the arithmetic dynamics of f t on rational points 1 (k) where k=(X) or ¯(X). An explicit relation between stability and canonical height is explained, with a proof that contains a piece of the Mordell–Weil theorem for elliptic curves over function fields. Our main goal is to pose some questions and conjectures about these families, guided by the principle of “unlikely intersections” from arithmetic geometry, as in [53]. We also include a proof that the hyperbolic postcritically-finite maps are Zariski dense in the moduli space 𝕄 d of rational maps of any given degree d>1. These notes are based on four lectures at KAWA 2015, in Pisa, Italy, designed for an audience specializing in complex analysis, expanding upon the main results of [6, 17, 14].

Publié le :
DOI : 10.5802/afst.1573
Laura De Marco 1

1 Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AFST_2018_6_27_2_389_0,
     author = {Laura De Marco},
     title = {Dynamical moduli spaces and elliptic curves},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {389--420},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {2},
     year = {2018},
     doi = {10.5802/afst.1573},
     mrnumber = {3831028},
     zbl = {1404.37047},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1573/}
}
TY  - JOUR
AU  - Laura De Marco
TI  - Dynamical moduli spaces and elliptic curves
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
SP  - 389
EP  - 420
VL  - 27
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1573/
DO  - 10.5802/afst.1573
LA  - en
ID  - AFST_2018_6_27_2_389_0
ER  - 
%0 Journal Article
%A Laura De Marco
%T Dynamical moduli spaces and elliptic curves
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2018
%P 389-420
%V 27
%N 2
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1573/
%R 10.5802/afst.1573
%G en
%F AFST_2018_6_27_2_389_0
Laura De Marco. Dynamical moduli spaces and elliptic curves. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 2, pp. 389-420. doi : 10.5802/afst.1573. https://afst.centre-mersenne.org/articles/10.5802/afst.1573/

[1] Lyubich, Mikhail Yur’evich Some typical properties of the dynamics of rational mappings, Usp. Mat. Nauk, Volume 38 (1983) no. 5, pp. 197-198 | MR | Zbl

[2] Lars V. Ahlfors Complex analysis. An introduction to the theory of analytic functions of one complex variable, International Series in pure and applied Mathematics, McGraw-Hill Book Company, 1979, xiv+331 pages | Zbl

[3] Yves André Finitude des couples d’invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math., Volume 505 (1998), pp. 203-208 | DOI | Zbl

[4] Matthew Baker A finiteness theorem for canonical heights attached to rational maps over function fields, J. Reine Angew. Math., Volume 626 (2009), pp. 205-233 | MR | Zbl

[5] Matthew Baker; Laura DeMarco Preperiodic points and unlikely intersections, Duke Math. J., Volume 159 (2011) no. 1, pp. 1-29 | DOI | MR | Zbl

[6] Matthew Baker; Laura DeMarco Special curves and postcritically-finite polynomials, Forum Math. Pi, Volume 1 (2013), e3, 35 pages (Article ID e3, 35 p.) | MR | Zbl

[7] Matthew Baker; Robert Rumely Equidistribution of small points, rational dynamics, and potential theory, Ann. Inst. Fourier, Volume 56 (2006) no. 3, pp. 625-688 | DOI | Numdam | MR | Zbl

[8] François Berteloot Bifurcation currents in holomorphic families of rational maps, Pluripotential theory (Lecture Notes in Math.), Volume 2075, Springer, 2013, pp. 1-93 | DOI | MR | Zbl

[9] Bodil Branner; John H. Hubbard The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math., Volume 160 (1988) no. 3-4, pp. 143-206 | DOI | MR | Zbl

[10] Gregory S. Call; Joseph H. Silverman Canonical heights on varieties with morphisms, Compos. Math., Volume 89 (1993) no. 2, pp. 163-205 | Numdam | MR | Zbl

[11] Antoine Chambert-Loir Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | Zbl

[12] David A. Cox Primes of the form x 2 +ny 2 . Fermat, class field theory, and complex multiplication, Pure and Applied Mathematics, John Wiley & Sons, 2013, xvi+356 pages | Zbl

[13] Laura De Marco; Xiaoguang Wang; Hexi Ye Bifurcation measures and quadratic rational maps, Proc. Lond. Math. Soc., Volume 111 (2015) no. 1, pp. 149-180 | DOI | MR | Zbl

[14] Laura De Marco; Xiaoguang Wang; Hexi Ye Torsion points and the Lattès family, Am. J. Math., Volume 138 (2016) no. 3, pp. 697-732 | DOI | Zbl

[15] Laura DeMarco Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett., Volume 8 (2001) no. 1-2, pp. 57-66 | DOI | MR | Zbl

[16] Laura DeMarco Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann., Volume 326 (2003) no. 1, pp. 43-73 | DOI | MR | Zbl

[17] Laura DeMarco Bifurcations, intersections, and heights, Algebra Number Theory, Volume 10 (2016) no. 5, pp. 1031-1056 | DOI | MR | Zbl

[18] Adrien Douady; John H. Hubbard A proof of Thurston’s topological characterization of rational functions, Acta Math., Volume 171 (1993) no. 2, pp. 263-297 | DOI | MR | Zbl

[19] Romain Dujardin The supports of higher bifurcation currents, Ann. Fac. Sci. Toulouse, Math., Volume 22 (2013) no. 3, pp. 445-464 | DOI | Numdam | MR | Zbl

[20] Romain Dujardin Bifurcation currents and equidistribution in parameter space, Frontiers in complex dynamics, Princeton University Press, 2014, pp. 515-566 | DOI | Zbl

[21] Romain Dujardin; Charles Favre Distribution of rational maps with a preperiodic critical point, Am. J. Math., Volume 130 (2008) no. 4, pp. 979-1032 | DOI | MR | Zbl

[22] Charles Favre; Thomas Gauthier Classification of special curves in the space of cubic polynomials, Int. Math. Res. Not., Volume 2018 (2018) no. 2, pp. 362-411 | DOI | MR | Zbl

[23] Charles Favre; Juan Rivera-Letelier Équidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann., Volume 335 (2006) no. 2, p. 311-36 corrigendum in ibid. 339 (2007), no. 4, p. 799-801 | DOI | Zbl

[24] Thomas Gauthier Strong bifurcation loci of full Hausdorff dimension, Ann. Sci. Éc. Norm. Supér., Volume 45 (2012) no. 6, pp. 947-984 | DOI | Numdam | MR | Zbl

[25] Thomas Gauthier Higher bifurcation currents, neutral cycles, and the Mandelbrot set, Indiana Univ. Math. J., Volume 63 (2014) no. 4, pp. 917-937 | DOI | MR | Zbl

[26] Dragos Ghioca; Liang-Chung Hsia; Thomas J. Tucker Preperiodic points for families of polynomials, Algebra Number Theory, Volume 7 (2013) no. 3, pp. 701-732 | DOI | MR | Zbl

[27] Dragos Ghioca; Liang-Chung Hsia; Thomas J. Tucker Preperiodic points for families of rational maps, Proc. Lond. Math. Soc., Volume 110 (2015) no. 2, pp. 395-427 | DOI | MR | Zbl

[28] Dragos Ghioca; Holly Krieger; Khoa D. Nguyen; Hexi Ye The dynamical André-Oort conjecture: unicritical polynomials, Duke Math. J., Volume 166 (2017) no. 1, pp. 1-25 | DOI | Zbl

[29] Dragos Ghioca; Hexi Ye A dynamical variant of the André-Oort conjecture, Int. Math. Res. Not., Volume 2018 (2018) no. 8, pp. 2447-2480 | DOI | Zbl

[30] Ricardo Mañé; Paulo Roberto Sad; Dennis P. Sullivan On the dynamics of rational maps, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983), pp. 193-217 | DOI | Numdam | MR | Zbl

[31] David Masser; Umberto Zannier Torsion anomalous points and families of elliptic curves, C. R. Math. Acad. Sci. Paris, Volume 346 (2008) no. 9-10, pp. 491-494 | DOI | MR | Zbl

[32] David Masser; Umberto Zannier Torsion anomalous points and families of elliptic curves, Am. J. Math., Volume 132 (2010) no. 6, pp. 1677-1691 | MR | Zbl

[33] David Masser; Umberto Zannier Torsion points on families of squares of elliptic curves, Math. Ann., Volume 352 (2012) no. 2, pp. 453-484 | DOI | MR | Zbl

[34] Curtis T. McMullen Families of rational maps and iterative root-finding algorithms, Ann. Math., Volume 125 (1987), pp. 467-493 | DOI | MR | Zbl

[35] Curtis T. McMullen Complex Dynamics and Renormalization, Annals of Mathematics Studies, 135, Princeton University Press, 1995, vii+214 pages | Zbl

[36] Curtis T. McMullen; Dennis P. Sullivan Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math., Volume 135 (1998) no. 2, pp. 351-395 | Zbl

[37] Alice Medvedev; Thomas Scanlon Invariant varieties for polynomial dynamical systems, Ann. Math., Volume 179 (2014) no. 1, pp. 81-177 | MR | Zbl

[38] John Milnor Geometry and dynamics of quadratic rational maps, Exp. Math., Volume 2 (1993) no. 1, pp. 37-83 | MR | Zbl

[39] John Milnor Dynamics in One Complex Variable, Annals of Mathematics Studies, 160, Princeton University Press, 2006, viii+304 pages | MR | Zbl

[40] John Milnor On Lattès maps, Dynamics on the Riemann sphere. A Bodil Branner Festschrift, European Mathematical Society, 2006, pp. 9-43 | Zbl

[41] Jonathan Pila O-minimality and the André-Oort conjecture for n , Ann. Math., Volume 173 (2011) no. 3, pp. 1779-1840 | Zbl

[42] Michel Raynaud Courbes sur une variété abélienne et points de torsion, Invent. Math., Volume 71 (1983), pp. 207-233 | Zbl

[43] Michel Raynaud Sous-variétés d’une variété abélienne et points de torsion, Arithmetic and geometry, Vol. I: Arithmetic (Progress in Mathematics), Volume 35, Birkhäuser, 1983, pp. 327-352 | DOI | Zbl

[44] Joseph Fels Ritt Prime and composite polynomials, Trans. Am. Math. Soc., Volume 23 (1922), pp. 51-66 | DOI | MR | Zbl

[45] Joseph H. Silverman (personal communication) | Numdam

[46] Joseph H. Silverman Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, 151, Springer, 1994, xiii+525 pages | MR | Zbl

[47] Joseph H. Silverman The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, 241, Springer, 2007, ix+511 pages | MR | Zbl

[48] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 2009, xx+513 pages | MR | Zbl

[49] Joseph H. Silverman Moduli spaces and arithmetic dynamics, CRM Monograph Series, 30, American Mathematical Society, 2012, vii+140 pages | MR | Zbl

[50] Amaury Thuillier Théorie du potentiel sur les courbes en géométrie analytique non archimedienne. Applications à la théorie d’Arakelov, Ph. D. Thesis, Université de Rennes 1 (France) (2005)

[51] Jacob Tsimerman A proof of the André-Oort conjecture for 𝒜 g (2015) (https://arxiv.org/abs/1506.01466)

[52] Xinyi Yuan Big line bundles over arithmetic varieties, Invent. Math., Volume 173 (2008) no. 3, pp. 603-649 | DOI | MR | Zbl

[53] Umberto Zannier Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, 181, Princeton University Press, 2012, xi+160 pages | MR | Zbl

Cité par Sources :