Markoff triples and strong approximation

Jean Bourgain; Alexander Gamburd; Peter Sarnak

doi:10.1016/j.crma.2015.12.006

Number theory/Group theory

Markoff triples and strong approximation
[Triplets de Markoff et approximation forte]

Présenté par : Jean Bourgain

Jean Bourgain ¹ ; Alexander Gamburd ² ; Peter Sarnak ^1,³

¹ IAS, USA
² The Graduate Center, CUNY, USA
³ Princeton University, USA

Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 131-135.

Résumé
Abstract

Nous explorons les propriétés de transitivité du groupe des morphismes engendré par les involutions de Vieta agissant sur les solutions en congruences de l'équation de Markoff ainsi que d'autres surfaces affines cubiques de type Markoff. Ces propriétés sont determinées par les orbites finies dans $\bar{Q}$ de ces actions, qui peuvent être determinées explicitement. Les resultats permettent d'établir une forme de l'approximation forte pour les points entiers sur ces surfaces et des applications du crible.

We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Markoff type affine cubic surfaces. These are dictated by the finite $\bar{Q}$ orbits of these actions and these can be determined effectively. The results are applied to give forms of strong approximation for integer points, and to sieving, on these surfaces.

Reçu le : 2015-11-16
Accepté le : 2015-11-16
Publié le : 2016-01-26

DOI : 10.1016/j.crma.2015.12.006

Affiliations des auteurs :

Jean Bourgain ¹ ; Alexander Gamburd ² ; Peter Sarnak ^{1, 3}

¹ IAS, USA
² The Graduate Center, CUNY, USA
³ Princeton University, USA

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     title = {Markoff triples and strong approximation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {131--135},
     publisher = {Elsevier},
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     number = {2},
     year = {2016},
     doi = {10.1016/j.crma.2015.12.006},
     language = {en},
}

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Jean Bourgain; Alexander Gamburd; Peter Sarnak. Markoff triples and strong approximation. Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 131-135. doi : 10.1016/j.crma.2015.12.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.006/

Bibliographie
Cité par

[1] Martin Aigner Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013

[2] E. Bombieri Continued fractions and the Markoff tree, Expo. Math., Volume 25 (2007) no. 3, pp. 187-213

[3] J. Bourgain A modular Szemeredi–Trotter theorem for hyperbolas, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012), pp. 793-796

[4] J. Bourgain; A. Gamburd Uniform expansion bounds for Cayley graphs of ${SL}_{2} (F_{p})$ , Ann. Math., Volume 167 (2008), pp. 625-642

[5] J. Bourgain; A. Gamburd; P. Sarnak Affine linear sieve, expanders and sum product, Invent. Math., Volume 179 (2010), pp. 559-644

[6] Jean Bourgain; Alex Gamburd; Peter Sarnak Generalization of Selberg's $\frac{3}{16}$ theorem and affine sieve, Acta Math., Volume 207 (2011) no. 2, pp. 255-290

[7] E. Breuillard; A. Gamburd Strong uniform expansion in ${SL}_{2} (F_{p})$ , Geom. Funct. Anal., Volume 20 (2010) no. 5, pp. 1201-1209

[8] S. Cantat; F. Loray Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation, Ann. Inst. Fourier (Grenoble), Volume 59 (2009), pp. 2927-2978

[9] Chang Mei-Chu Elements of large order in prime finite fields, Bull. Aust. Math. Soc., Volume 88 (2013), pp. 169-176

[10] M.-C. Chang; B. Kerr; I. Shparlinski; U. Zannier Elements of large orders on varieties over prime finite fields, J. Théor. Nr. Bordx., Volume 26 (2014), pp. 579-594

[11] P. Corvaja; U. Zannier On integral points on surfaces, Ann. Math. (2), Volume 160 (2004), pp. 705-726

[12] P. Corvaja; U. Zannier Greatest common divisors of $u - 1$ , $v - 1$ in positive characteristic and rational points on curves over finite fields, J. Eur. Math. Soc., Volume 15 (2013), pp. 1927-1942

[13] B. Dubrovin; M. Mazzocco Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math., Volume 141 (2000), pp. 55-147

[14] M.H. El-Huti Cubic surfaces of Markov type, Math. USSR Sb., Volume 22 (1974) no. 3, pp. 333-348

[15] G. Frobenius Über die Markoffschen Zahlen, Preuss. Akad. Wiss. Sitzungsbericht (1913), pp. 458-487

[16] A. Gamburd; I. Pak Expansion of product replacement graphs, Combinatorica, Volume 26 (2006) no. 4, pp. 411-429

[17] R. Gilman Finite quotients of the automorphism group of a free group, Can. J. Math., Volume 29 (1977), pp. 541-551

[18] W. Goldman The modular group action on real $SL (2)$ -characters of a one-holed torus, Geom. Topol., Volume 7 (2003), pp. 443-486

[19] R. Heath-Brown; S. Konyagin New bounds for Gauss sums derived from k-th powers and for Heilbronn's exponential sum, Q. J. Math. (2000), pp. 221-235

[20] C. Hooley Applications of Sieve Methods to the Theory of Numbers, Cambridge Tracts in Mathematics, vol. 70, 1976

[21] M. Laurent Exponential Diophantine equations, C. R. Acad. Sci. Paris, Ser. I, Volume 296 (1983), pp. 945-947

[22] O. Lisovyy; Y. Tykhyy Algebraic solutions of the sixth Painlevé equation, J. Geom. Phys., Volume 85 (2014), pp. 124-163

[23] Yu.I. Manin Cubic Forms, 1974

[24] A. Markoff Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 15 (1879), pp. 381-409

[25] A. Markoff Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 17 (1880), pp. 379-399

[26] C.R. Matthews Counting points modulo p for some finitely generated subgroups of algebraic groups, Bull. Lond. Math. Soc. (3), Volume 14 (1982), pp. 149-154

[27] C. Matthews; L. Vaserstein; B. Weisfeiler Congruence properties of Zariski dense groups, Proc. Lond. Math. Soc. (3), Volume 48 (1984), pp. 514-532

[28] D. McCullough; M. Wanderley Nielsen equivalence of generating pairs in $SL (2, q)$ , Glasg. Math. J., Volume 55 (2013), pp. 481-509

[29] Greg McShane; Igor Rivin Simple curves on hyperbolic tori, C. R. Acad. Sci. Paris, Ser. I, Volume 320 (1995), pp. 1523-1528

[30] M. Mirzakhani Counting mapping class group orbits on hyperbolic surfaces, 2015 | arXiv

[31] P. Sarnak; S. Adams Betti numbers of congruence groups, Isr. J. Math., Volume 88 (1994), pp. 31-72 (with an appendix by Z. Rudnick)

[32] P. Sarnak; A. Saleh-Golsefidy The affine sieve, J. Amer. Math. Soc., Volume 26 (2013) no. 4, pp. 1085-1105

[33] S.A. Stepanov The number of points of a hyperelliptic curve over a prime field, Math. USSR, Izv., Volume 3 (1969) no. 5, pp. 1103-1114

[34] P. Vojta A generalization of theorems of faltings and Thue–Siegel–Roth–Wirsing, J. Amer. Math. Soc., Volume 25 (1992), pp. 763-804

[35] A. Weil On the Riemann hypothesis in function fields, Proc. Natl. Acad. Sci. USA, Volume 27 (1941), pp. 345-347

[36] D. Zagier On the number of Markoff numbers below a given bound, Math. Comput., Volume 39 (1982) no. 160, pp. 709-723

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