Comptes Rendus
Number theory/Group theory
Markoff triples and strong approximation
Comptes Rendus. Mathématique, Volume 354 (2016) no. 2, pp. 131-135.

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 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.

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 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2015.12.006
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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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/

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