no. 2
Number theory/Group theory
Markoff triples and strong approximation
[Triplets de Markoff et approximation forte]

Présenté par : Jean Bourgain

Bourgain, Jean1 ; Gamburd, Alexander2 ; Sarnak, Peter1, 3
1 IAS, USA
2 The Graduate Center, CUNY, USA
3 Princeton University, USA
Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 131-135.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.12.006
Bourgain, Jean 1 ; Gamburd, Alexander 2 ; Sarnak, Peter 1, 3

1 IAS, USA
2 The Graduate Center, CUNY, USA
3 Princeton University, USA 
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Bourgain, Jean; Gamburd, Alexander; Sarnak, Peter. Markoff triples and strong approximation. Comptes Rendus. Mathématique, Tome 354 (2016) no. 2, pp. 131-135. doi : 10.1016/j.crma.2015.12.006. http://www.numdam.org/articles/10.1016/j.crma.2015.12.006/

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