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H 2 Integral points on Markoff type cubic surfaces and dynamics

Auteurs : Sarnak, Peter (Auteur de la Conférence)
CIRM (Editeur )

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integral points on hypersurfaces 3 and higher dimensions cubic surfaces Markoff surfaces and dynamics diophantine analysis of Markoff surfaces integral points on a fixed surface and strong approximation connection to Painlevé strong approximation - the basic conjecture results towards the main conjecture Markoff numbers outline of some points in the proofs

Résumé : Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the corresponding nonlinear group of morphims of affine three space.

Codes MSC :
11G05 - Elliptic curves over global fields
37A45 - Relations of ergodic theory with number theory and harmonic analysis

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de publication : 06/01/17
    Date de captation : 12/12/16
    Collection : Research talks ; Dynamical Systems and Ordinary Differential Equations ; Number Theory
    Format : MP4 (.mp4) - HD
    Durée : 01:02:29
    Domaine : Dynamical Systems & ODE ; Number Theory
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2016-12-12_Sarnak.mp4

Informations sur la rencontre

Nom de la rencontre : Jean-Morlet Chair: Ergodic theory and its connections with arithmetic and combinatorics / Chaire Jean Morlet : Théorie ergodique et ses connexions avec l'arithmétique et la combinatoire
Organisateurs de la rencontre : Cassaigne, Julien ; Ferenczi, Sébastien ; Hubert, Pascal ; Kulaga-Przymus, Joanna ; Lemanczyk, Mariusz
Dates : 12/12/16 - 16/12/16
Année de la rencontre : 2016
URL Congrès : http://lemanczyk-ferenczi.weebly.com/con...

Citation Data

DOI : 10.24350/CIRM.V.19100603
Cite this video as: Sarnak, Peter (2016). Integral points on Markoff type cubic surfaces and dynamics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19100603
URI : http://dx.doi.org/10.24350/CIRM.V.19100603

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