10.24350/CIRM.V.19100603
Sarnak, Peter
0000 0000 6305 658X
Princeton Institute for Advanced Study, USA
Integral points on Markoff type cubic surfaces and dynamics
CIRM
2016
11G05
37A45
Systèmes Dynamiques & EDO
Théorie des Nombres
Hennenfent, Guillaume
2017-01-06T00:00:00
2016-12-12T00:00:00
ENG
video conference
19100603
http://library.cirm-math.fr/Record.htm?record=19282915124910001979
http://videos.cirm-math.fr/2016-12-12_Sarnak.mp4
https://youtu.be/1R6NE3ouMtg
http://library.cirm-math.fr/19100603.vtt
http://lemanczyk-ferenczi.weebly.com/conference.html
MP4
CC BY NC ND
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.
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