Integral points on Markoff type cubic surfaces and dynamics

Chapters

Appears in collections : 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, Exposés de recherche

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.

Information about the video

Date of recording 12/12/2016
Date of publication 06/01/2017
Institution CIRM
Licence CC BY NC ND
Language English
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19100603
Cite this video Sarnak, Peter (12/12/2016). Integral points on Markoff type cubic surfaces and dynamics. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19100603
URL https://dx.doi.org/10.24350/CIRM.V.19100603

Domain(s)

Bibliography

Athreya, J., Bufetov, A., Eskin, A., & Mirzakhani, M. (2012). Lattice point asymptotics and volume growth on Teichmüller space. Duke Mathematical Journal, 161(6), 1055-1111 - http://dx.doi.org/10.1215/00127094-1548443
Bombieri, E. (2007). Continued fractions and the Markoff tree. Expositiones Mathematicae, 25(3), 187-213 - http://dx.doi.org/10.1016/j.exmath.2006.10.002
Bourgain, J., Gamburd, A., & Sarnak, P. (2015). Markoff Triples and Strong Approximation. - https://arxiv.org/abs/1505.06411
Bourgain, J. (2012). A modular Szemeredi-Trotter theorem for hyperbolas. Comptes Rendus. Mathématique. Académie des Sciences, Paris, 350, 793-796 - http://dx.doi.org/10.1016/j.crma.2012.09.011
Cantat, S., & Loray, F. (2009). Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation. Annales de l'Institut Fourier, 59, 7, 2957-2978 - http://dx.doi.org/10.5802/aif.2512
Chang, M.C. (2013). Elements of large order in prime finite fields. Bulletin of the Australian Mathematical Society, 88, 169-176 - http://dx.doi.org/10.1017/S0004972712000810
Cohn, H. (1971). Representation of Markoff's binary quadratic forms by geodesics on a perforated torus. Acta Arithmetica, 18, 125-136 - http://eudml.org/doc/204972
Corvaja, P. & Zannier, U. (2013). Greatest common divisors of $u-1,v-1$ in positive characteristic and rational points on curves over finite fields. Journal of the European Mathematical Society (JEMS), 15(5), 1927-1942 - http://dx.doi.org/10.4171/JEMS/409
Dubrovin, B., & Mazzocco, M. (2000). Monodromy of certain Painlevé-VI transcendents and reflection groups. Inventiones Mathematicae, 141, 55-147 - http://dx.doi.org/10.1007/s002220000065
Goldman, W. (2003). The modular group action on real $SL(2)$-characters of a one-holed torus. Geometry & Topology, 7, 443-486 - http://dx.doi.org/10.2140/gt.2003.7.443
Gorodentsev, A. & Rudakov, A. (1987). Exceptional vector bundles on projective spaces. Duke Mathematical Journal, 54, 115-130 - http://dx.doi.org/10.1215/S0012-7094-87-05409-3
Hacking, P., & Prokhorov, Y. (2010). Smoothable del Pezzo surfaces with quotient singularities. Compositio Mathematica, 146(1), 169-192 - http://dx.doi.org/10.1112/S0010437X09004370
Laurent, M. (1983). Exponential diophantine equations. Comptes Rendus de l'Académie des Sciences. Série I, 296, 945-947 - https://www.zbmath.org/?q=an:0533.10011
Lisovyy, O., & Tykhyy, Y. (2008). Algebraic solutions of the sixth Painleve equation. - https://arxiv.org/abs/0809.4873
Matthews, C.R., Vaserstein, L.N., & Weisfeiler, B. (1984). Congruence properties of Zariski dense groups. I. Proceedings of the London Mathematical Society. Third Series, 48, 514-532 - http://dx.doi.org/10.1112/plms/s3-48.3.514
Mccullough, D., & Wanderley, M. (2013). Nielsen equivalence of generating pairs in $SL(2,q)$. Glasgow Mathematical Journal, 55, 481-509 - http://dx.doi.org/10.1017/S0017089512000675
McShane, G., & Rivin, I. (1995). A norm on homology of surfaces and counting simple geodesics. IMRN. International Mathematics Research Notices, 2, 61-69 - http://dx.doi.org/10.1155/S1073792895000055
Salehi, A., & Sarnak, P. (2013). The affine sieve. Journal of the American Mathematical Society, 4, 1085-1105 - http://dx.doi.org/10.1090/S0894-0347-2013-00764-X
Stepanov, S.A. (1969). On the number of points of a hyperelliptic curve over a finite prime field. Mathematics of the USSR. Izvestija, 3(5), 1103-1114 - http://dx.doi.org/10.1070/IM1969v003n05ABEH000834
Zagier, D. (1982). On the number of Markoff numbers below a given bound. Mathematics of Computation, 39, 709-723 - http://dx.doi.org/10.2307/2007348