Cubic surfaces of Markoff type
Matthew de Courcy-Ireland (Stockholm University)
Abstract: The Markoff surface is a cubic surface with the special feature that it is only quadratic in each variable separately. Exchanging the two roots of such a quadratic produces new solutions from old, which enabled A. A. Markoff (senior) to find all the integer solutions. More recently, since work of J. Bourgain, A. Gamburd, and P. Sarnak, it has become possible to understand how the integer solutions are related to the solutions modulo primes. Given a large prime modulus, all solutions to the congruence can be shown to lift to integer solutions by combining their work with a complementary result of W. Y. Chen, which has recently been given a new proof by D. E. Martin. The talk will survey some of these developments, including some work in progress joint with Matthew Litman and Yuma Mizuno where we adapt Martin's proof to a wider family of surfaces.
algebraic geometrynumber theory
Audience: researchers in the topic
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| Organizers: | Michael Bennett, Philipp Habegger, Alina Ostafe* |
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