BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:William Chen (Columbia University) DTSTART:20210205T163000Z DTEND:20210205T173000Z DTSTAMP:20260423T010131Z UID:LAGA-AGAA/7 DESCRIPTION:Title: Markoff triples\, Nielsen equivalence\, and nonabelian level structures< /a>\nby William Chen (Columbia University) as part of Séminaire de géom étrie arithmétique et motivique (Paris Nord)\n\n\nAbstract\nFollowing Bo urgain\, Gamburd\, and Sarnak\, we say that the Markoff equation $x^2 + y^ 2 + z^2 - 3xyz = 0$ satisfies strong approximation at a prime $p$ if its i ntegral points surject onto its $\\mathbf F_p$ points. In 2016\, Bourgain\ , Gamburd\, and Sarnak were able to establish strong approximation at all but a sparse (but infinite) set of primes\, and conjectured that it holds at all primes. Building on their results\, in this talk I will explain how to establish strong approximation for all but a finite and effectively co mputable set of primes\, thus reducing the conjecture to a finite computat ion. The key result amounts to establishing a congruence on the degree of a certain line bundle on the moduli stack of elliptic curves with $\\SL(2\ ,p)$-structures. To make contact with the Markoff equation\, we use the fa ct that the Markoff surface is a level set of the character variety for $\ \SL(2)$ representations of the fundamental group of a punctured torus\, an d that the strong approximation conjecture can be expressed in terms of th e mapping class group action on the character variety\, which in turn also determines the geometry of the moduli stack of elliptic curves with $\\SL (2\,p)$-structures. As time allows we will also describe some applications .\n LOCATION:https://researchseminars.org/talk/LAGA-AGAA/7/ END:VEVENT END:VCALENDAR