Uniform bounds on rational points on curves of low Mordell-Weil rank

Algebra Seminar
Wednesday, February 18, 2015 - 15:05
1 hour (actually 50 minutes)
Skiles 006
University of Waterloo
In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves.  I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform.  These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.