Arithmetic Geometry and Number Theory RTG Seminar: Bounding Points on Curves using $p$-Adic Hodge Theory

Seminar | November 19 | 3:10-5 p.m. | 784 Evans Hall

 Brian Lawrence, University of Chicago

 Department of Mathematics

Methods of $p$-adic analysis provide the most powerful tools to bound the set of rational points on a curve. The earliest work in this direction was the method of Chabauty; in many cases this is already enough to enumerate (with proof) the rational points. Much more recently, work of Kim on the unipotent fundamental group has led to computational breakthroughs by Balakrishnan, Dogra et al. The method of Chabauty works only when $r < g$ (here $r$ is the rank of the Jacobian, and $g$ the genus of the curve); Kim's method has been applied to the case $r = g$, though it is expected to apply in general. I will discuss a new method for bounding points on curves, using instead a reductive representation of the fundamental group. This new method may apply to all curves, but it presents substantial computational difficulties.