Arithmetic Geometry and Number Theory RTG Seminar: The unipotent Albanese map and rational points on varieties

Seminar | May 3 | 3-5 p.m. | 748 Evans Hall

 Daniel Hast, Rice University

 Department of Mathematics

Given a curve of genus at least $2$ over a number field, Faltings' theorem tells us that its set of rational points is finite. Provably computing the set of rational points remains a major open problem, as does the question of whether the number of rational points can be uniformly bounded. We will survey some recent progress and ongoing work using the Chabauty–Kim method, which uses the fundamental group to construct $p$-adic analytic functions that vanish on the set of rational points. In particular, we present a new proof of Faltings' theorem for superelliptic curves over the rational numbers (due to joint work with Jordan Ellenberg), and a conditional generalization of the Chabauty–Kim method to number fields and higher dimensions.