Title: Rational points and unipotent fundamental groups
Speaker: Daniel Hast (Rice University)
Abstract: Given a curve of genus at least 2 over a number field, what can we say about its set of rational points? Faltings' theorem tells us that this set is finite, but many questions remain about how to obtain good bounds on the number of rational points and how to provably list all rational points. We will survey some recent progress and ongoing work on these questions using Kim's non-abelian Chabauty 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 $\mathbb{Q}$, due to joint work with Jordan Ellenberg. We will also discuss a conditional generalization of this strategy from $\mathbb{Q}$-points to points in any real number field.
Seminar URL: https://research.math.osu.edu/agseminar/
