Counting and Finding Rational Points on Surfaces

A celebrated result of Coleman gives an explicit version of Chabauty’s theorem, bounding the number of rational points on curves over number fields via the study of zeros of p-adic analytic functions. While many developments have extended and refined this result, obtaining analogous explicit bounds for higher-dimensional subvarieties of abelian varieties remains a major challenge.
In this talk, I will sketch the proof of such an explicit bound for surfaces contained in abelian varieties — a step toward a higher-dimensional Chabauty–Coleman method. This is joint work with Héctor Pastén.
I will also describe an application of this method to a computational problem: determining an upper bound for the number of unexpected quadratic points on hyperelliptic curves of genus 3 defined over Q. I will illustrate the method through an explicit example where this set can be computed. This is joint work with Jennifer Balakrishnan.

Jerson has also kindly agreed to give a pretalk from 2:00-2:45pm in SC 507.