A Chabauty–Coleman Bound for 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 through 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, representing a step toward a higher-dimensional Chabauty--Coleman method. This is joint work with Hector Pasten.

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 and Picard curves of genus 3 defined over $\mathbb{Q}$. I will illustrate the method through an explicit example in which this set can be computed. This is joint work with Jennifer Balakrishnan and part of an ongoing project with Riya Parankimamvila and Steffen Müller.