p-adic integrals on a curve with good reduction enjoy (surprisingly) a unique analytic continuation. In case of bad reduction the Berkovich analytification is generally no longer contractable, and the continuation is no longer unique. Stoll recently gave an explicit comparison between various approaches to continuation (due to Colmez and Berkovich), with a striking arithmetic consequence -- uniformity of rational points on hyperelliptic curves (of fixed genus g and rank < g-3).

I will describe a characterization of Stoll's work via tropicalizations and non-archimedian analytic geometry and explain two applications: an extension of Stoll's uniformity result to non-hyperelliptic curves (with a central role played by the combinatorics of linear systems on graphs), and an application to the uniform Manin-Mumford conjecture. This is joint work with Eric Katz and Joe Rabinoff.