From Diophantine equations to p-adic analytic geometry

A Diophantine equation is a polynomial equation in several variables, generally with integer coefficients, like x3 + y3 = z3. Provably finding all integer solutions of a Diophantine equation is a storied mathematical problem that is easy to state and notoriously difficult to solve. The method of Chabauty--Coleman is one particularly successful technique for ruling out extraneous solutions of a certain class of Diophantine equations. The method is p-adic in nature, and involves producing p-adic analytic functions that vanish on all integer-valued solutions. I will discuss work with Katz and Zureick-Brown on finding uniform bounds on the number of rational points on a curve of fixed genus, defined over a number field, subject to a (conjecturally weak) restriction on its Jacobian. The same technique also makes progress on the uniform Manin-Mumford conjecture on the size of torsion packets on curves of fixed genus.