Rational points on curves and p-adic periods

Faltings' Theorem says that the number of rational points on curves of higher genus is finite, and non-abelian Chabauty provides a blueprint both for proving this finiteness and computing these finite sets using p-adic analytic "Chabauty-Kim" functions. Until recently, computations were possible only in the special cases of classical and quadratic Chabauty, which apply only under certain hypotheses on the curve.

We review previous work joint with I. Dan-Cohen going beyond these special cases for the related finite Diophantine set of S-integral points on the projective line minus three points. We then discuss recent work joint with M. Lüdtke computing a Chabauty-Kim function in a non-rational case, that of S-integral points on a punctured elliptic curve. The computations are based on a theory of p-adic periods developed jointly with Dan-Cohen.