Variations on quadratic Chabauty

Let C be a curve over the rationals of genus g at least 2.
By Faltings' theorem, we know that C has finitely many rational
points. When the Mordell-Weil rank of the Jacobian of C is less than g, the
Chabauty-Coleman method can often be used to find these rational
points through the construction of certain p-adic integrals.

When the rank is equal to g, we can use the theory of p-adic height
pairings to produce p-adic double integrals that allow us to find
integral points on curves. In particular, I will discuss how to carry
out this ``quadratic Chabauty'' method on hyperelliptic curves over
number fields (joint work with Amnon Besser and Steffen Mueller) and
present related ideas to find rational points on bielliptic genus 2
curves (joint work with Netan Dogra).