*Abstract: Let C be a smooth projective curve defined over the rational numbers with genus at least 2. It was conjectured by Mordell and proved by Faltings that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points, and in practice, given a curve, provably finding its set of rational points can be quite difficult.In the case when the Mordell-Weil rank of the Jacobian of C is less than the genus, the Chabauty-Coleman method can be used to find rational points, using the construction of certain p-adic line integrals. Nevertheless, the situation in higher rank is still rather mysterious. I will discuss some new techniques that apply in the case when the rank is equal to the genus.*

*Tea @ 3:45 in Cupples I, room 200*

*Host: Xiang Tang*