To participate in this seminar, please register here: https://www.msri.org/seminars/25206

By Faltings's Theorem, formerly known as the Mordell Conjecture, a smooth projective curve of genus at least 2 that is defined over a number field K has at most finitely many K-rational points. Votja later gave a second proof. Many authors, including de Diego, Parshin, Rémond, Vojta, proved upper bounds for the number of K-rational points. In this talk I will discuss joint work with Vesselin Dimitrov and Ziyang Gao. We show that the number of points on the curve is bounded as a function of K, the genus, and the rank of the Mordell-Weil group of the curve's Jacobian. We follow Vojta's approach and complement it by bounding the number of "small points" using a new lower bound for the Néron-Tate height.