# The Number of Rational Points on a Curve of Genus > 1

## Location

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

Bombieri, de Diego, Parshin, Rémond, Vojta, proved upper bounds for

the number of K-rational points. I will discuss joint work with

Vesselin Dimitrov and Ziyang Gao where we prove that the number of

points on the curve is bounded from above 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 to the Mordell Conjecture and answer a

question of Mazur. I will explain the new feature: an inequality for

the Néron-Tate height in a family of abelian varieties. It allows us

to bound from above the number of points when the modular height of

the curve is sufficiently large.