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