Uniform boundedness conjecture for rational points

In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field $K$ and a positive integer $g\geq 2$ that there exists a number $N(K,g)$ depending only on $K$ and $g$ such that for any algebraic curve $C$ defined over $K$ having genus equal to $g$ has at most $N(K,g)$ $K$-rational points. This is a refinement of Faltings's theorem, which asserts that the set of $K$-rational points $C(K)$ is necessarily finite.

A variant of the conjecture, due to Mazur, asserts that there should be a number $N(K,g,r)$ such that for any algebraic curve $C$ defined over $K$ having genus $g$ and whose Jacobian variety $J_{C}$ has Mordell-Weil rank over $K$ equal to $r$, the number of $K$-rational points of $C$ is at most $N(K,g,r)$. This variant of the conjecture is known as Mazur's Conjecture B and was resolved by Dimitrov, Gao, and Habegger in 2020.

Progress

The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur . They proved that, assuming that the conjecture holds if one assumes the Bombieri-Lang conjecture. Further progress was made by Michael Stoll who proved that Mazur's Conjecture B holds for hyperelliptic curves with the additional hypothesis that $r\leq g-3$. Stoll's result was further refined by Katz, Rabinoff, and Zureick-Brown in 2015. Both of these works rely on Chabauty's method. In contrast, the work of Dimitrov, Gao, and Habegger do not use Chabauty's method but instead relies on earlier work of Gao and Habegger on the geometric Bogomolov conjecture.