Uniform boundedness conjecture for rational points

The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur.^[1] They proved that the conjecture holds if one assumes the Bombieri–Lang conjecture.

Mazur's conjecture B is a weaker variant of the uniform boundedness conjecture that asserts that there should be a number ${\displaystyle N(K,g,r)}$  such that for any algebraic curve ${\displaystyle C}$  defined over ${\displaystyle K}$  having genus ${\displaystyle g}$  and whose Jacobian variety ${\displaystyle J_{C}}$  has Mordell–Weil rank over ${\displaystyle K}$  equal to ${\displaystyle r}$ , the number of ${\displaystyle K}$ -rational points of ${\displaystyle C}$  is at most ${\displaystyle N(K,g,r)}$ .

Michael Stoll proved that Mazur's conjecture B holds for hyperelliptic curves with the additional hypothesis that ${\displaystyle r\leq g-3}$ .^[2] Stoll's result was further refined by Katz, Rabinoff, and Zureick-Brown in 2015.^[3] Both of these works rely on Chabauty's method.

Mazur's conjecture B was resolved by Dimitrov, Gao, and Habegger in 2021 using the earlier work of Gao and Habegger on the geometric Bogomolov conjecture instead of Chabauty's method.^[4]