Uniform boundedness conjecture for rational points

Mazur's conjecture B

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 ${\displaystyle r\le g-3}$. Stoll's result was further refined by Katz, Rabinoff, and Zureick-Brown in 2015. Both of these works rely on Chabauty's method.

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.