Uniform boundedness conjecture for rational points

For other uniform boundedness conjectures, see Uniform boundedness conjecture.

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

Progress

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

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]