Uniform boundedness conjecture for rational points

In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field ${\displaystyle K}$ and a positive integer ${\displaystyle g\geq 2}$ that 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[]

A variant of the conjecture, due to Mazur, 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 ${\displaystyle N(K,g,r)}$. This variant of the conjecture is known as Mazur's Conjecture B.

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, , and in 2015.^[3] Both of these works rely on Chabauty's method.

Mazur's Conjecture B was resolved by , , and Habegger in a preprint in 2020 which has since appeared in the Annals of Mathematics using the earlier work of Gao and Habegger on the geometric Bogomolov conjecture instead of Chabauty's method.^[4]

References[]