In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field and a positive integer that there exists a number depending only on and such that for any algebraic curve defined over having genus equal to has at most -rational points. This is a refinement of Faltings's theorem, which asserts that the set of -rational points is necessarily finite.
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.
A variant of the conjecture, due to Mazur, asserts that there should be a number such that for any algebraic curve defined over having genus and whose Jacobian variety has Mordell–Weil rank over equal to , the number of -rational points of is at most . 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 .[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 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]