In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field and a positive integer , 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.

Mazur's conjecture B is a weaker variant of the uniform boundedness conjecture that 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 .

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 2021 using the earlier work of Gao and Habegger on the geometric Bogomolov conjecture instead of Chabauty's method.^[4]