In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field
and a positive integer
such that for any algebraic curve
having genus equal to
This is a refinement of Faltings's theorem, which asserts that the set of
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
and whose Jacobian variety
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.