Uniform boundedness conjecture for rational points

In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field formula_1 and a positive integer formula_2 that there exists a number formula_3 depending only on formula_1 and formula_5 such that for any algebraic curve formula_6 defined over formula_1 having genus equal to formula_5 has at most formula_3 formula_1-rational points. This is a refinement of Faltings's theorem, which asserts that the set of formula_1-rational points formula_12 is necessarily finite.
Progress.
The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur. They proved that, assuming 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 formula_13 such that for any algebraic curve formula_6 defined over formula_1 having genus formula_5 and whose Jacobian variety formula_17 has Mordell–Weil rank over formula_1 equal to formula_19, the number of formula_1-rational points of formula_6 is at most formula_13. 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 formula_23. Stoll's result was further refined by Katz, Rabinoff, and Zureick-Brown in 2015. Both of these works rely on Chabauty's method.