Uniform boundedness conjecture for rational points explained
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.
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
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]
Notes and References
- Lucia . Caporaso . Joe . Harris . Barry . Mazur . Uniformity of rational points . . 10 . 1 . 1997 . 1–35 . 10.1090/S0894-0347-97-00195-1. free .
- Michael . Stoll . Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank . . 21 . 3 . 2019 . 923–956 . 10.4171/JEMS/857 . free . 1307.1773 .
- Eric . Katz . Joseph . Rabinoff . David . Zureick-Brown . Uniform bounds for the number of rational points on curves of small Mordell–Weil rank . . 165 . 16 . 2016 . 3189–3240 . 10.1215/00127094-3673558 . 1504.00694 . 42267487 .
- Vessilin . Dimitrov . Ziyang . Gao . Philipp . Habegger. Uniformity in Mordell–Lang for curves . . 194 . 2021. 237–298 . 10.4007/annals.2021.194.1.4. 210932420 .
