Uniform boundedness conjecture for rational points
In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field [math]\displaystyle{ K }[/math] and a positive integer [math]\displaystyle{ g \geq 2 }[/math] that there exists a number [math]\displaystyle{ N(K,g) }[/math] depending only on [math]\displaystyle{ K }[/math] and [math]\displaystyle{ g }[/math] such that for any algebraic curve [math]\displaystyle{ C }[/math] defined over [math]\displaystyle{ K }[/math] having genus equal to [math]\displaystyle{ g }[/math] has at most [math]\displaystyle{ N(K,g) }[/math] [math]\displaystyle{ K }[/math]-rational points. This is a refinement of Faltings's theorem, which asserts that the set of [math]\displaystyle{ K }[/math]-rational points [math]\displaystyle{ C(K) }[/math] 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
A variant of the conjecture, due to Mazur, asserts that there should be a number [math]\displaystyle{ N(K,g,r) }[/math] such that for any algebraic curve [math]\displaystyle{ C }[/math] defined over [math]\displaystyle{ K }[/math] having genus [math]\displaystyle{ g }[/math] and whose Jacobian variety [math]\displaystyle{ J_C }[/math] has Mordell–Weil rank over [math]\displaystyle{ K }[/math] equal to [math]\displaystyle{ r }[/math], the number of [math]\displaystyle{ K }[/math]-rational points of [math]\displaystyle{ C }[/math] is at most [math]\displaystyle{ N(K,g,r) }[/math]. 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 [math]\displaystyle{ r \leq g - 3 }[/math].[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]
References
- ↑ Caporaso, Lucia; Harris, Joe; Mazur, Barry (1997). "Uniformity of rational points". Journal of the American Mathematical Society 10 (1): 1–35. doi:10.1090/S0894-0347-97-00195-1.
- ↑ Stoll, Michael (2019). "Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank". Journal of the European Mathematical Society 21 (3): 923–956. doi:10.4171/JEMS/857.
- ↑ Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David (2016). "Uniform bounds for the number of rational points on curves of small Mordell–Weil rank". Duke Mathematical Journal 165 (16): 3189–3240. doi:10.1215/00127094-3673558.
- ↑ Dimitrov, Vessilin; Gao, Ziyang; Habegger, Philipp (2021). "Uniformity in Mordell–Lang for curves". Annals of Mathematics 194: 237–298. doi:10.4007/annals.2021.194.1.4. https://hal.sorbonne-universite.fr/hal-03374335/file/Dimitrov%20et%20al.%20-%202021%20-%20Uniformity%20in%20Mordell%E2%80%93Lang%20for%20curves.pdf.
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