Uniform boundedness conjecture for rational points#Mazur's conjecture B

{{Short description|Mathematics conjecture about rational points on algebraic curves}}

{{For|other uniform boundedness conjectures|Uniform boundedness conjecture (disambiguation){{!}}Uniform boundedness conjecture}}

In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field $K$ and a positive integer $g \geq 2$ , there exists a number $N(K,g)$ depending only on $K$ and $g$ such that for any algebraic curve $C$ defined over $K$ having genus equal to $g$ has at most $N(K,g)$ $K$ -rational points. This is a refinement of Faltings's theorem, which asserts that the set of $K$ -rational points $C(K)$ is necessarily finite.

Progress

The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur.{{cite journal |first1=Lucia |last1=Caporaso |first2=Joe |last2=Harris |first3=Barry |last3=Mazur |title=Uniformity of rational points |journal=Journal of the American Mathematical Society |volume=10 |issue=1 |year=1997 |pages=1–35 |doi=10.1090/S0894-0347-97-00195-1|doi-access=free }} 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 $N(K,g,r)$ such that for any algebraic curve $C$ defined over $K$ having genus $g$ and whose Jacobian variety $J_C$ has Mordell–Weil rank over $K$ equal to $r$ , the number of $K$ -rational points of $C$ is at most $N(K,g,r)$ .

Michael Stoll proved that Mazur's conjecture B holds for hyperelliptic curves with the additional hypothesis that $r \leq g - 3$ .{{cite journal |first=Michael |last=Stoll |title=Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank |journal=Journal of the European Mathematical Society |volume=21 |issue=3 |year=2019 |pages=923–956 |doi=10.4171/JEMS/857 |doi-access=free |arxiv=1307.1773 }} Stoll's result was further refined by Katz, Rabinoff, and Zureick-Brown in 2015.{{cite journal |first1=Eric |last1=Katz |first2=Joseph |last2=Rabinoff |first3=David |last3=Zureick-Brown |title=Uniform bounds for the number of rational points on curves of small Mordell–Weil rank |journal=Duke Mathematical Journal |volume=165 |issue=16 |year=2016 |pages=3189–3240 |doi=10.1215/00127094-3673558 |arxiv=1504.00694 |s2cid=42267487 }} 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.{{cite journal |first1=Vessilin |last1=Dimitrov |first2=Ziyang |last2=Gao |first3=Philipp |last3=Habegger|title=Uniformity in Mordell–Lang for curves |journal = Annals of Mathematics | volume = 194 | year=2021| pages=237–298 |doi=10.4007/annals.2021.194.1.4|arxiv=2001.10276 |s2cid=210932420 |url=https://hal.sorbonne-universite.fr/hal-03374335/file/Dimitrov%20et%20al.%20-%202021%20-%20Uniformity%20in%20Mordell%E2%80%93Lang%20for%20curves.pdf }}

References

Category:Conjectures

Category:Arithmetic geometry