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Uniform boundedness conjecture for rational points

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In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field 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

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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

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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]

References

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  1. ^ 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.
  2. ^ 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. arXiv:1307.1773. doi:10.4171/JEMS/857.
  3. ^ 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. arXiv:1504.00694. doi:10.1215/00127094-3673558. S2CID 42267487.
  4. ^ Dimitrov, Vessilin; Gao, Ziyang; Habegger, Philipp (2021). "Uniformity in Mordell–Lang for curves" (PDF). Annals of Mathematics. 194: 237–298. doi:10.4007/annals.2021.194.1.4. S2CID 210932420.