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The bound in Merel's solution to the Uniform Boundedness conjecture is not explicit, as it relies on Falting's work on the Mordell conjecture. I think this still is the case.

But there are known explicit bounds for the largest prime divisor. The best one seems to be $(1+3^{d/2})^2$, where $d$ is the degree of the number field, due de Oesterlé (1994!). But as far as I known, the proof of such a bound remains unpublished.

Quoting a relatively recent survey on the topic for context ("Torsion subgroups of elliptic curves over number fields" by Andrew Sutherland, 2012):

Oesterlé's bound plays a critical role in several of the results discussed here; it is quite unfortunate that no proof has been published. The work of Parent in implies that Oesterlé's bound holds for all suffciently large d, but we are typically interested in particular small values of d (e.g. d = 5; 6; 7). There is current work in progress aimed at addressing this gap in the literature [6].

[6] Maarten Derickx, e-mail regarding in nitely many rational points on a modular curve of degree Q-gonality 2, December 2012.

  • My question is, has this gap been filled? Alternatively, has Oesterlé's work been superseded?
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Yes, this is published as appendix A to chapter 3 in Derickx' PhD thesis available here: https://openaccess.leidenuniv.nl/handle/1887/43186 . The thesis contains, of course, many more interesting results.

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    $\begingroup$ 1-step link: openaccess.leidenuniv.nl/bitstream/handle/1887/43186/03.pdf . It looks like a preprint: M. Derickx, S. Kamieny, W. Stein, M. Stoll. Torsion points on elliptic curves over number fields of small degree. If once there's a reference it could be added here. $\endgroup$ – YCor Jun 11 '17 at 15:48

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