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Uniform boundedness in terms of ramification

Abstract

Let \(d\ge 1\) be fixed. Let F be a number field of degree d, and let E / F be an elliptic curve. Let \(E(F)_{\text {tors}}\) be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E / F, such that the size of \(E(F)_{\text {tors}}\) is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of \(E(F)_{\text {tors}}\). In 1996, Parent proved a bound (also exponential in d) for the largest p-power order of a torsion point that may appear in \(E(F)_{\text {tors}}\). It has been conjectured, however, that there is a bound for the size of \(E(F)_{\text {tors}}\) that is polynomial in d. In this article we show that under certain hypotheses there is a linear bound for the largest p-power order of a torsion point defined over F, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers F over (p).

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Author's contributions

The author would like to thank Kevin Buzzard, Pete Clark, Brian Conrad, Harris Daniels, Benjamin Lundell, Robert Pollack, James Stankewicz, Jeremy Teitelbaum, Ravi Ramakrishna, John Voight, Felipe Voloch and David Zywina for their helpful suggestions and comments. In addition, the author would like to express his gratitude to the anonymous referees for very detailed reports, and pointing out a crucial oversight in an earlier version of the paper.

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Correspondence to Álvaro Lozano-Robledo.

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Lozano-Robledo, Á. Uniform boundedness in terms of ramification. Res. number theory 4, 6 (2018). https://doi.org/10.1007/s40993-018-0095-0

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Mathematics Subject Classification

  • Primary 11G05
  • Secondary 14H52