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1 November 2016 Uniform bounds for the number of rational points on curves of small Mordell–Weil rank
Eric Katz, Joseph Rabinoff, David Zureick-Brown
Duke Math. J. 165(16): 3189-3240 (1 November 2016). DOI: 10.1215/00127094-3673558

Abstract

Let X be a curve of genus g2 over a number field F of degree d=[F:Q]. The conjectural existence of a uniform bound N(g,d) on the number #X(F) of F-rational points of X is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the existence of a uniform bound Ntors,(g,d) on the number of geometric torsion points of the Jacobian J of X which lie on the image of X under an Abel–Jacobi map. For fixed X, the finiteness of this quantity is the Manin–Mumford conjecture, which was proved by Raynaud.

We give an explicit uniform bound on #X(F) when X has Mordell–Weil rank rg3. This generalizes recent work of Stoll on uniform bounds for hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F-rational torsion points of J lying on the image of X under an Abel–Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate.

Our methods combine Chabauty–Coleman’s p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.

Citation

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Eric Katz. Joseph Rabinoff. David Zureick-Brown. "Uniform bounds for the number of rational points on curves of small Mordell–Weil rank." Duke Math. J. 165 (16) 3189 - 3240, 1 November 2016. https://doi.org/10.1215/00127094-3673558

Information

Received: 6 June 2015; Revised: 19 December 2015; Published: 1 November 2016
First available in Project Euclid: 14 October 2016

zbMATH: 06666955
MathSciNet: MR3566201
Digital Object Identifier: 10.1215/00127094-3673558

Subjects:
Primary: 14G05
Secondary: 14T05

Rights: Copyright © 2016 Duke University Press

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Vol.165 • No. 16 • 1 November 2016
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