Duke Mathematical Journal

Uniform bounds for the number of rational points on curves of small Mordell–Weil rank

Eric Katz, Joseph Rabinoff, and David Zureick-Brown

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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.

Article information

Duke Math. J., Volume 165, Number 16 (2016), 3189-3240.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G05: Rational points
Secondary: 14T05: Tropical geometry [See also 12K10, 14M25, 14N10, 52B20]

Chabauty Jacobian Manin–Mumford non-Archimedean $p$-adic integration rational points tropical geometry


Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David. Uniform bounds for the number of rational points on curves of small Mordell–Weil rank. Duke Math. J. 165 (2016), no. 16, 3189--3240. doi:10.1215/00127094-3673558. https://projecteuclid.org/euclid.dmj/1476450482

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