Math @ Duke

Publications [#345769] of Joseph D Rabinoff
Papers Published
 Katz, E; Rabinoff, J; ZureickBrown, D, Uniform bounds for the number of rational points on curves of small mordellweil rank,
Duke Mathematical Journal, vol. 165 no. 16
(January, 2016),
pp. 31893240 [doi]
(last updated on 2021/08/04)
Abstract: Let X be a curve of genus g ≥ 2 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 Frational points of X is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the BombieriLang 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 AbelJacobi map. For fixed X, the finiteness of this quantity is the ManinMumford conjecture, which was proved by Raynaud. We give an explicit uniform bound on #X(F) when X has MordellWeil rank r ≤ g 3. 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 Frational torsion points of J lying on the image of X under an AbelJacobi 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 ChabautyColeman's padic integration, nonArchimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.


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