Let be a curve of genus over a number field of degree . The conjectural existence of a uniform bound on the number of -rational points of 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 on the number of geometric torsion points of the Jacobian of which lie on the image of under an Abel–Jacobi map. For fixed , the finiteness of this quantity is the Manin–Mumford conjecture, which was proved by Raynaud.
We give an explicit uniform bound on when has Mordell–Weil rank . 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 -rational torsion points of lying on the image of under an Abel–Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of lying on when the reduction type of is highly degenerate.
Our methods combine Chabauty–Coleman’s -adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.
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