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

By Eric Katz, Joseph Rabinoff, David Zureick-Brown. 2015

80 citations recorded by the source corpus.

Let $X$ be a curve of genus $g\geq 2$ 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 $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by [CHM97] to follow from the Bombieri--Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{{\rm tors},\dagger}(g,d)$ 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 $X$ this quantity was conjectured to be finite by Manin--Mumford, and was proved to be so by Raynaud [Ray83]. We give an explicit uniform bound on $\#X(F)$ when $X$ has Mordell--Weil rank $r\leq g-3$. This generalizes recent work of Stoll on uniform bounds on 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5986ca1d70574529c77ff1ac6b493807395d410f

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Citations in the corpus

Degeneration of Linear Series from the Tropical Point of View and Applications (2015, methodology): “If K is not discretely valued, this is still true with the right definition of the sheaf Ω(1)C/R (see [KRZB15]).”
Uniformity in Mordell–Lang for curves (2020, unknown): “For example, if Jac(C)(F ) has rank at most g − 3, Stoll [Sto19] showed that #C(F ) is bounded solely in terms of F and g if C is hyperelliptic; Katz–Rabinoff–Zureick-Brown [KRZB16] later, under the same rank hypothesis, removed the hyperelliptic hypothesis and improved the dependence on F to [F : Q”
Tropical independence II: The maximal rank conjecture for quadrics (2015, methodology): “Combined with techniques from p-adic integration, this method also leads to uniform bounds on rational points for curves of fixed genus with small Mordell–Weil rank [KRZB15].”