David Zureick-Brown : Uniformity of rational points and tropical geometry

Posted by Nathan Jones , part of the Number Theory Seminar.

At

April 28, 2015, 11 a.m.

In

SEO 427

Abstract

Let $X$ be a curve of genus $g$ over a number field $F$ of degree $d = [F:\mathbb{Q}]$. 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, to follow from the Bomberi--Lang conjecture. We prove a special case of this conjecture -- we give an explicit uniform bound when $X$ has Mordell--Weil rank $r\leq g-3$. This generalizes recent work of Stoll on uniform bounds for hyperelliptic 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 harmonic analysis on Berkovich curves, and the theory of linear systems and divisors on metric graphs. This is joint work with Joe Rabinoff and Eric Katz.