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 F-rational points of X is an …

A metrized complex of algebraic curves over an algebraically closed field κ κ is, roughly
speaking, a finite metric graph Γ Γ together with a collection of marked complete nonsingular …

We discuss linear series on tropical curves and their relation to classical algebraic geometry,
describe the main techniques of the subject, and survey some of the recent major …

Consider the smooth projective models 𝐶 of curves 𝑦²= 𝑓 (𝑥) with 𝑓 (𝑥)∊ ℤ [𝑥] monic and
separable of degree 2𝑔+ 1. We prove that for 𝑔≥ 3, a positive fraction of these have only …

We show that there is a bound depending only on g, r and [K: Q] for the number of K-rational
points on a hyperelliptic curve C of genus g over a number field K such that the Mordell–Weil …

Canonical heights on genus-2 Jacobians Page 1 Algebra & Number Theory msp Volume 10
2016 No. 10 Canonical heights on genus-2 Jacobians Jan Steffen Müller and Michael Stoll …

Ogg's celebrated Torsion conjecture--as it relates to modular curves--can be paraphrased as
saying that rational points (on the modular curves that parametrize torsion points on elliptic …

One of the most important results in Diophantine geometry is the finiteness of the number of
rational points on nice curves of genus at least two. However, there are no practical methods …

We study the Selmer varieties of smooth projective curves of genus at least two defined over
which geometrically dominate a curve with CM Jacobian. We extend a result of Coates and …

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St.
Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set …