1. Rational points. A classical conjecture of Mordell states that a curve of genus ^ 2 over the rational numbers has only a finite number of rational points. Let K be a finitely generated field over the rational numbers. Then the same statement should hold for a curve defined over K, and a specialization argument due to Néron shows in fact that this latter statement is implied by the correspond...

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell-Weil group of large height. As an application we show that every elliptic curve of prime conductor in the SteinWatkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.

The main result of this note is that there are at most seven rational points (including the one at infinity) on the curve CA with the affine equation y = x+A (where A is a tenth power free integer) when the Mordell-Weil rank of the Jacobian of CA is one. This bound is attained for A = 18 .

Let E be an elliptic curve over the rationals. A crucial step in determining a Mordell-Weil basis for E is to exhibit some positive lower bound λ > 0 for the canonical height ĥ on non-torsion points. We give a new method for determining such a lower bound, which does not involve any searching for points.

In this paper, we consider a family of elliptic curves over Q with 2-torsion part Z2. We prove that, for every such elliptic curve, a positive proportion of quadratic twists have Mordell–Weil rank 0. Mathematics Subject Classifications (2000). 11G05, 11L40, 14H52.

Let φ : P → P be a morphism of degree d ≥ 2 defined over C. The dynamical Mordell–Lang conjecture says that the intersection of an orbit Oφ(P ) and a subvariety X ⊂ P is usually finite. We consider the number of linear subvarieties L ⊂ P such that the intersection Oφ(P ) ∩ L is “larger than expected.” When φ is the d-power map and the coordinates of P are multiplicatively independent, we prove ...

Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve overQwith the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato’s Euler system. CONTENTS

Shioda described in his article [6] a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over kptq. In this article we find all elliptic curves over kptq for which his method is applicable. For these curves we also compute the maximal Mordell-Weil rank.

Watkins conjectured that for an elliptic curve $E$ over $\mathbb {Q}$ of Mordell-Weil rank $r$, the modular degree is divisible by $2^r$. If has non-trivial rational $2$-torsion, we prove conjecture all quadratic twists squarefree integers with sufficiently many prime factors.

These notes are based on lectures given at the “Arithmetic of Hyperelliptic Curves” workshop, Ohrid, Macedonia, 28 August–5 September 2014. They offer a brief (if somewhat imprecise) sketch of various methods for computing the set of rational points on a curve, focusing on Chabauty and the Mordell–Weil sieve.