This article deals with a constructive aspect of Hilbert’s seventeenth problem : producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. To do this we use a reformulation of this problem in terms of hyperelliptic curves due to Huisman and Mahé and we follow a method from Cassels, Ellison...

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for prime number p, in the p-adic Lie group jacobian, closure Mordell-Weil with curve. If rank less than then this has never failed. Minhyong Kim's non-abelian Chabauty programme aims remove condition rank. The simplest case, called quadratic Chabauty, was developed by Bala...

Let k be a global field, k a separable closure of k, and Gk the absolute Galois group Gal(k/k) of k over k. For every σ ∈ Gk, let k σ be the fixed subfield of k under σ. Let E/k be an elliptic curve over k. We show that for each σ ∈ Gk, the Mordell-Weil group E(k σ ) has infinite rank in the following two cases. Firstly when k is a global function field of odd characteristic and E is parametriz...

Using the Shioda–Tate theorem and an adaptation of Silverman’s specialization theorem, we reduce Mordell–Weil ranks for abelian varieties over fields finitely generated infinite [Formula: see text] to Néron–Severi recently proved by Ambrosi in positive characteristic. More precisely, prove that after a blow-up base surface text], all vertical curves fibration with from complement sparse subset ...

We exhibit several families of elliptic curves with torsion group isomorphic to Z/6Z and generic rank at least 3. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo. We mention the details of some of them and we add other examples developed more recently by Dujella and Peral, and MacLeod. Then we apply an algorithm of Gusi¢ and Tadi¢ a...

Given an elliptic curve E1 over a number field K and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2Selmer groups of E1, E2 and A, we can obtain information about X(E1/K)[2] and we give examples where...