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...

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 ...

In the last lecture we proved that the torsion subgroup of the rational points on an elliptic curve E/Q is finite. In this lecture we will prove a special case of Mordell’s theorem, which states that E(Q) is finitely generated. By the structure theorem for finitely generated abelian groups, this implies E(Q) ' Z ⊕ T, where Zr is a free abelian group of rank r, and T is the (necessarily finite) ...

We rst develop a notion of quadratic form on a locally compact abelian group. Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over Fp. A random subspace chosen with respect to this measure is discrete with probability 1, and the dimension of its intersection with a xed compact open maximal isotro...

Let $E$ be an elliptic curve with positive rank over a number field $K$ and let $p$ odd prime number. $K_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $K_n$ denote its $n$-th layer. The Mordell--Weil is said to constant in tower if for all $n$, $E(K_n)$ equal $E(K)$. We apply techniques Iwasawa theory obtain explicit conditions above sense. then indicate potential applications Hilbert's ten...

We study the arithmetic structure of elliptic curves over k(t), where k is an algebraically closed field. In [Shi86] Shioda shows how one may determine rank of the Néron-Severi group of a Delsarte surface–a surface that may be defined by four monomial terms. To this end, he describes an explicit method of computing the Lefschetz number of a Delsarte surface. He proves the universal bound of 56 ...