Descent by 3-isogeny and 3-rank of quadratic fields. (English) Zbl 0804.11040

Top, Jaap

Gouvêa, Fernando Q. (ed.) et al., Advances in number theory. The proceedings of the third conference of the Canadian Number Theory Association, held at Queen’s University, Kingston, Canada, August 18-24, 1991. Oxford: Clarendon Press. 303-317 (1993).

The author studies the elliptic curves \(E_ t\) given by \(Y^ 2 = X^ 3 + a(t) (X-b)^ 2,\) where \(b \in \overline \mathbb{Q}^*\) and \(a(t) \in \overline \mathbb{Q} [t]\) has degree 2. When neither \(a(t)\) nor \(4a(t) + 27b\) are squares in \(\overline \mathbb{Q}(t)\), he computes the Mordell-Weil lattice \(E_ t (\overline \mathbb{Q}(t))\). It turns out to be isomorphic to a certain lattice of rank 3.
The author points out several (well-known) connections between 3-class groups of quadratic number fields and the Mordell-Weil groups of elliptic curves over \(\mathbb{Q}\), \(\mathbb{Q}(t)\) and \(\overline \mathbb{Q}(t)\).
For the entire collection see [Zbl 0773.00021].

Reviewer: R.Schoof (Povo)

MSC:

11G05	Elliptic curves over global fields
11R29	Class numbers, class groups, discriminants
14H52	Elliptic curves
11R11	Quadratic extensions

Keywords:

elliptic curves; Mordell-Weil lattice; 3-class groups; Mordell-Weil groups