Groupes de Selmer et corps cubiques. (Selmer group and cubic fields). (French) Zbl 0601.14027

Satgé, Philippe

J. Number Theory 23, 294-317 (1986).

The object of this article is the computation of the Selmer group S of the elliptic curve \(A: Y^ 2Z=X^ 3+kZ^ 3\) for any rational integer \(k\neq 0\) with respect to the isogeny \(\lambda\) of degree 3 whose kernel is generated by the 3-division point (0:\(\sqrt{k}:1)\). In theorem 1.14 S is described as a subgroup of Hom(Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}}(\sqrt{k})),{\mathbb{Z}}/3{\mathbb{Z}})\) in terms of a list of local conditions imposed on the cubic field extensions which correspond to the Galois characters. Lateron this is applied under certain simplifying assumptions (in theorem 2.6) to describe the Selmer group by a system of linear equations over \({\mathbb{F}}_ 3={\mathbb{Z}}/3{\mathbb{Z}}\) thus leading to an explicit formula for the \({\mathbb{F}}_ 3\)-dimension of S in terms of congruences. This generalizes classical results by Selmer and Cassels. Finally the results are applied to show that certain elliptic curves have no rational point of infinite order, which improves work of Mordell.

Reviewer: C.-G.Schmidt

MSC:

14H45	Special algebraic curves and curves of low genus
14H52	Elliptic curves
11R16	Cubic and quartic extensions

Keywords:

Selmer group; elliptic curve

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