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

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


14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
11R16 Cubic and quartic extensions
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[1] Birch, B. J.; Stephens, N. M., The parity of the rank of the Mordell-Weil group, Topology, 5, 295-299 (1966) · Zbl 0146.42401
[2] Cassels, J. W.S, On a conjecture of Selmer, J. Reine Angew. Math., 202, 52-99 (1959) · Zbl 0090.03005
[3] Cassels, J. W.S, On the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math., 217, 180-199 (1965) · Zbl 0241.14017
[4] Hasse, H., Arithmetische Theorie der Kubischen Zahlkörper, Math. Z., 31, 565-582 (1930) · JFM 56.0167.02
[5] Hecke, E., (Vorlesungen über die Theorie der Algebraischen Zahlen (1923), Akademische Verlag: Akademische Verlag Leipzig) · JFM 49.0106.10
[6] Mordell, L. J., (Diophantine Equations (1969), Academic Press: Academic Press New York) · Zbl 0188.34503
[7] Stage, Ph, Une généralisation du calcul de Selmer, dans, (Séminaire de Théorie des Nombres de Paris. Séminaire de Théorie des Nombres de Paris, 1981-1982 (1983), Birkhäuser-Verlag: Birkhäuser-Verlag Boston/Basel/Stuttgart)
[8] Satge, Ph, Corps de Discriminant donné, (Thèse de 3ième cycle (1972), Université Paris XI) · Zbl 0798.14010
[9] Selmer, E. S., The diophantine equation \(ax^3 + by^3 + cz^3 = 0\), Acta Math., 85, 203-362 (1951) · Zbl 0042.26905
[10] Serre, J. P., (Corps Locaux (1968), Hermann: Hermann Paris)
[11] Tate, J., Algorithm for determining the type of a singular fiber in an elliptic pencil, (Lectures Notes in Mathematics, Vol. 476 (1975), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York) · Zbl 1214.14020
[12] Velu, J., Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris, 238-241 (1971) · Zbl 0225.14014
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