Geodesic
On the Galois cohomology of elliptic curves defined over a local field
Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 477-488 
O. N. Vvedenskii. On the Galois cohomology of elliptic curves defined over a local field. Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 477-488. http://geodesic.mathdoc.fr/item/SM_1970_12_3_a8/
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     title = {On~the {Galois} cohomology of elliptic curves defined over a~local field},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this article we calculate the group of principal homogeneous spaces over elliptic curves defined over a complete discretely normed field with algebraically closed residue field of characteristic $p>3$ and belonging to types $(c\,4)$, $(c\,5)$ in the classification of A. Neron. The result of our calculations refutes Neron's earlier statement that the group of principal homogeneous spaces over curves of type $(c)$ is trivial. Moreover the calculation of the fundamental group of the proalgebraic group of the points on these curves that are rational over the ground field supports (in this case) the conjecture of I. R. Shafarevich concerning the duality of the group of principal homogeneous spaces and the character group of the fundamental group. Bibliography: 7 titles.

[1] A. Néron, “Modéles minimaux des espaces principaux homogénes sur les courbes elliptiques. Local fields”, Proc. of a conference on local fields, 1966, Berlin–New York, 1967, 66–77 | MR | Zbl

[2] I. R. Shafarevich, “Glavnye odnorodnye prostranstva, opredelennye nad polem funktsii”, Trudy Matem. in-ta im. V. A. Steklova, 64 (1961), 316–346

[3] A. P. Ogg, “Cohomology of abelian varieties over function fields”, Ann. Math., 76:2 (1962), 185–212 | DOI | MR | Zbl

[4] J.-P. Serre, “Groupes proalgeibriques”, Publ. Math. IHES, 1961, no. 7, 341–403 | Zbl

[5] A. Néron, “Modéles minimaux des varietes abéliennes sur les corps locaux et globaux”, Publ. Math. IHES, 1964, no. 21, 361–482 | MR

[6] J.-P. Serre, “Sur les corps locaux à corps résiduel algebriquement clos”, Bull. Soc. Math. France, 89:1 (1961), 105–154 | MR | Zbl

[7] O. N. Vvedenskii, “Dvoistvennost v ellipticheskikh krivykh nad lokalnym polem. I, II”, Izv. AN SSSR, seriya matem., 28 (1964), 1091–1112 ; 30 (1966), 891–922 | MR | Zbl | MR | Zbl