Geodesic
On elliptic curves over pseudolocal fields
Sbornik. Mathematics, Tome 38 (1981) no. 1, pp. 83-94 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper it is shown that for pseudolocal fields there is a natural analog of the Tate–Shafarevich duality for elliptic curves, taking the following form: Theorem. If $A$ is an elliptic curve defined over the pseudolocal field $k$, whose residue field has characteristic not equal to $2$ or $3$, then the Tate–Shafarevich pairing $$ H^1(k,A)\times A_k\to Q/Z $$ is left nondegenerate. Bibliography: 11 titles. 
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V. I. Andriichuk. On elliptic curves over pseudolocal fields. Sbornik. Mathematics, Tome 38 (1981) no. 1, pp. 83-94. http://geodesic.mathdoc.fr/item/SM_1981_38_1_a5/

[1] E. Artin, J. Tate, Class field theory, Harvard, 1961

[2] J. Ax, “The elementary theory of finite fields”, Ann. Math., 88:2 (1968), 239–271 | DOI | MR | Zbl

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

[4] O. N. Vvedenskii, “O lokalnykh “polyakh klassov” ellipticheskikh krivykh”, Izv. AN SSSR, seriya matem., 37 (1973), 20–88 | MR

[5] J. S. Milne, “Wail–Châtelet groups over local fields”, Ann. scient. École Norm. Sup., 4e serie, 3:3 (1970), 273–284 | MR | Zbl

[6] A. Neron, “Modéles minimaux des variétés abeliennes sur les corps locaux et globaux”, Publ. Math. IHES, 1974, no. 21

[7] Zh.-P. Serr, Kogomologii Galua, izd-vo “Mir”, Moskva, 1968 | MR

[8] J.-P. Serre, Corps Locaux, Hermann, Paris, 1962 | MR | Zbl

[9] I. R. Shafarevich, “Gruppa glavnykh odnorodnykh algebraicheskikh mnogoobrazii”, DAN SSSR, 124:1 (1959), 42–43 | Zbl

[10] S. Shatz, “The cohomology of certain elliptic carves over local and quasi-local fields”, Illinois J. Math., 11 (1967), 234–241 | MR | Zbl

[11] J. Tate, “WC-group over $p$-adic fields”, Sem. Bourbaki, 1956, no. 156