On quasi-local ``class fields'' of elliptic curves.~I
Izvestiya. Mathematics , Tome 10 (1976) no. 5, pp. 913-936.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider those elliptic curves defined over a quasi-local field with all finite Galois cohomology finite. The main result is the extension, with preservation of nondegeneracy of Shafarevich–Ogg pairing for curves of the given class, to the components of the paired groups corresponding to the characteristic of the residue field. Bibliography: 32 titles.
@article{IM2_1976_10_5_a1,
     author = {O. N. Vvedenskii},
     title = {On quasi-local ``class fields'' of elliptic {curves.~I}},
     journal = {Izvestiya. Mathematics },
     pages = {913--936},
     publisher = {mathdoc},
     volume = {10},
     number = {5},
     year = {1976},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1976_10_5_a1/}
}
TY  - JOUR
AU  - O. N. Vvedenskii
TI  - On quasi-local ``class fields'' of elliptic curves.~I
JO  - Izvestiya. Mathematics 
PY  - 1976
SP  - 913
EP  - 936
VL  - 10
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1976_10_5_a1/
LA  - en
ID  - IM2_1976_10_5_a1
ER  - 
%0 Journal Article
%A O. N. Vvedenskii
%T On quasi-local ``class fields'' of elliptic curves.~I
%J Izvestiya. Mathematics 
%D 1976
%P 913-936
%V 10
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1976_10_5_a1/
%G en
%F IM2_1976_10_5_a1
O. N. Vvedenskii. On quasi-local ``class fields'' of elliptic curves.~I. Izvestiya. Mathematics , Tome 10 (1976) no. 5, pp. 913-936. http://geodesic.mathdoc.fr/item/IM2_1976_10_5_a1/

[1] Artin M., Grothendieck topologies, Harvard, 1962

[2] Artin M., Grothendieck A., Cohomologie étalés des schémas, Séminaire de Géometrie Algébrique, IHES, Bures-sur-Ivette, 1963–1964

[3] Deligne P., Mumford D., “The irreducibility of the space of curves of given genus”, Publ. Math. IHES, 1969, no. 36, 75–109 | MR | Zbl

[4] Demazure M., Grothendieck A., Schémas en groupes, Śeminaire de Géometrie Algébrique,, IHES, Bures-sur-Ivette, 1963–1964

[5] Faddeev D. K., “K teorii algebr nad polem algebraicheskikh funktsii odnoi peremennoi”, Vestnik LGU, 1957, no. 7, 45–52 | MR

[6] Greenberg M., “Schemata over local rings”, Ann. Math., 73:3 (1961), 624–648 | DOI | MR | Zbl

[7] Lang S., “Divisors and endomorphisms on an abelian variety”, Amer. Journ. Math., 79:2 (1957), 319–330 | DOI | MR | Zbl

[8] Lang S., Abelian varieties, New York, London, 1959 | MR

[9] Mazur B., “Notes on etale cohomology of number fields”, Ann. Scient. Ec. Norm. Sup.(4), 6 (1973), 521–556 | MR

[10] Neron A., “Modèles minimaux des variétés abéliennes sur les corps locaux et globaux”, Publ. Math. IHES, 1964, no. 21 | MR

[11] Neron A., “Modèles minimaux des espaces principaux homogènes sur les courbes elliptiques”, Local fields (Proc. of a conference on local fields, 1966), Berlin, New York, 1967, 66–77 | MR | Zbl

[12] Neron A., “Modèles minimaux et différentiels”, Symposia Mathematica, 111, Inst. Nazion. Alta Matem., Bologna, 1970, 279–293 | MR

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

[14] Raynaud M., “Caractéristique d'Euler–Poincare d'un faisceau et cohomologie des variétés abéliennes”, Sém. Bourbaki, v. 286, 1964–1965 | Zbl

[15] Raynaud M., “Specialisation du founcteur de Picard”, Publ. Mat. IHES, 1970, no. 38, 27–76 | MR | Zbl

[16] Shafarevich I. R., “Glavnye odnorodnye prostranstva, opredelennye nad polem funktsii”, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 64, 1961, 316–346

[17] Shatz S. S., “Cohomology of artinian group schemes over local fields”, Ann. Math., 79:3 (1964), 411–449 | DOI | MR | Zbl

[18] Shatz S. S., “The cohomological dimension of certain Grothendieck topologies”, Ann. Math., 83:3 (1966), 572–595 | DOI | MR | Zbl

[19] Serre J.-P., “Groupes proalgébriques”, Publ. Math. IHES, 1960, no. 7, 1–66 | MR

[20] Serre J.-P., “Sur les corps locauax à corps résiduell algébriquement clos”, Bull. Soc. Math. France, 89:2 (1961), 105–154 | MR | Zbl

[21] Tate J., $WC$-groups over $p$-adic fields, Sém. Bourbaki, 156, 1957 | MR

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

[23] Vvedenskii O. N., “O kogomologiyakh Galua ellipticheskikh krivykh, opredelennykh nad lokalnym polem”, Matem. sb., 83:3 (1970), 474–484 | MR | Zbl

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

[25] Vvedenskii O. N., “O “universalnykh normakh” formalnykh grupp, opredelennykh nad koltsom tselykh lokalnogo polya”, Izv. AN SSSR. Ser. matem., 37 (1973), 737–751 | MR

[26] Shatz S. S., “Principal homogeneous spaces for finite group schemes”, Proc. AMS, 22:3 (1969), 678–680 | DOI | MR | Zbl

[27] Tate J., “Duality theorems in Galois cohomology over number fields”, Proc. Intern. Congress Math., Stockholm, 1962, 288–295 | MR

[28] Serre J. P., Geometrie algebrique, preprint Proc. Intern. Congress Math., Stockholm, 1962 | MR

[29] Vvedenskii O. M., “Pro $p$-komponenti kogomologii abelevikh mnogovidiv, viznachenikh nad kvazi-globalnimi i kvazi-lokalnimi polyami kharakteristiki $p0$”, Dokl. AN URSR, ser. A, 1974, no. 7, 579–581 | MR | Zbl

[30] Vvedenskii O. M., “Pro kvazi-lokalni “polya klasiv” eliptichnikh krivykh tipu II”, Dokl. AN URSR, ser. A, 1974, no. 9, 774–777 | MR

[31] Vvedenskii O. M., “Pro kvzi-lokalni “polya klasiv” eliptichnikh krivykh tipu II i deyakikh krivykh tipu (s)”, Dokl. AN URSR, ser. A, 1975, no. 5

[32] Vvedenskii O. N., “O dvoistvennosti v ellipticheskikh krivykh nad kvazi-lokalnym polem”, Dokl. AN SSSR, 219:6 (1974), 1291–1293 | MR