Geodesic
Formal groups and the norm residue symbol
Izvestiya. Mathematics, Tome 15 (1980) no. 2, pp. 289-348 
V. A. Kolyvagin. Formal groups and the norm residue symbol. Izvestiya. Mathematics, Tome 15 (1980) no. 2, pp. 289-348. http://geodesic.mathdoc.fr/item/IM2_1980_15_2_a4/
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     author = {V. A. Kolyvagin},
     title = {Formal groups and the norm residue symbol},
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     volume = {15},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1980_15_2_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper investigates the canonical pairing associated with a one-dimensional formal group law $F$ over the ring of integers of a finite extension of $\mathbf Q_p$ and an isogeny $f\colon F\to F$, just as the Hilbert symbol is associated with the multiplicative law and the isogeny "raising to the $p$th power". Formulas are obtained which generalize the formulas of Artin–Hasse, Iwasawa, and Wiles. The formulas describe the values of the symbol in terms of $p$-adic differentiation, the logarithm of the formal group law, the norm, and the trace. Bibliography: 8 titles.

[1] Leng S., Algebra, Mir, M., 1968

[2] Dzh. Kassels, A. Frelikh (red.), Algebraicheskaya teoriya chisel, Mir, M., 1969

[3] Iwasawa K., “On explisit formulas for the norm residue sumbol”, J. Math. Soc. Japan, 20:1,2 (1968), 151–164 | MR

[4] Wiles A., “Higher resiprosity lawers”, Ann. Math., 107:2 (1978), 235–254 | DOI | MR | Zbl

[5] Shafarevich I. R., “Obschii zakon vzaimnosti”, Matem. sb., 26:1 (1950), 113–146 | Zbl

[6] Vostokov S. V., “Yavnaya forma zakona vzaimnosti”, Izv. AN SSSR. Ser. matem., 42:6 (1978), 1288–1321 | MR | Zbl

[7] Kolyvagin V. A., “Simvol normennogo vycheta v lokalnykh polyakh”, Uspekhi matem. nauk, 1978, no. 6, 217–218 | MR | Zbl

[8] Vostokov S. V., “Yavnaya formula sparivaniya v formalnykh modulyakh”, Dokl. AN SSSR, 241:2 (1978), 275–278 | MR | Zbl