Geodesic
Explicit formulae for the Hilbert symbol of a~formal group over the Witt vectors
Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 463-515

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In this paper an explicit formula is obtained for a generalisation of the Hilbert symbol, associated with an arbitrary formal group of finite height, defined over the ring of Witt vectors with coefficients in a perfect field of characteristic $p>0$. This formula becomes the Bruckner–Vostokov formula in the case of a multiplicative formal group. The proof is based on an application of Fontaine's theory of $p$-adic periods of formal groups, the Fontaine–Wintenberg field-of-norms functor, and Witt's explicit reciprocity law in characteristic $p$. 
@article{IM2_1997_61_3_a0,
     author = {V. A. Abrashkin},
     title = {Explicit formulae for the {Hilbert} symbol of a~formal group over the {Witt} vectors},
     journal = {Izvestiya. Mathematics },
     pages = {463--515},
     publisher = {mathdoc},
     volume = {61},
     number = {3},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_3_a0/}
}
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V. A. Abrashkin. Explicit formulae for the Hilbert symbol of a~formal group over the Witt vectors. Izvestiya. Mathematics , Tome 61 (1997) no. 3, pp. 463-515. http://geodesic.mathdoc.fr/item/IM2_1997_61_3_a0/