Geodesic
An explicit formula for the~Hilbert symbol for Honda
Sbornik. Mathematics, Tome 194 (2003) no. 2, pp. 165-197

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Based on the pairing on Cartier curves explicitly constructed in the previous paper of the authors, an explicit formula for the Hilbert symbol is constructed in a multidimensional local field of characteristic zero with residue field of positive characteristic on the formal module of a one-dimensional Honda formal group. In the proof a Shafarevich basis on the formal module is constructed, and so-called integer $\mu$-modules in two-dimensional local rings of a special form ( $\mu$-rings) are studied. 
@article{SM_2003_194_2_a0,
     author = {S. V. Vostokov and F. Lorenz},
     title = {An explicit  formula for {the~Hilbert} symbol  for  {Honda}},
     journal = {Sbornik. Mathematics},
     pages = {165--197},
     publisher = {mathdoc},
     volume = {194},
     number = {2},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_2_a0/}
}
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S. V. Vostokov; F. Lorenz. An explicit  formula for the~Hilbert symbol  for  Honda. Sbornik. Mathematics, Tome 194 (2003) no. 2, pp. 165-197. http://geodesic.mathdoc.fr/item/SM_2003_194_2_a0/