An explicit formula for the Hilbert symbol of a formal group
Copyright policy )An explicit formula for the Hilbert symbol of a formal group
A Brückner-Vostokov formula for the Hilbert symbol of a formal group was established by Abrashkin under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ( ϕ , Γ )-modules and a cohomological interpretation of Abrashkin’s technique. To do this, we build ( ϕ , Γ )-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the cup-product and the Kummer map.
11F80, Class field theory; \(p\)-adic formal groups, \((\phi, \Gamma)\)-modules, Mathematics - Number Theory, Galois cohomology, Formal groups, \(p\)-divisible groups, explicit reciprocity law, \(p\)-adic representations, 14F30, 11S31, 11S23, formal groups, 11S25, FOS: Mathematics, 11F80; 11S25; 14L05; 11S31; 11S23; 14F30, Number Theory (math.NT), 14L05
11F80, Class field theory; \(p\)-adic formal groups, \((\phi, \Gamma)\)-modules, Mathematics - Number Theory, Galois cohomology, Formal groups, \(p\)-divisible groups, explicit reciprocity law, \(p\)-adic representations, 14F30, 11S31, 11S23, formal groups, 11S25, FOS: Mathematics, 11F80; 11S25; 14L05; 11S31; 11S23; 14F30, Number Theory (math.NT), 14L05
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