The Field of Norms Functor and the Hilbert Symbol
662KbAbstractThe classical Hilbert symbol of a higher local field containing a primitive th root of unity is a pairing, describing Kummer extensions of exponent. In this thesis we define a generalised Hilbert symbol and prove a formula for it. Our approach has several ingredients. The field of norms functor of Scholl associates to any strictly deeply ramified tower a field of characteristic. Separable extensions of correspond functorially to extensions of, giving rise to. We define morphisms which are compatible with the norms for every. Using these, we show that field of norms functor commutes with the reciprocity maps and constructed replica hermes flats
by Fesenko. Imitating Fontaine’s approach, we obtain an invariant form of Parshin’s formula for the Witt pairing in characteristic. The main lemma’ relates Kummer extensions of and Witt extensions of, allowing us to derive a formula for the generalised Hilbert symbol, where is the adic completion of.