Title:

The field of norms functor and the Hilbert symbol

The classical Hilbert symbol of a higher local field $F$ containing a primitive $p^M$th root of unity $\zeta_M$ is a pairing $F^*/(F^*)^{p^M}\times K_N(F)/p^M \to \mu_{p^M}$, describing Kummer extensions of exponent $p^M$. 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 $F_.$ a field $\c F$ of characteristic $p$. Separable extensions of $\cal F$ correspond functorially to extensions of $F_.$, giving rise to $\Gamma_{\cal F}\cong \Gamma_{F_{\infty}}\subset \Gamma_F$. We define morphisms $\cal N_{\cal F/F_n}: K_N^t(\cal F)/p^M \to K_N^t(F_n)/p^M$ which are compatible with the norms $N_{F_{n+m}/F_n}$ for every $m$. Using these, we show that field of norms functor commutes with the reciprocity maps $\Psi_{\cal F}: K_N^t(\cal F) \to \Gamma_{\cal F}^{ab}$ and $\Psi_{F_n}: K_N^t(F_n) \to \Gamma_{F_n}^{ab}$ constructed by Fesenko. Imitating Fontaine's approach, we obtain an invariant form of Parshin's formula for the Witt pairing in characteristic $p$. The `main lemma' relates Kummer extensions of $F$ and Witt extensions of $\cal F$, allowing us to derive a formula for the generalised Hilbert symbol $\hat F_{\infty}^* \times K_N(\cal F) \to \mu_{p^M}$, where $\hat F_{\infty}$ is the $p$adic completion of $\varinjlim_n F_n$.
