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^*)^pM\times K_N(F)/p^M \to μ_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/Fn}: K_N^t(\cal F)/p^M \to K_N^t(F_n)/p^M which are compatible with the norms N_Fn+m/Fn 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_Fn: K_N^t(F_n) \to \Gamma_Fn^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 μ_p^M, where \hat F_\infty is the $p$-adic completion of \varinjlim_n F_n.

Name dc:type.qualificationname

PhD

Level dc:type.qualificationlevel

doctoral

Grantor dc:publisher.institution

Durham University

Year dc:date.issued

2010

Author dc:creator

• Jenni, Ruth Christine