The Field of Norms Functor and the Hilbert Symbol

JENNI, RUTH,CHRISTINE (2010) The Field of Norms Functor and the Hilbert Symbol. Doctoral thesis, Durham University.

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Abstract

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 $\varinjlim_n F_n$ .

Item Type:	Thesis (Doctoral)
Award:	Doctor of Philosophy
Keywords:	Algebraic Number Theory; Local Fields; Hilbert Symbol; Field of Norms functor
Faculty and Department:	Faculty of Science > Mathematical Sciences, Department of
Thesis Date:	2010
Copyright:	Copyright of this thesis is held by the author
Deposited On:	29 Oct 2010 15:29

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The Field of Norms Functor and the Hilbert Symbol

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