On the Hahn-Witt series and their generalizations

Abstract

In this paper we study the field of Hahn-Witt series $HW({\overline{\mathbb{F}}}_p)$ with residue field ${\overline{\mathbb{F}}}_p$ (also known as a $p$-adic Malcev-Neumann field \cite{La86, P93}), and its generalizations. Informally, the Hahn-Witt series are possibly infinite linear combinations of rational powers of $p,$ in which the coefficients are Teichmüller representatives, and the set of exponents is well-ordered. They form an algebraically closed extension of ${\mathbb{Q}}_p,$ with a canonical automorphism $\phi ,$ coming from the absolute Frobenius of ${\overline{\mathbb{F}}}_p.$ We prove that the action of $\phi$ on the $p$-power roots of unity is given by $\phi (\zeta )={\zeta }^{-1},$ answering a question of Kontsevich. More generally, we consider the $\pi$-typical Hahn-Witt series $H{W}_{(K,\pi )}({\overline{\mathbb{F}}}_q)$, where $\pi$ is a uniformizer in a local field $K$ with residue field ${\mathbb{F}}_q.$ Again, this field is an algebraically closed extension of $K,$ and it has a canonical automorphism ${\phi }_{\pi },$ coming from the relative Frobenius of ${\overline{\mathbb{F}}}_q$ over ${\mathbb{F}}_q.$ We prove that the action of ${\phi }_{\pi }$ on the maximal abelian extension $K^{ab}$ corresponds via local class field theory to the uniformizer $-\pi \in K^{\ast }.$

19 pages