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 Teichm\"uller representatives, and the set of exponents is well-ordered. They form an algebraically closed extension of $\mathbb{Q}_p,$ with a canonical automorphism $\varphi,$ coming from the absolute Frobenius of $\overline{\mathbb{F}}_p.$ We prove that the action of $\varphi$ on the $p$-power roots of unity is given by $\varphi(\zeta)=\zeta^{-1},$ answering a question of Kontsevich. More generally, we consider the $\pi$-typical Hahn-Witt series $HW_{(K,\pi)}(\overline{\mathbb{F}}_q)$, 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 $\varphi_{\pi},$ coming from the relative Frobenius of $\overline{\mathbb{F}}_q$ over $\mathbb{F}_q.$ We prove that the action of $\varphi_{\pi}$ on the maximal abelian extension $K^{ab}$ corresponds via local class field theory to the uniformizer $-\pi\in K^*.$