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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.19163 |
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| _version_ | 1866929402071220224 |
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| author | Efimov, Alexander I. |
| author_facet | Efimov, Alexander I. |
| contents | 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 $φ,$ coming from the absolute Frobenius of $\overline{\mathbb{F}}_p.$ We prove that the action of $φ$ on the $p$-power roots of unity is given by $φ(ζ)=ζ^{-1},$ answering a question of Kontsevich.
More generally, we consider the $π$-typical Hahn-Witt series $HW_{(K,π)}(\overline{\mathbb{F}}_q)$, where $π$ 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 $φ_π,$ coming from the relative Frobenius of $\overline{\mathbb{F}}_q$ over $\mathbb{F}_q.$ We prove that the action of $φ_π$ on the maximal abelian extension $K^{ab}$ corresponds via local class field theory to the uniformizer $-π\in K^*.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Hahn-Witt series and their generalizations Efimov, Alexander I. Number Theory Commutative Algebra 11S31 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 $φ,$ coming from the absolute Frobenius of $\overline{\mathbb{F}}_p.$ We prove that the action of $φ$ on the $p$-power roots of unity is given by $φ(ζ)=ζ^{-1},$ answering a question of Kontsevich. More generally, we consider the $π$-typical Hahn-Witt series $HW_{(K,π)}(\overline{\mathbb{F}}_q)$, where $π$ 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 $φ_π,$ coming from the relative Frobenius of $\overline{\mathbb{F}}_q$ over $\mathbb{F}_q.$ We prove that the action of $φ_π$ on the maximal abelian extension $K^{ab}$ corresponds via local class field theory to the uniformizer $-π\in K^*.$ |
| title | On the Hahn-Witt series and their generalizations |
| topic | Number Theory Commutative Algebra 11S31 |
| url | https://arxiv.org/abs/2406.19163 |