Elementary analytic methods in higher ramification theory
type: Item
instance of: scholarly article
title: Elementary analytic methods in higher ramification theory
zbMATH Open document ID: 1271.11105
author: Jonathan Lubin
published in: Journal of Number Theory
publication date: 2013-01-18T00:00:00Z
review text: Let \(k\) be a local field with finite residue field. In this paper the author rederives several classical results in the higher ramification theory of extensions of \(k\), as developed for instance in Chapter IV of [\textit{J.-P. Serre}, Local fields. Paris: Hermann (1962; Zbl 0137.02601)]. Among the results proved are Herbrand's theorem, the Hasse-Arf theorem, and functoriality of the Hasse-Herbrand transition function in towers of extensions. Let \(K/k\) be a finite separable totally ramified extension, let \(\pi\) be a uniformizer for \(K\), and let \(f(X)\) be the minimum polynomial for \(\pi\) over \(k\). The approach taken here is based on consideration of the translated polynomial \(f(X+\pi)\). The Newton polygon and copolygon of \(f(X+\pi)\) encode the ramification data of \(K/k\). This allows for an alternative definition for the transition function \(\phi_k^K(x)\), and leads to a conceptual proof of the formula \(\phi_k^L=\phi_k^K\circ\phi_K^L\) for a tower of extensions \(L\supset K\supset k\). Sen's theorem [\textit{S. Sen}, Ann. Math. (2) 90, 33--46 (1969; Zbl 0199.36301)] is usually viewed as a generalization of the Hasse-Arf theorem, but here it is proved first and then used in the proof of the Hasse-Arf theorem.
reviewed by: Kevin Keating
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S05
zbMATH DE Number: 6126848
zbMATH Keywords: Ramification theory
zbMATH Keywords: Hasse-Herbrand functions
zbMATH Keywords: Herbrand's theorem
zbMATH Keywords: Hasse-Arf theorem
zbMATH Keywords: Sen's theorem
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.02.017
OpenAlex ID: W2067506326
cites work: Q2702127
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cites work: On the ramification breaks
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DOI: 10.1016/J.JNT.2012.02.017
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description: scientific article; zbMATH DE number 6126848