Elementary analytic methods in higher ramification theory
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.
scientific article; zbMATH DE number 6126848
scientific article