Ramification of Some Automorphisms of Local Fields

Ramification of Higher Local Fields

Ramification groups of local fields

We recall the definition and basic properties of ramification groups. We compute it in the most elementary cases. One is the classical case, i.e., the case where the residue field is perfect. The other is the abelian and the equal characterictic case. Plan 1. Ramification groups. 2. Classical case. 3. Rank 1 case. (Possible) Applications. 1. Grothendieck-Ogg-Shafarevich formula, conductor formu...

Ramification of Wild Automorphisms of Laurent Series Fields

We study the structure of the Nottingham group N (Fp) of power series f ∈ X · Fp[[X]] with f (0) = 1 where p > 2 is prime, which is isomorphic to the group of wild automorphisms of Fp((X)). We concern ourselves with power series g ∈ N (Fp) with g 6= X for all m ≥ 1 (that is, power series of infinite order in N (Fp)), and we determine a necessary and sufficient criterion for such g ∈ N (Fp) havi...

Ramification of Local Fields with Imperfect Residue Fields II

In [1], a filtration by ramification groups and its logarithmic version are defined on the absolute Galois group of a complete discrete valuation field without assuming that the residue field is perfect. In this paper, we study the graded pieces of these filtrations and show that they are abelian except possibly in the absolutely unramified and non-logarithmic case. 2000 Mathematics Subject Cla...

Ramification of local fields with imperfect residue fields I

Let K be a complete discrete valuation field, and let G be the Galois group of a separable closure Ω. Classically the ramification filtration of G is defined in the case where the residue field of K is perfect ([5], Chapter IV). In this paper, we define without any assumption on the residue field, two ramification filtrations of G and study some of their properties. Our first filtration, (G)a∈Q...