A ramification filtration of the Galois group of a local field

Abstract

Let K be a field of formal Laurent series with coefficients in a finite field of characteristic p, G <p -the maximal quotient of Gal(K sep /K) of period p and nilpotent class < p and {G (v) <p } v 0 -its filtration by ramification subgroups in the upper numbering. Let G <p = G(L) be the identification of nilpotent Artin-Schreier theory: here G(L) is the group obtained from a suitable profinite Lie F p -algebra L via the Campbell-Hausdorff composition law. We develop a new technique to characterize the ideals<p and to find their generators. Given v 0 > 0 we construct epimorphism of Lie algebrasη † : L −→L † and the action Ω U of the formal group α p = Spec F p [U ], U p = 0, of order p onL † . Suppose dΩ U = B † U , where B † ∈ DiffL † , and L † [v 0 ] is the ideal ofL † generated by the elements of B † (L † ). The main result of the paper states that L (v0) = (η † ) −1L † [v 0 ]. In the last sections we relate this result to the explicit construction of generators of L (v0) obtained earlier by the author and develop its more efficient version.

Groups of automorphisms of local fields of period p and nilpotent class < p, I

“…In [1,2,3] the author introduced new techniques (nilpotent ArtinSchreier theory) which allowed us to study p-extensions of characteristic p with Galois groups of nilpotent class < p. Such groups come from Lie algebras via classical equivalence L → G(L) of the categories of Lie F p -algebras and p-groups of period p of the same nilpotent class s 0 < p, [11]. This equivalence can be briefly explained as follows.…”

Одно Теоретико-Групповое Свойство Фильтрации Ветвления

“…она не имеет соотношений, которые могут быть выражены через элементы собст венного подмножества некоторого минимального множества образующих группы r(p)/r(p)( v°) . Заметим, что в случае, когда основное поле К имеет характерис тику р, эти соотношения по модулю подгруппы коммутаторов порядка ^ р были описаны в терминах образующих группы Г(р) в работах [1]- [3].…”

A semi-stable case of the Shafarevich Conjecture

“…In Subsection 3 we obtain the ramification estimates for the Galois modules H from the image of V * : if v > 2 − 1/p then the higher ramification subgroups Γ (v) K act trivially on H. We also obtain the ramification estimate for the Galois modules which are associated with the modulo p subquotients of crystalline representations with HT weights from [0, p) and prove that both estimates are sharp. The methods we use here are close to the methods from [8,9,10]; one can use also our constructions to show that the estimates from [24] are sharp if e = n = 1. In Subsection 4 we explain the construction of our modification of Breuil's functor V f t .…”

“…In Subsections 3.3-3.5 we shall give a proof based on our characteristic p approach from [8,9,10]. One can also apply the methods from [24].…”

“…Such r * can be always written in the form r * = m/(q − 1), where m ∈ N is prime to p and q is an integral power of p. For the following Lemma cf. [8], Subsection 1.5. Lemma 3.3.…”