2 00 1 Ramification in C - Extensions of Local Fields of Characteristic 0

We construct explicitly APF extensions of finite extensions of Qp for which the Galois group is not a p-adic Lie group and which do not have any open subgroup with Zp-quotient. Let K be a finite extension of Qp and L/K be a Galois totally ramified pro-pextension. If the Galois group of L/K is a p-adic Lie group it is known from Sen ([Sen]) that the sequence of upper ramification breaks of L/K is unbounded and that the higher ramification groups are open in Gal(L/K), i.e. in the terminology of [FW], the extension is arithmetically profinite (APF). The more general question concerning the existence of APF extensions of finite extensions of Qp with a Galois group which is a free pro-p-group remains open (the answer is positive in characteristic p, see [Fe]) and we are interested in obtaining methods to construct APF extensions with a non p-adic Lie Galois group. Coates and Greenberg ([CG]) raised the question whether the notion of unbounded sequence of upper ramification breaks (extensions deeply ramified) depends on the existence of an open subgroup of the Galois group having a quotient isomorphic to Zp. Fesenko answered this question negatively in [Fe] by showing the existence of a finitely generated pro-p-group T such that no open subgroup contains a Zp-quotient. He showed that the group T can be realized a the Galois group of a finite extension of Qp. In this paper we look at pro-p-groups such that the lower p-series define a filtration of open subgroups. We show that this type of group can be realized as the Galois group of a deeply ramified extension (and arithmetically profinite). Not only the reason for being deeply ramified is not due to the presence of a Zp-extension but we show that this notion is independent of the nature of the Galois group, i.e. there are totally ramified extensions with such a Galois group which are not deeply ramified. We give a method to construct explicitly APF extensions with a Galois group such as we described above. The method consists in taking a tower of extension Ki/Ki−1 such that Ki+2/Ki−1 is a Galois extension of order p 3 with Galois group having two generators ai and ai+1 and the Galois group of Ki+2/Ki+1 is generated by the commutator [ai+1, ai]. We show how we can control asymptotically the ramification in M = ∪iKi/K. We call the normal closure of such an extension a C-extension (the ”C” standing for commutators) Supported by a Marie Curie Fellowship from the European Union.