ON DEEPLY RAMIFIED EXTENSIONS

Abstract

Recently J. Coates and R. Greenberg introduced a new important class of extensions of local number fields with finite residue field which they call deeply ramified fields. These extensions play an essential role in their study of the arithmetics of abelian varieties over local fields with finite residue fields [1].

The first aim of this paper is to provide an ‘elementary’ treatment of deeply ramified extensions of local fields with arbitrary perfect residue fields using a method different from the original approach of Coates and Greenberg. Equivalent properties (1), (3)–(8) of deeply ramified extensions in Section 1 are due to them, and for their proofs a presentation of the different as an integral was involved. We translate the most important constructions into the language of the Hasse–Herbrand function (Section 1) and then apply the methods of [3, Chapter 3], where the Hasse–Herbrand function is defined in terms of the norm map. Some of the properties of deeply ramified extensions (two implications in the language of this paper) have been already studied by M. Matignon [7] and J. Fresnel, M. Matignon [5] for different purposes (see Remark 1.6 and the beginning of Section 2).

The second aim of this text is to expose relations among classes of deeply ramified extensions, that of arithmetically profinite extensions (Subsections 2.1–2.4) and that of p-adic Lie extensions (Subsections 2.5–2.7). For local fields with finite residue field we give in Subsection 2.2 an example of a Galois deeply ramified extension with infinite residue extension in which every Galois deeply ramified subextension is not arithmetically profinite; and in Subsection 2.4 we give an example of a Galois deeply ramified extension with finite residue field extension and a nondiscrete set of breaks (that means that this extension is not arithmetically profinite). The main result is that for local fields with finite residue field the class of Galois deeply ramified extensions with finite residue extension and a discrete set of breaks coincides with the class of infinite Galois arithmetically profinite extensions (Proposition 2.3). However, in the case of the fields with infinite residue fields Proposition 2.3 does not hold (Subsection 2.1). In Subsection 2.5 we construct an example that shows that the class of infinite Galois totally ramified arithmetically profinite extensions is strictly larger than the class of the most natural arithmetic origin – the class of totally ramified p-adic Lie extensions. Subsection 2.6 contains an example of a Galois deeply and totally ramified extension L of a local field F with finite residue field such that the norm group of L/E is of finite index in E*.