Ivan Fesenko

Local fields and their extensions

Complete discrete valuation fields Extensions of discrete valuation fields The norm map Local class field theory I Local class field theory II The group of units of local number fields Explicit formulas for the Hilbert symbol Explicit formulas for the Hilbert pairing on formal groups The Milnor

Analysis on arithmetic schemes. II

K

Where the Wild Things Are: Ramification Groups and the Nottingham Group

-groups of a local field Bibliography List of notations Index.

ON DEEPLY RAMIFIED EXTENSIONS

We construct adelic objects for rank two integral structures on arithmetic surfaces and develop measure and integration theory, as well as elements of harmonic analysis. Using the topological Milnor K 2 -delic and K 1 × K 1 -delic objects associated to an arithmetic surface, an adelic zeta integral is defined. Its unramified version is closely related to the square of the zeta function of the surface. For a proper regular model of an elliptic curve over a global field, a two-dimensional version of the theory of Tate and Iwasawa is derived. Using adelic analytic duality and a two-dimensional theta formula, the study of the zeta integral is reduced to the study of a boundary integral term. The work includes first applications to three fundamental properties of the zeta function: its meromorphic continuation and functional equation and a hypothesis on its mean periodicity; the location of its poles and a hypothesis on the permanence of the sign of the fourth logarithmic derivative of a boundary function; and its pole at the central point where the boundary integral explicitly relates the analytic and arithmetic ranks.

Complete Discrete Valuation Fields. Abelian Local Class Field Theories

Group theorists have recently been interested in a pro-p group known as the Nottingham group. This group \(\mathcal{N}\left( {{\mathbb{F}_p}} \right)\)is defined as the set of power series \(t + {a_2}{t^2} + \ldots \in t\mathbb{F}p\left[ {\left[ t \right]} \right]\), the group operation being substitution of one power series in another. Interest in this group reached a peak with the proof by Rachel Camina of the following:

Local fields and their extensions : a constructive approach

Recently J. Coates and R. Greenberg have 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-definitions (1), (3)–(8) of deeply ramified extensions in the first section 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 methods of the third Chapter of [3], where the HasseHerbrand function is defined in terms of the norm map. Some of 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 the second section). 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 (subsection 2.5-2.7). For local fields with finite residue field we give in (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 (2.4) – 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) doesn’t hold (subsection 2.1). In (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

On just infinite pro-p-groups and arithmetically profinite extensions of local fields

This chapter discusses local fields. Local behavior of a 1-dimensional scheme X near a “nice” point x is described by the local ring, whose completion is a complete discrete valuation ring with residue field k ( x ). The class of complete discrete valuation fields is closely connected with global fields—algebraic number and rational function fields. Local class field theory is one of the highest tops of classical algebraic number theory. It establishes a 1–1 correspondence between abelian extensions of a complete discrete valuation field F whose residue field is finite and subgroups in the multiplicative group F *. In the equal-characteristic cases for an arbitrary field K , there exists a complete discrete valuation field F , whose residue field is isomorphic to K . For the unequal-characteristic case: if K is a field of characteristic p , then there is a complete discrete valuation field F of characteristic 0 with prime element p and residue field K .