Invitation to higher local fields, Part I, section 1: Higher dimensional local fields

We describe our online database of finite extensions of the p-adic numbers, and how it can be used to facilitate local analysis of number fields.

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.

This is an introduction to noncommutative local reciprocity maps for totally ramified Galois extensions with arithmetically profinite group. These maps in general are not homomorphisms but Galois cycles; a description of their image and kernel is included.

This is is a survey of applications of Fontaine's theory of p-adic representations of local fields (Phi-Gamma-modules) to Galois cohomology of local fields and explicit formulas for the Hilbert symbol in relation with two-dimensional local objects.

The objective of this article is to give an introduction to p-adic analysis along the lines of Tate's thesis, as well as incorporating material of a more recent vintage, for example Weil groups.

An exponential homomorphism for a complete discrete valuation field of characteristic zero which relates differential forms and the Milnor K-groups of the field is studied. An application to explicit formulas is included.

Suppose $\mathcal K$ is $N$-dimensional local field of characteristic $p$, $\mathcal G =\mathop{Gal}(\mathcal K_{sep}/\mathcal K)$, $\mathcal G_{<p}$ is the maximal quotient of $\mathcal G$ of period $p$ and nilpotent class $<p$ and $\mathcal K_{<p}\subset \mathcal K_{sep}$ is such that $\mathop{Gal}(\mathcal K_{<p}/\mathcal K)=\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to identify $\mathcal G_{<p}$ with the group $G(\mathcal L)$ obtained from a profinite Lie $\...

This paper has been withdrawn, as it is superseded by arXiv:0806.2122 (Bloch-Kato exponential maps for local fields with imperfect residue fields), which is a more recent version of the same paper.

The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.

In the theory of local fields we have the well-known filtration of unit groups. In this short paper we compute the first cohomology groups of unit gorups for a finite Galois extension of local fields. We show that these cohomology groups are closely related to the ramification indices.