Georges Gras

Remarks on K2 of number fields

Abstract Let K2Z where Z is the ring of integers of a number field k. We define a subgroup H20k of index 2r in K2Z (r is the number of real conjugates of k) and study its triviality. For instance, for p = 2 or 3 and for a Galois p-extension k of the base field Q , the triviality of the p-part of H20k depends only on the ramification in k/ Q ; this permits in practice to give the list of such fields. The general case is similar (and effective as soon as classes and units of the base field are known). We verify also that the Birch-Tate conjecture is true at p = 2 for all real abelian fields such that the order of H20k is odd.

Pseudo-mesures p- adiques associées aux fonctions L deQ

It is shown that some standard results concerning the p-adic L- functions, Lp(χ), ofQ(p-divisibilities of 1/2Lp(χ, s), and congruences for 1/2Lp(χ, t)−1/2Lp(χ, s), s, t∈ℤp) are direct consequences of a general structural theorem, based only on the functional properties of the p-adic pseudo-measures and distributions attached to these Lp-functions (essentially the “eulerian” ones). The method suggests that all such divisibilities and congruences are obtained systematically by this way, and are the best possible (in a standard point of view). In particular, these results improve significantly all the known ones.

Basic Tools and Notations

This chapter gives the definitions of the objects which will be used throughout this book. We are thus led to give the main general notations.

Abelian Extensions with Restricted Ramification — Abelian Closure

This chapter deals with the correspondence of class field theory both for finite and infinite extensions; this second aspect, obtained by limiting processes, will enable us to understand the structure of the maximal abelian extension of a number field K (Section 4 of the present chapter). Indeed, since any finite abelian extension of K is contained in a ray class field K(m)res, we have \({\overline K ^{ab}}\, = \,\mathop U\limits_m \,K{(m)^{res}}\), where m ranges in the set of moduli of K.

Reciprocity Maps Existence Theorems

The fundamental results given in this chapter do not necessarily form a sequence of logical steps for a proof of class field theory, but are written and commented so as to be used. This is so true that, as we will see several times, a classical proof consists in deducing local class field theory from global class field theory, as was initiated by Hasse and Schmidt in 1930, and in particular to base some local computations on global arguments (a typical example being the global computation of a local Hilbert symbol in 7.5); however here, in the description of the results, we will go from local to global, which seems more natural.

Cyclic Extensions with Prescribed Ramification

In this chapter we give an approach to the study of ramification in \({\bar K^{ab}}\left[ {{p^e}} \right]/K\), the maximal pro-p-subextension of\({\bar K^{ab}}/K\) with exponent p e , in particular through the study of the ramification possibilities for cyclic extensions of degree p e of K. We will apply these results to the case of the maximal tamely ramified abelian extension \(H_{ta}^{res}/K\) whose structure is always complicated as soon as the invariants \({{\rm A}^{res}}\) or E res are nontrivial. Concerning this, we will have to make an assumption on the group \({\left( {{{\rm A}^{res}}} \right)_p}\) when e ≥ 2, but the case e = 1 can be solved without any assumption.

Invariant Class Groups in p-Ramification Genus Theory

If the arithmetical invariants of K are known, in other words if class field theory over K is explicit, the situation for a finite extension L is a priori completely different, and one usually studies the corresponding invariants of L using several means. This chapter explains the two classical approaches: invariant classes formulas and genus theory.