Invariant generalized ideal classes – structure theorems for p-class groups in p-extensions

Abstract

We give, in Sections 2 and 3, an english translation of: Classes généralisées invariantes, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: Class Field Theory: from theory to practice, SMM, Springer-Verlag, 2 nd corrected printing 2005. We recall, in Section 4, some structure theorems for finite Zp[G]-modules (G ≃ Z/p Z) obtained in: Sur les ℓ-classes d'idéaux dans les extensions cycliques relatives de degré premier ℓ, Annales de l 'Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a p-class group in a cyclic extension of degree p. In Section 5, we apply this to the study of the structure of relative p-class groups of Abelian extensions of prime to p degree, using the Thaine-Ribet-Mazur-Wiles-Kolyvagin "principal theorem", and the notion of "admissible sets of prime numbers" in a cyclic extension of degree p, from: Sur la structure des groupes de classes relatives, Annales de l'Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the p-ramification theory (as dual form of non-ramification theory) and which have become standard in a p-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject.We shall see the field L given, via Class Field Theory, by some Artin group of K (e.g., the Hilbert class field H + K of K associated with the group of principal ideals, in the narrow sense, any ray class field H + K,m associated with a ray group modulo a modulus m of k, in the narrow sense, or more generally any subfield L of these canonical fields, defining Gal(H + K,m /L) by means of a sub-G-module H of the generalized class group Cℓ + K,m ≃ Gal(H + K,m /K)). We intend to give, from the arithmetic of k and elementary local normic computations in K/k, an explicit formula for #Gal(L/K) G = #(Cℓ + K,m /H) G .This order is the degree, over K, of the maximal subfield of L (denoted L ab ) which is Abelian over k.

Algorithmic complexity of Greenberg’s conjecture

“…The results of the paper may be described as follows in two parts: (A) From results of [4,5,6]. The formal algorithm, determining #C kn (whence giving the Iwasawa invariants), computes inductively the classical filtration (C i kn ) i≥0 , where…”

Practice of the Incomplete $p$-Ramification Over a Number Field -- History of Abelian $p$-Ramification

“…(i) We know the fixed point formula in a p-extension L/K (under the conjecture of Leopoldt), but, even in a p-cyclic extension with Galois group G, and contrary to the case of p-class groups (as done in [128] after a very long history), we do not know how to compute the filtration (M i ) i≥0 , of M := T K,P , defined inductively by: (ii) The explicit computation of the p-rank, r K,S (1.3), of A K,S /T K,S for S ⊆ P, is available only in favorable Galois cases with an algebraic reasoning on the canonical representation Q p log S (E K ) given by the Herbrand theorem on units under Leopoldt's conjecture (see § 2.4).…”

“…See the bibliographies of these articles to expand the list of contributions. Of course it is not possible to evoke all the studied families of pro-p-groups having some logical links with S-ramification (with more general sets S regarding P) as, for instance, that of "mild groups" introduced by Labute [12] (and [13] for the case p = 2) dealing with the numbers of generators d(G) and of relations r(G) and defined as follows: (i) We know the fixed point formula in a p-extension L/K (under the conjecture of Leopoldt), but, even in a p-cyclic extension with Galois group G, and contrary to the case of p-class groups (as done in [128] after a very long history), we do not know how to compute the filtration (M i ) i≥0 , of M := T K,P , defined inductively by: M 0 = 1 and M i+1 /M i := (M/M i ) G , for all i ≥ 0.…”

On the $λ$-stability of p-class groups along cyclic p-towers of a number field

“…For known results (all relative to λ = 0), one may cite Fukuda [2] using Iwasawa's theory, Li-Ouyang-Xu-Zhang [10, § 3] working in a non-abelian Galois context, in Kummer towers, via the use of the fixed points formulas [3,4], then Mizusawa [12] above Z 2 -extensions, and Mizusawa-Yamamoto [13] for generalizations, including ramification and splitting conditions, via the Galois theory of pro-p-groups.…”