The arithmetic of \(\ell\)-extensions.
(L’arithmétique des \(\ell\)-extensions.)

*(French)*Zbl 0601.12002
Publ. Math. Fac. Sci. Besançon, Théor. Nombres Années 1984/85-1985/86, No. 1, 349 pp. (1986).

This thesis is devoted to some aspects of algebraic number theory concerning the main fundamental invariants of number fields, like class groups or \(S\)-class groups, units or \(S\)-units, tame and Hilbert kernels of \(K_ 2\), various Galois groups attached to \(S\)-ramification theory, all this in comparison with Leopoldt and Gross conjectures.

The main originality of this work is to deal with finite or infinite class field theory, restricted essentially to the abelian pro-\(\ell\)-groups attached to a number field \(K\), for a fixed prime \(\ell\). By this way, the author is able to include the Iwasawa theory aspects, without other specific techniques. More precisely, the techniques introduced are mainly the two following ones:

(i) In chap. I, the description of the Galois group \(G_ K\) of the maximal abelian \(\ell\)-extension of the number field \(K\), as the quotient of the ”\(\ell\)-idèles”: \[ {\mathcal J}_ K=\prod\varprojlim_n K^{\times}_ p/K_ p^{\times \ell^ n} \] (where the product, over the set of places \({\mathfrak p}\) of \(K\), is the restricted product corresponding to the groups of \(\ell\)-units \(\varprojlim_n U_{{\mathfrak p}}/U_{{\mathfrak p}}^{\ell^ n})\), by the subgroup of ”principal \(\ell\)-idèles”: \({\mathcal R}_ K={\mathbb Z}_{\ell}\otimes K^{\times}.\)

This quotient, \({\mathcal J}_ K/{\mathcal R}_ K\), is a topological \({\mathbb Z}_{\ell}\)-module which gives rise to a more convenient description of standard class field theory, for abelian pro-\(\ell\)- extensions (Galois groups, normic properties, genera theory...). This point of view is first applied to give some general results on duality between the Kummerian description of abelian pro-\(\ell\)-extensions and the theory of J. Tate of the \(K_ 2\) [Invent. Math. 36, 257–274 (1976; Zbl 0359.12011)];

(ii) in chap. II, the author gives the definition of the group of ”\(\ell\)-infinitesimal” numbers of \(K\), defined as the kernel of the canonical map: \[ {\mathbb Z}_{\ell}\otimes K^{\times}\quad \to \prod_{{\mathfrak L}| \ell}\varprojlim_nK^{\times}_{{\mathfrak L}}/K_{{\mathfrak L}}^{\times \ell^ n} \] (generalization of Leopoldt kernel about the subgroup \(E_ K\subset K^{\times}\) of units of \(K\)). This formalism is well adapted to the previous description of class field theory and gives rise to a general conjecture about monogenic Galois submodules of \(K^{\times}\); it contains Gross and Leopoldt’s classical conjectures; the author proves his conjecture when \(K/{\mathbb Q}\) is abelian and, more generally, when \({\mathbb Q}_{\ell}[\text{Gal}(K/ {\mathbb Q})]\) is a direct product of fields.

In the sect. 2 of chap. II the author gives the arithmetical properties of various infinitesimal class groups and interprets for instance, for \(S\cap T=\emptyset\), the Galois group of the maximal \(T\)-ramified, \(S\)- decomposed abelian \(\ell\)-extension of \(K\), and gives results of duality in this situation of ”\(\ell\)-class-field theory”.

These infinitesimal class groups give rise to a theory of invariant classes (in a Galois extension), and to a genera theory given by the author in full generality.

In the chap. III, the author gives probably the largest generalization of genera theory which may be desired for the applications in number theory, and everyone will find the material for its specific need.

Chap. IV deals with Iwasawa theory aspects (when the \({\mathbb Z}_ p\)- extension of \(K\) is metabelian over a subfield of \(K\)); the author gives a new approach for the general case of noetherian \(\Lambda\)-modules \(X\), where \(\Lambda ={\mathbb Z}_ p[[ T]]\); he uses first the classical results of J.-P. Serre [Sémin. Bourbaki 11 (1958/59), No. 174 (1959; Zbl 0119.27603)] to give the general structure of \(X\), and he introduces the concept of parametrized sequences of finite \({\mathbb Z}_ p\)-modules whose main interest is to give generalizations of Iwasawa’s formulas. For the main applications see the author [Sémin. Théor. Nombres, Univ. Bordeaux I 1983/84, Exp. No. 23 (1984; Zbl 0545.12006), and Publ. Math. Fac. Sci. Besançon, Théor. Nombres 1983/84 (1984; Zbl 0567.12008)].

The redaction of this thesis is rather dense and the results are given in full generality; moreover this text of 350 pages is written in French, so it may be conjectured that many results of this work (main or more technical ones) will be ignored during a certain time.

It is not possible to give here all the connections with known results or to put more precisely how this work improves them (the bibliography has 170 references, and the author explains, in general with many care, these relations) but we can briefly evoke some specific works of: Coates, Kuz’min, Nguyen Quang Do, Wingberg, the reviewer (on duality involving Tate theory on \(K_ 2\), on \(S\)-ramification theory), Bertrandias-Payan, Federer, Gillard, Heider (on Kummer theory, Leopoldt or Gross conjectures, capitulation of classes), Emsalem, Waldschmidt (on transcendental aspects), Coates, Greenberg, Kisilevsky (on various aspects of Iwasawa theory), Browkin, Hurrelbrink, Kolster, the reviewer (on class field theory aspects of \(K_ 2\)).

The main originality of this work is to deal with finite or infinite class field theory, restricted essentially to the abelian pro-\(\ell\)-groups attached to a number field \(K\), for a fixed prime \(\ell\). By this way, the author is able to include the Iwasawa theory aspects, without other specific techniques. More precisely, the techniques introduced are mainly the two following ones:

(i) In chap. I, the description of the Galois group \(G_ K\) of the maximal abelian \(\ell\)-extension of the number field \(K\), as the quotient of the ”\(\ell\)-idèles”: \[ {\mathcal J}_ K=\prod\varprojlim_n K^{\times}_ p/K_ p^{\times \ell^ n} \] (where the product, over the set of places \({\mathfrak p}\) of \(K\), is the restricted product corresponding to the groups of \(\ell\)-units \(\varprojlim_n U_{{\mathfrak p}}/U_{{\mathfrak p}}^{\ell^ n})\), by the subgroup of ”principal \(\ell\)-idèles”: \({\mathcal R}_ K={\mathbb Z}_{\ell}\otimes K^{\times}.\)

This quotient, \({\mathcal J}_ K/{\mathcal R}_ K\), is a topological \({\mathbb Z}_{\ell}\)-module which gives rise to a more convenient description of standard class field theory, for abelian pro-\(\ell\)- extensions (Galois groups, normic properties, genera theory...). This point of view is first applied to give some general results on duality between the Kummerian description of abelian pro-\(\ell\)-extensions and the theory of J. Tate of the \(K_ 2\) [Invent. Math. 36, 257–274 (1976; Zbl 0359.12011)];

(ii) in chap. II, the author gives the definition of the group of ”\(\ell\)-infinitesimal” numbers of \(K\), defined as the kernel of the canonical map: \[ {\mathbb Z}_{\ell}\otimes K^{\times}\quad \to \prod_{{\mathfrak L}| \ell}\varprojlim_nK^{\times}_{{\mathfrak L}}/K_{{\mathfrak L}}^{\times \ell^ n} \] (generalization of Leopoldt kernel about the subgroup \(E_ K\subset K^{\times}\) of units of \(K\)). This formalism is well adapted to the previous description of class field theory and gives rise to a general conjecture about monogenic Galois submodules of \(K^{\times}\); it contains Gross and Leopoldt’s classical conjectures; the author proves his conjecture when \(K/{\mathbb Q}\) is abelian and, more generally, when \({\mathbb Q}_{\ell}[\text{Gal}(K/ {\mathbb Q})]\) is a direct product of fields.

In the sect. 2 of chap. II the author gives the arithmetical properties of various infinitesimal class groups and interprets for instance, for \(S\cap T=\emptyset\), the Galois group of the maximal \(T\)-ramified, \(S\)- decomposed abelian \(\ell\)-extension of \(K\), and gives results of duality in this situation of ”\(\ell\)-class-field theory”.

These infinitesimal class groups give rise to a theory of invariant classes (in a Galois extension), and to a genera theory given by the author in full generality.

In the chap. III, the author gives probably the largest generalization of genera theory which may be desired for the applications in number theory, and everyone will find the material for its specific need.

Chap. IV deals with Iwasawa theory aspects (when the \({\mathbb Z}_ p\)- extension of \(K\) is metabelian over a subfield of \(K\)); the author gives a new approach for the general case of noetherian \(\Lambda\)-modules \(X\), where \(\Lambda ={\mathbb Z}_ p[[ T]]\); he uses first the classical results of J.-P. Serre [Sémin. Bourbaki 11 (1958/59), No. 174 (1959; Zbl 0119.27603)] to give the general structure of \(X\), and he introduces the concept of parametrized sequences of finite \({\mathbb Z}_ p\)-modules whose main interest is to give generalizations of Iwasawa’s formulas. For the main applications see the author [Sémin. Théor. Nombres, Univ. Bordeaux I 1983/84, Exp. No. 23 (1984; Zbl 0545.12006), and Publ. Math. Fac. Sci. Besançon, Théor. Nombres 1983/84 (1984; Zbl 0567.12008)].

The redaction of this thesis is rather dense and the results are given in full generality; moreover this text of 350 pages is written in French, so it may be conjectured that many results of this work (main or more technical ones) will be ignored during a certain time.

It is not possible to give here all the connections with known results or to put more precisely how this work improves them (the bibliography has 170 references, and the author explains, in general with many care, these relations) but we can briefly evoke some specific works of: Coates, Kuz’min, Nguyen Quang Do, Wingberg, the reviewer (on duality involving Tate theory on \(K_ 2\), on \(S\)-ramification theory), Bertrandias-Payan, Federer, Gillard, Heider (on Kummer theory, Leopoldt or Gross conjectures, capitulation of classes), Emsalem, Waldschmidt (on transcendental aspects), Coates, Greenberg, Kisilevsky (on various aspects of Iwasawa theory), Browkin, Hurrelbrink, Kolster, the reviewer (on class field theory aspects of \(K_ 2\)).

Reviewer: G. Gras

##### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R23 | Iwasawa theory |

11R37 | Class field theory |

11R18 | Cyclotomic extensions |

11R70 | \(K\)-theory of global fields |

11S15 | Ramification and extension theory |