On capitulation cokernels in Iwasawa theory M Le Floc'h A Movahhedi T Nguyen Quang Do

Description

Niveau: Supérieur, Doctorat, Bac+8
On capitulation cokernels in Iwasawa theory M. Le Floc'h A. Movahhedi T. Nguyen Quang Do March 1, 2004 Abstract For a number field F and an odd prime p, we study the “capi- tulation cokernels” coker(A?n ? A??n∞ ) associated with the (p)-class groups of the cyclotomic Zp-extension of F . We prove that these cokernels stabilize and we characterize their direct limit in Iwasawa theoretic terms, thus generalizing previous partial results obtained by H. Ichimura. This problem is intimately related to Greenberg's Conjecture. Introduction Let F be a number field and p an odd prime. Let F∞ be the cyclotomic Zp- extension of F , with finite layers Fn for all integers n, and let us write as usual ?n := Gal(F∞/Fn). In this introduction (and only therein, since this is not standard vocabulary), the natural maps An ? A?n∞ and A?n ? A??n∞ between the p-primary part of the class group (resp. (p)-class group) of Fn and the ?n- fixed points of A∞ := lim??An (resp. A ? ∞ := lim??A ? n) will be called capitulation maps. Their study is an interesting problem in Iwasawa theory, especially in connection with Greenberg's Conjecture, which predicts the triviality of A∞ and A?∞ in the totally real case.

• gross conjecture


• group


• let z


• cokernels stabilize


• retrieve all


• galois group


• xfn galois


• all ? ?


• prime ideal


• bertrandias-payan over

Sujets

• Galois group
• Prime ideal

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 21
• Langue: English

Extrait

On

capitulation cokernels in

M. Le Floc’h

Iwasawa

theory

A. Movahhedi T. Nguyen Quang Do

March 1, 2004

Abstract

For a number fieldFand an odd primep, we study the “capi- tulation cokernels” coker(A 0 n →A 0 Γ ∞ n ) associated with the (p)-class groups of the cyclotomicZ p -extension ofF prove that these. We cokernels stabilize and we characterize their direct limit in Iwasawa theoretic terms, thus generalizing previous partial results obtained by H. Ichimura. This problem is intimately related to Greenberg’s Conjecture.

Introduction

LetFbe a number field andpan odd prime. LetF ∞ be the cyclotomicZ p - extension ofF, with finite layersF n for all integersn, and let us write as usual Γ n := Gal(F ∞ /F n introduction (and only therein, since this is not). In this standard vocabulary), the natural mapsA n →A Γ ∞ n andA 0 n →A 0 Γ ∞ n between thep-primary part of the class group (resp. (p)-class group) ofF n and the Γ n - fixed points ofA ∞ := l − i → mA n (resp.A ∞ 0 := l − i → mA 0 n ) will be calledcapitulation maps . Theirstudy is an interesting problem in Iwasawa theory, especially in connection with Greenberg’s Conjecture, which predicts the triviality ofA ∞ andA ∞ 0 in the totally real case. capitulation kernels have been studied The intensively ([G1], [GJ], [Iw1], [Ku], etc.) but – strangely enough – the same is not true for the cokernels. To the best of our knowledge, the capitulation cokernels have been touched upon by Iwasawa in [Iw2], p. 198, but almost all the substantial results obtained so far are contained in papers by H. Ichimura ([I1, I2, I3]; see also [S]), who only deals with totally real abelian fields of degree prime top, or totally real fields satisfying Leopoldt’s Conjecture in whichpis totally split. → In this article, we shall only consider the capitulation mapsj n :A 0 n A 0 Γ ∞ n , assuming the finiteness ofA 0 Γ ∞ n  good One(the Gross Conjecture).

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reason for this is that, except in the totally real case (whereA ∞ =A 0 ∞ mod- ulo Leopoldt’s Conjecture), the groupsA Γ n areingeinfitW.etreneonla ∞ completely solve the problem of describing the asymptotical behaviour of the capitulation cokernels in the general case (and of course we retrieve all previously known results). More precisely, in Section 1, we show that the ca- pitulation cokernels stabilize and their direct limit measures the asymptotical deviation between the class groupsA 0 n and the co-descent modules (X 0 ∞ ) Γ n , whereX 0 ∞ = li ←− mA 0 n (Theorem 1.4). section 2, we show that the stabi- In lization starts from leveln 0 , wheren 0 is the smallest integernsuch that the natural map (X 0 ∞ ) Γ n →A 0 n becomes surjective; supposing (for simplification) thatFcontains a primitivep-th root of unity, we characterize lim cokerj n as −→ the Kummer dual of a certain Galois module related to embedding problems in Iwasawa theory (Theorem 2.4). In Section 3, we take another look at this module, using Gross kernels (Theorem 3.2). Finally, the abelian semi-simple case (i.e., whenFis abelian of degree prime topoverQ) is studied in detail: explicit formulæ and numerical examples are given in Section 4.

First, we set up some notations. | | p p-adic valuation; γtopological generator of Γ; Λ =Z p [[Γ]] = ∼ Z p [[T]] the Iwasawa algebra, the isomorphism above being obtained by mappingγ +to 1T; Σ,Splaces ofFoverp(resp. overpand∞); A 0 n p-primary part of the (p)-class group ofF n ; A 0 ∞ = l − i → mA 0 n p-primary part of the (p)-class group ofF ∞ ; M n ,M ∞ maximalp-ramified abelian pro-p-extension ofF n (resp. ofF ∞ ); L ∞ maximal unramified abelian pro-p-extension ofF ∞ ; L 0 n ,L 0 ∞ maximal unramified abelian pro-p-extension ofF n (resp. ofF ∞ ) in which every prime overpsplits completely; U n 0 group of (p)-units inF n ; U ∞ 0 = l − i → mU n 0 group of (p)-units inF ∞ ; N 0 ∞ field generated overF ∞ by thep n -th roots of allε∈U ∞ 0 for alln≥0; F nBP ,F ∞ BP field of Bertrandias-Payan overF n (resp.F ∞ ),i.e., the compositum of allp-extensions ofF n (resp.F ∞ ) which are infinitely embeddable in cyclicp-extensions; G S (F n ) Galois group of the maximumS-ramified extension ofF n ;

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X F n Galois group of the maximumS-ramified abelian pro-p-extension ofF n ; X ∞ = Gal(M ∞ /F ∞ ); X ∞ = Gal(L ∞ /F ∞ ); 0 X ∞ 0 = Gal(L ∞ /F ∞ ); N ∞ 00 =N ∞ 0 ∩F ∞ BP ; Z 0 ∞ = Gal(N ∞ 0 /F ∞ ); BP ∞ = Gal(F BP /F ∞ ); ∞ W ∞ = Gal(M ∞ /F ∞ BP ); T ∞ fixed field of tor Λ X ∞ ; The fields above fit into the following diagram:

N ∞ 0

N 00 ∞

T ∞

F ∞

1 Asymptotical results

M ∞

F ∞ BP

In this section, we shall use the theory of adjoints (and co-adjoints) to give an asymptotical description of the cokernels cokerj n . For further details about adjoints, see [Iw1] or [W]. Throughout this paper, we will only consider finitely generated Λ-modules. LetZbe a (finitely generated) torsion Λ- module. For each prime idealpof height one in Λ, letZ p :=Z⊗ Λ Λ p , where Λ p is the localization of Λ atp. The kernel of the canonical map Z→ ⊕ p Z p is the maximal finite Λ-submodule ofZ, which we shall write Z 0 , and the cokernel is by definition theco-adjointofZ, denotedβ(Z). Let α(Z) := Hom Z p (β(Z)Q p /Z p ) which we make into a Λ-module by defining (σ.f)(y) :=f(σ −1 .y) forσ∈Λ,y∈β(Z) andf∈α(Z). We then see that α(Z) is a finitely generated Λ-torsion module, called theadjointofZ. We recall two important facts about adjoints that we shall subsequently use: an adjoint has no non-trivial finite submodules andα(Z) is pseudo- isomorphic toZ (−1) , where (−1) that the Galois action has been in-means

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