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

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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


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Language English
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(A0nA0Γn) associated with the (p)-class groups of the cyclotomicZp-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. LetFbe the cyclotomicZp-extension ofF, with finite layersFnfor all integersn, and let us write as usual Γn:= Gal(F/Fnintroduction (and only therein, since this is not). In this standard vocabulary), the natural mapsAnAΓnandA0nA0Γnbetween thep-primary part of the class group (resp. (p)-class group) ofFnand the Γn-fixed points ofA:= limAn(resp.A0:= limA0n) will be calledcapitulation maps . Theirstudy is an interesting problem in Iwasawa theory, especially in connection with Greenberg’s Conjecture, which predicts the triviality ofAandA0in 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 mapsjn:A0n A0Γn, assuming the finiteness ofA0Γn good One(the Gross Conjecture).
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reason for this is that, except in the totally real case (whereA=A0mod-ulo Leopoldt’s Conjecture), the groupsAΓnareingeintW.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 groupsA0nand the co-descent modules (X0)Γn, whereX0= li←−mA0n(Theorem 1.4). section 2, we show that the stabi- In lization starts from leveln0, wheren0is the smallest integernsuch that the natural map (X0)ΓnA0nbecomes surjective; supposing (for simplification) thatFcontains a primitivep-th root of unity, we characterize lim cokerjnas −→ 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. | |pp-adic valuation; γtopological generator of Γ; Λ =Zp[[Γ]] =Zp[[T]] the Iwasawa algebra, the isomorphism above being obtained by mappingγ +to 1T; Σ,Splaces ofFoverp(resp. overpand); A0np-primary part of the (p)-class group ofFn; A0= limA0np-primary part of the (p)-class group ofF; Mn,Mmaximalp-ramified abelian pro-p-extension ofFn (resp. ofF); Lmaximal unramified abelian pro-p-extension ofF; L0n,L0maximal unramified abelian pro-p-extension ofFn (resp. ofF) in which every prime overpsplits completely; Un0group of (p)-units inFn; U0= limUn0group of (p)-units inF; N0field generated overFby thepn-th roots of allεU0for alln0; FnBP,FBPfield of Bertrandias-Payan overFn(resp.F),i.e., the compositum of allp-extensions ofFn(resp.F) which are infinitely embeddable in cyclicp-extensions; GS(Fn) Galois group of the maximumS-ramified extension ofFn;
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XFnGalois group of the maximumS-ramified abelian pro-p-extension ofFn; X= Gal(M/F); X= Gal(L/F); 0 X0= Gal(L/F); N00=N0FBP; Z0= Gal(N0/F); BP= Gal(FBP/F); W= Gal(M/FBP); Tfixed field of torΛX; The fields above fit into the following diagram:
N0
N00
T
F
1 Asymptotical results
M
FBP
In this section, we shall use the theory of adjoints (and co-adjoints) to give an asymptotical description of the cokernels cokerjn. 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, letZp:=ZΛΛp, where Λpis the localization of Λ atp. The kernel of the canonical map Z→ ⊕pZpis the maximal finite Λ-submodule ofZ, which we shall write Z0, and the cokernel is by definition theco-adjointofZ, denotedβ(Z). Let α(Z) := HomZp(β(Z)Qp/Zp) which we make into a Λ-module by defining (σ.f)(y) :=f(σ1.y) forσΛ,yβ(Z) andfα(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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