Takashi Hara
In this paper, we prove the Iwasawa main conjecture of totally real fields for certain specific non-commutative $p$-adic Lie extensions, using the integral logarithms introduced by Oliver and Taylor. Our result gives certain generalization of Kazuya Kato's proof of the main conjecture for Galois extensions of Heisenberg type.
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The purpose of this paper is to prove the main conjecture of non-commutative Iwasawa theory for p-adic Lie extensions, for an odd prime p, of totally real number fields assuming that the Iwasawa mu invariant of a certain totally real number field vanishes.
We completely calculate the Fitting ideal of the classical $p$-ramified Iwasawa module for any abelian extension $K/k$ of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results by the first and third authors where we had to assume that only $p$-adic places may ramify in $K/k$. One of the important ingredients is the computation of some complexes in appropriate derived categories.
We prove a `Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of non-commutative Iwasawa algebras. We use these results to describe the precise connection between the main conjecture of non-commutative Iwasawa theory (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture.
We study special values of L-functions of elliptic curves over Q twisted by Artin representations that factor through a false Tate curve extension $Q(μ_p^\infty,\sqrt[p^\infty]{m})/Q$. In this setting, we explain how to compute L-functions and the corresponding Iwasawa-theoretic invariants of non-abelian twists of elliptic curves. Our results provide both theoretical and computational evidence for the main conjecture of non-commutative Iwasawa theory.
Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\mathbbZ_p$ . First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $Λ(G)$, of $G$. We also prove some results about certain localisation of $Λ(G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\infty}$ be an admissible $p$-adic Lie extension of $F$ with Galois group $G$. The computation of the Whitehead groups are used to …
We discuss the formalism of Iwasawa theory descent in the setting of the localized K_1-groups of Fukaya and Kato. We then prove interpolation formulas for the `leading terms' of the global Zeta isomorphisms that are associated to certain Tate motives and of the p-adic L-functions that are associated to certain critical motives.
We give a survey on the relationship between non-commutative Iwasawa theory and the Equivariant Tamagawa Number Conjecture following the work of (in alphabetical order) Burns/Flach, Coates/Sujatha, Fukaya/Kato, Huber/Kings and others.
Main theorem of [Buyukboduk, arXiv:0706.0377v1] suggests that it should be possible to lift the Kolyvagin systems of Stark units constructed in [Buyukboduk, arXiv:math/0703426v1] to a Kolyvagin system over the cyclotomic Iwasawa algebra. This is what we prove in this paper. This construction gives the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement of local Iwasawa theory. This s…
We discuss the triviality of some classical Iwasawa modules in connection with the notion of $\ell$-rationality for totally $\ell$-adic number fields.
We establish the Iwasawa main conjecture for semi-stable abelian varieties over a function field of characteristic $p$ under certain restrictive assumptions. Namely we consider $p$-torsion free $p$-adic Lie extensions of the base field which contain the constant $\mathbb Z_p$-extension and are everywhere unramified. Under the classical $μ=0$ hypothesis we give a proof which mainly relies on the interpretation of the Selmer complex in terms of $p$-adic cohomology [TV] together with the trace formulas of [EL1].
Assuming that Iwasawa's $μ_{K/k}$-invariant vanishes, we prove the 'main conjecture' of equivariant Iwasawa theory, at odd prime numbers $l$, for arbitrary extensions $K/k$ of totally real number fields, up to its uniqueness assertion.
We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for $\ell$-adic representations of the Galois group of a function field of characteristic $p$. We also prove a functional equation for the resulting non-commutative $L$-functions. As corollaries, we obtain non-commutative generalisations of the main conjecture for Picard-$1$-motives of Greither and Popescu and a main conjecture for abelian varieties over function fields in precise analogy to the $\operatorname{GL}_2$ main conjecture of Coates, Fukaya, Kato, Sujatha and Venjakob.
It is well known that, for any finitely generated torsion module M over the Iwasawa algebra Z_p [[Γ ]], where Γ is isomorphic to Z_p, there exists a continuous p-adic character ρ of Γ such that, for every open subgroup U of Γ, the group of U-coinvariants M(ρ)_U is finite; here M( ρ) denotes the twist of M by ρ. This twisting lemma was already applied to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a non commutative generalization of this twisting lemma replacing torsion modules over Z_p [[ Γ ]] by ce…
In this paper, we construct a higher rank Euler system for the multiplicative group over a totally real field by using the Iwasawa main conjecture proved by Wiles. A key ingredient of the construction is to generalize the notion of the characteristic ideal. Under certain technical assumptions, we prove that all higher Fitting ideals of a certain $p$-ramified Iwasawa module are described by analytic invariants canonically associated with Stickelberger elements.
We formulate a new equivariant Main Conjecture in Iwasawa theory of number fields and study its properties. This is done for arbitrary one-dimensional $p$-adic Lie extensions $L_\infty/K$ containing the cyclotomic $\mathbb{Z}_p$-extension $K_\infty$ of the base field. As opposed to existing conjectures in the area, no requirement that $L_\infty/K$ be abelian or that $L_\infty$ be totally real is imposed. We prove the independence of the Main Conjecture of essentially all of its parameters and explore its functorial behaviour. It is furthermore shown that, to a large extent, this new conjecture…
The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kat…
Let p be an odd prime. We give an unconditional proof of the equivariant Iwasawa main conjecture for totally real fields for an infinite class of one-dimensional non-abelian p-adic Lie extensions. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant.
The article is withdrawn. There is a gap in the proof of the vanishing of Iwasawa mu-invariant for the cyclotomic Z_p extension of a CM field, abelian over a totally real number field.
In this paper we use ideas of the non-abelian Iwasawa main conjecture to prove a result about the first Galois cohomology of continuous Galois modules V_p(j) for large Tate-twist j and V_p a Q_p vector space. We show that under a technical hypothesis that H^1(K, V_p(j)) is generated by twists of norm-compatible units in an infinite tower of number fields with elements in V_p(j-1). This confirms a consequence of the non-abelian Iwasawa main conjecture. Using the "Bloch-Kato-conjecture" a similar result is proven for motivic cohomology with finite coefficients.
Let $K$ be a totally real number field of degree $r$. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{2}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. The goal of this paper is to compare the characteristic ideal of the $χ$-quotient of the projective limit of the narrow class groups to the $χ$-quotient of the projective limit of the $r$-th exterior power of totally positive units modulo a subgroup of Rubin-Stark units, for some $\overline{\mathbb{Q}_{2}}$-irreducible characters $χ$ of $\mathrm{Gal}(L_{\infty}/K_{\infty})$.
An Integral Equivariant Refinement of the Iwasawa Main Conjecture for Totally Real Fields
本文在全实数域的 Abel、CM 扩张下，通过计算非空 $T$ 对应的 $X_S^{T,-}$ 的 Fitting ideal，给出了 Wiles 全实域 Iwasawa 主猜想的一个整等变细化，并扩展了多项已有猜想与结果。
On characteristic elements modulo $p$ in non-commutative Iwasawa theory
本文在非交换岩泽理论中引入特征元的模 $p$ 概念并研究其性质，进而应用于模形式张量积的 Greenberg Selmer 群，其中 $p$ 为其中一个模形式的 Eisenstein 素数。
We produce generalizations of Iwasawa's `Riemann-Hurwitz' formula for number fields. These generalizations apply to cyclic extensions of number fields of degree p^n for any positive integer n. We first deduce some congruences and inequalities and then use these formulas to establish a vanishing criterion for Iwasawa λ-invariants which generalizes a result of Takashi Fukuda et. al. for totally real number fields.
Rarity of pseudo-null Iwasawa modules for $p$-adic Lie extensions
该研究得到全实域$p$-进李扩张下$p$-分歧岩泽模伪零性的充要条件，应用于CM域非分歧岩泽模负分量的相关问题，证明这类伪零性十分稀有。
On natural complexes in Iwasawa theory
本文针对非全实数域扩张的等变岩泽主猜想中的两类自然复形展开研究，证明在弱Leopoldt猜想成立的条件下，二者在岩泽模导出范畴中同构，推进了岩泽理论相关猜想的研究。
The main conjectures in Iwasawa theory predict the relationship between the Iwasawa modules and the $p$-adic $L$-functions. Using a certain proved formulation of the main conjecture, Greither and Kurihara described explicitly the (initial) Fitting ideals of the Iwasawa modules for the cyclotomic $\mathbb{Z}_p$-extensions of finite abelian extensions of totally real fields. In this paper, we generalize the algebraic theory behind their work by developing the theory of ``shifts of Fitting invariants.'' As applications to Iwasawa theory, we obtain a noncommutative version and a two-variable versi…
Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We reduce the conjectured triviality of the reduced Whitehead group SK_1(QG) of the algebra QG=Quot(ΛG) with the Iwasawa algebra ΛG = Z_l[[G]] to the case of pro-l Galois groups G and finite unramified coefficient extensions.
On non-commutative Iwasawa theory and derivatives of Euler systems
本文利用约化行列式函子描述非交换代数的相对 $K_0$-群，结合 zeta element 推广，提出了非交换 $p$-进岩泽理论的主猜想及 Rubin-Stark 欧拉系统的高阶导数公式，为等变 Tamagawa 数猜想提供了广泛推广的证据。
In an earlier paper the author proved one divisibility of Perrin- Riou's Iwasawa main conjecture for Heegner points on elliptic curves. In the present paper, that result is generalized to abelian varieties of GL2-type (i.e. abelian varieties with real multiplication defined over totally real fields) under the hypothesis that the abelian variety is associated to a Hilbert modular form via a construction of Zhang.
The paper starts out from pseudomeasures (in the sense of Serre) which hold the arithmetic properties of the abelian $l$-adic Artin $L$-functions over totally real number fields. In order to generalize to non-abelian $l$-adic $L$-functions, these abelian pseudomeasures must satisfy congruences which are introduced but not yet known to be true. The relation to the ``equivariant main conjecture'' of Iwasawa theory is discussed.
Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We give a full description of the algebraic structure of the semisimple algebra QG=Quot(ΛG) for pro-l Galois groups G with ΛG = Z_l[[G]] the Iwasawa algebra. We moreover compute the cohomological dimension of the centres of the Wedderburn components of QG and state some results on the completion of QG.
In this article I generalise previous computations (by K. Kato, T. Hara and myself) of K_1 (only up to p-power torsion) of p-adic group rings of finite non-abelian p-groups in terms of p-adic group rings of abelian subquotients of the group. Such computation have applications in non-commutative Iwasawa theory due to a strategy proposed by D. Burns, K. Kato (and a modification by T. Hara) for deducing non-commutative main conjectures from commutative main conjectures and certain congruences between special L-values. However, I do not say anything about Iwasawa theory in this article. The detail…
It is well known that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb Z_p[[Γ]]$, where $Γ\cong \mathbb Z_p$, there exists a continuous $p$-adic character $ρ$ of $Γ$ such that, for the twist $M(ρ)$ of $M$, the $Γ_n := Γ^{p^n}$ Euler characteristic, i.e. $χ(Γ_n, M(ρ))$, is finite for every $n$. We prove a generalization of this result by considering modules over the Iwasawa algebra of a general $p$-adic Lie group $G$, instead of $Γ$. We relate this twisted Euler characteristic to the evaluation of the {\it Akashi series} at the twist and in turn use it to indicate…
We describe an explicit `higher rank' Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of general number fields. We then show that this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture. As a first application of this approach, we use it to prove new cases of the conjecture for Tate motives over natural families of abelian CM extensions of totally real fields for which the relevant $p$-adic $L$-functions possess trivial zeroes.
We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet'…
In this short note we partially answer a question of Fukaya and Kato by constructing a $q$-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non-commutative p-adic zeta function.
We discuss three different formulations of the equivariant Iwasawa main conjecture attached to an extension K/k of totally real fields with Galois group G, where k is a number field and G is a p-adic Lie group of dimension 1 for an odd prime p. All these formulations are equivalent and hold if Iwasawa's μ-invariant vanishes. Under mild hypotheses, we use this to prove non-abelian generalizations of Brumer's conjecture, the Brumer-Stark conjecture and a strong version of the Coates-Sinnott conjecture provided that μ= 0.
We show that for an odd prime p, the p-primary parts of refinements of the (imprimitive) non-abelian Brumer and Brumer-Stark conjectures are implied by the equivariant Iwasawa main conjecture (EIMC) for totally real fields. Crucially, this result does not depend on the vanishing of the relevant Iwasawa mu-invariant. In combination with the authors' previous work on the EIMC, this leads to unconditional proofs of the non-abelian Brumer and Brumer-Stark conjectures in many new cases.
On the non-commutative Iwasawa main conjecture for voltage covers of graphs
本文研究取值于一致 $p$-进 Lie 群的电压赋值所产生的图电压覆盖，证明 Picard 群射影极限的非交换 Iwasawa 主猜想，并给出 Jacobian 情形及 $\mathfrak{M}_H(G)$ 性质的判别条件。
The goal of this paper is to illustrate different approaches to understand Euler characteristics in the setting of totally real commutative and non-commutative Iwasawa theory. In addition to this, and in the spirit of Hesselholt and Mitchell, we show how these objects interact with homotopy theoretic invariants such as ($K(1)$-local) $K$-theory and Topological Cyclic homology.
In this paper, we prove a result on the $2$-adic logarithm of the fundamental unit of the field $\mathbb{Q}(\sqrt[4]{-q}) $, where $q\equiv 3\bmod 4$ is a prime. When $q\equiv 15\bmod 16$, this result confirms a speculation of Coates-Li and has consequences for certain Iwasawa modules arising in their work.
Kurihara established a refinement of the minus-part of the Iwasawa main conjecture for totally real number fields using the higher Fitting ideals. In this paper, by using Kurihara's methods and Mazur-Rubin theory, we study the higher Fitting ideals of the plus-part of Iwasawa modules associated the cyclotomic Z_p-extension of abelian fields for an odd prime number p. We define the higher cyclotomic ideals C_i for any non-negative integer i, which are ideals of the Iwasawa algebra defined by the Kolyvagin derivative classes of circular units, and prove that they give upper and lower bounds of t…
In this paper the new techniques and results concerning the structure theory of modules over non-commutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions K of number fields k "up to pseudo-isomorphism". In particular, a close relationship is revealed between the Selmer group of abelian varieties, the Galois group of the maximal abelian unramified p-extension of K as well as the Galois group of the maximal abelian outside S unramified p-extension where S is a finite set of certain places of k. Moreover, we determine the Galois module structure …
In this paper, we study the Iwasawa theory of a motive whose Hodge-Tate weights are $0$ or $1$ (thence in practice, of a motive associated to an abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number field that is either totally real or CM. In particular, under certain technical assumptions, we construct Sprung-type Coleman maps on the local Iwasawa cohomology groups and use them to define integral $p$-adic $L$-functions and (one unconditionally and other conjecturally) cotorsion Selmer groups. This allows us to reformulate Perrin-Riou's main conjecture in terms of the…
This paper contains a complete proof of Fukaya's and Kato's epsilon$-isomorphism conjecture in [23] for invertible Λ-modules (the case of V = V_0(r) where V_0 is unramified of dimension 1). Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of G_{Q_p} in [27], but apart from fixing some sign-ambiguities in (loc.\ cit.) we use the theory of (ϕ,Γ)-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of [23] and extend i…
On Non-Noetherian Iwasawa Theory
本文证明了一类非Noetherian交换环上有限生成挠模的一般结构定理，并首次将其应用于研究有限域上曲线的$\mathbb{G}_m$的Weil-étale上同调群。
We construct a new class of Iwasawa modules, which are the number field analogues of the p-adic realizations of the Picard 1-motives constructed by Deligne in the 1970s and studied extensively from a Galois module structure point of view in our recent work. We prove that the new Iwasawa modules are of projective dimension 1 over the appropriate profinite group rings. In the abelian case, we prove an Equivariant Main Conjecture, identifying the first Fitting ideal of the Iwasawa module in question over the appropriate profinite group ring with the principal ideal generated by a certain equivari…
We prove a unicity result for the $L$-functions appearing in the non-commutative Iwasawa main conjecture over totally real fields. We then consider continuous representations $ρ$ of the absolute Galois group of a totally real field $F$ on adic rings in the sense of Fukaya and Kato. Using our unicity result, we show that there exists a unique sensible definition of a non-commutative $L$-function for any such $ρ$ that factors through the Galois group of a possibly infinite totally real extension. We also consider the case of CM-extensions and discuss the relation with the equivariant main conjec…
We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilb…
Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. In this article, we extend results of \cite{Kazim109} relating characteristic ideal of the $χ$-quotient of the projective limit of the ideal class groups to the $χ$-quotient of the projective limit of the $r$-th exterior power of units modulo Rubin-Stark units, in the non semi-simple case, for some $\overline{\mathbb{Q}_{p}}$-irreductible ch…
An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications
本文无条件证明了 Galois 群含交换 Sylow $p$-子群的一维 $p$-进 Lie 扩张下全实数域的等变岩泽主猜想，并由此导出 Coates-Sinnott 猜想及等变玉川数猜想的新结果。
On zeta elements and functional equations for Tate motives over totally real fields
本文研究了完全实数域上塔特动机的伊瓦萨理论，构造了插值L函数在正整数处值的zeta元素，并建立了与L函数值相关的双 dual 上同调群中的canonical元素。
Let $Λ$ (isomorphic to $\mathbb{Z}_p[[T]]$) denote the usual Iwasawa algebra and $G$ denote the Galois group of a finite Galois extension $L/K$ of totally real fields. When the non-primitive Iwasawa module over the cyclotomic $\mathbb{Z}_p$-extension has a free resolution of length one over the group ring $Λ[G]$, we prove that the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive $p$-adic $L$-function (which is an element of a $K_1$-group) in a maximal $Λ$-order. This integrality result involves a careful study of the Dieudonné det…
We develop the topological polylogarithm which provides an integral version of Nori's Eisenstein cohomology classes for $GL_n(\mathbf{Z})$ and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of $L$-functions of totally real fields and a construction of the associated $p$-adic $L$-function. Using a result of Graf, we also apply this to prove some integrality and $p$-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.
The goal of this article is two-fold: First, to prove a (two-variable) main conjecture for a CM field $F$ without assuming the $p$-ordinary hypothesis of Katz, making use of what we call the Rubin-Stark $\mathcal{L}$-restricted Kolyvagin systems which is constructed out of the conjectural Rubin-Stark Euler system of rank $g$. (We are also able to obtain weaker unconditional results in this direction.) Second objective is to prove the Park-Shahabi plus/minus main conjecture for a CM elliptic curve $E$ defined over a general totally real field again using (a twist of the) Rubin-Stark Kolyvagin s…
We improve upon the recent keystone result of Dasgupta-Kakde on the $\Bbb Z[G(H/F)]^-$-Fitting ideals of certain Selmer modules $Sel_S^T(H)^-$ associated to an abelian, CM extension $H/F$ of a totally real number field $F$ and use this to compute the $\Bbb Z_p[[G(H_\infty/F)]]^-$-Fitting ideal of the Iwasawa module analogues $Sel_S^T(H_\infty)_p^-$ of these Selmer modules, where $H_\infty$ is the cyclotomic $\Bbb Z_p$-extension of $H$, for an odd prime $p$. Our main Iwasawa theoretic result states that the $\Bbb Z_p[[G(H_\infty/F]]^-$-module $Sel_S^T(H_\infty)_p^-$ is of projective dimension $…
In this paper we develop a new method to study Iwasawa theory and Eisenstein families for unitary groups $\mathrm{U}(r,s)$ of general signature over a totally real field $F$. As a consequence we prove that for a motive corresponding to a regular algebraic cuspidal automorphic representation $π$ on $\mathrm{U}(r,s)_{/F}$ which is ordinary at $p$, twisted by a Hecke character, if its Selmer group has rank $0$, then the corresponding central $L$-value is nonzero. This generalizes a result of Skinner-Urban in their ICM 2006 report in the special case when $F=\mathbb{Q}$ and the motive is conjugate…
Vanishing of $\mathrm{Sha}(A/K)[\mathfrak{P}^\infty]$ and its consequences for the anticyclotomic Iwasawa theory of $\mathrm{GL}_2$-abelian varieties
本文将椭圆曲线Shafarevich-Tate群p部分在虚二次域上消失的Kolyvagin经典结果，推广到全实域上GL₂型模阿贝尔簇及其CM扩域，结合已有Iwasawa理论，证明K的反循环多Zₚ扩张层上存在类似消失性。
On the Torsion Congruence for Zeta Functions of Totally Real Fields
本研究利用Shintani锥分解探究全实域ζ函数的特殊值，证明素数次域扩张下其特殊值的一类同余关系，该结果蕴含Ritter-Weiss的挠同余，对全实域非交换岩泽主猜想的证明具有关键意义。
We investigate the group of universal norms attached to the cyclotomic Z {\ell}-tower of a totally real number field in connection with Grenberg's conjecture on Iwasawa invariants of such a field.
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