Let $p$ be a prime number, and $G$ a compact $p$-adic Lie group. We recall that the Iwasawa algebra $Λ(G)$ is defined to be the completed group ring of $G$ over the ring of $p$-adic integers. Interesting examples of finitely generated modules over $Λ(G),$ in which $G$ is the image of Galois in the automorphism group of a $p$-adic Galois representation, abound in arithmetic geometry. The study of such $Λ(G)$-modules arising from arithmetic geometry can be thought of as a natural generalization of Iwasawa theory. One of the cornerstones of classical Iwasawa theory is the fact that, when $G$ is the additive group of $p$-adic integers, a good structure theory for finitely generated $Λ(G)$-modules is known, up to pseudo-isomorphism. The aim of the present paper is to extend as much as possible of this commutative structure theory to the non-commuta tive case.
We give a necessary and sufficient "Galois descent" condition to the freeness of the Iwasawa module built from Sinnott's circular units. Then we describe explicit examples for which this condition is not fulfilled.
For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions over Iwasawa algebra. In a previous paper of the authors, under several assumptions for an imaginary quadratic field, we obtain a necessary and sufficient condition for the Iwasawa module to be cyclic as a module over the Iwasawa algebla. Our main result is to give methods for computation and numerical examples about the results. We remark that our results do not need the assumption that Greenberg's generalized conjecture holds.
Let $p$ be a prime number. We show that, there exists an infinite family of imaginary abelian fields such that, the Iwasawa module of the maximal multiple ${\Bbb Z}_p$-extension is non trivial and pseudo-null for each field in the family. We also discuss on an application to non-abelian Iwasawa theory in the sense of Ozaki.
In Part I we review some specific properties of the $Λ$-modules in Iwasawa theory, which add structure to the general properties of Noetherian $Λ$-torsion modules. Part II deals with Kummer theory and gives a detailed construction of the Iwasawa linear space. This provides a new, simpler proof of the conjectures of Leopoldt and Gross for CM extensions. Using a construction of Thaine, we then prove that $λ^+ = 0$ in these fields, thus proving a part of Greenberg's conjecture - the fact $μ^+ = 0$ still has to be shown. In the Appendices we give some elementary partial proofs of the main facts pr…
Let H be a torsionfree compact p-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra Λ_H of H has trivial centre and use this result to classify the prime c-ideals in the Iwasawa algebra Λ_G of G := H \times \Zp. We also show that a finitely generated torsion Λ_G-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.
The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller Iwasawa modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show…
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…
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 associate to a filtered $φ$-module $D$ a sub-$Z_p[[x]]$-module of convergent series on the open unit disk in which the $p$-adic $L$-functions of the Galois representation associated to $D$ and the $Z_p$-cyclotomic extension live (if they exist). This generalizes the already known case where $D$ is associated to an elliptic curve or a modular form.
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…
Noncommutative Iwasawa theory has created a lot of interest in Whitehead groups of Iwasawa algebras of compact p-adic Lie groups with a quotient isomorphic to the additive group of p-adic integers. In this paper we compute Whitehead groups of Iwasawa algebra of a pro-p compact p-adic Lie group of dimension one. We also give results on Whitehead groups of the localisation of such Iwasawa algebras at the canonical Ore set defined by Coates, Fukaya, Kato, Sujatha and Venjakob.
We consider the family of CM-fields which are pro-p p-adic Lie extensions of number fields of dimension at least two, which contain the cyclotomic Z_p-extension, and which are ramified at only finitely many primes. We show that the Galois groups of the maximal unramified abelian pro-p extensions of these fields are not always pseudo-null as Iwasawa modules for the Iwasawa algebras of the given p-adic Lie groups. The proof uses Kida's formula for the growth of lambda-invariants in cyclotomic Z_p-extensions of CM-fields. In fact, we give a new proof of Kida's formula which includes a slight weak…
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.
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 …
For a prime number p, we denote by K the cyclotomic Z_p-extension of a number field k. For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension of K unramified outside S. This paper treats the case where S does not contain p and k is the rational number field or an imaginary quadratic field. In this case, we prove the explicit formulae for the free ranks of the S-ramified Iwasawa modules as abelian pro-p groups, by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of a…
We begin a study of m-th Chern classes and m-th characteristic symbols for Iwasawa modules which are supported in codimension at least m. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least 1. We apply this to an Iwasawa module constructed from an inverse limit of p-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteris…
For a CM-field $K$ and an odd prime number $p$, let $\widetilde K'$ be a certain multiple $\mathbb{Z}_p$-extension of $K$. In this paper, we study several basic properties of the unramified Iwasawa module $X_{\widetilde K'}$ of $\widetilde K'$ as a $\mathbb{Z}_p[[{\rm Gal}(\widetilde K'/K)]]$-module. Our first main result is a description of the order of a Galois coinvariant of $X_{\widetilde K'}$ in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic $\mathbb{Z}_p$-extension of $K$ under a certain assumption on the splitting of primes above $p$. Second …
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…
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…
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…
This paper is lead by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, R of a p-adic analytic group G. For G without any p-torsion element we prove that R is an Auslander regular ring. This result enables us to give a good definition for pseudo-null R-modules. Then the category of R-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the p-primary part of a finitely generated R-module. A local duality theorem as well as the Auslander-Buchsbaum equality are further …
Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.
Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(φ,Γ)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting.
The aim of this article is to study Bloch-Kato's exponential map and Perrin-Riou's "big" exponential map purely in terms of (ϕ,Γ)-modules over the Robba ring. We first generalize the definition of Bloch-Kato's exponential map for all the (ϕ,Γ)-modules without using Fontaine's rings B_{cris}, B_{dR} of p-adic periods and then we generalize the construction of Perrin-Riou's "big" exponential map for all the de Rham (ϕ,Γ)-modules and prove that this map interpolates our Bloch-Kato's exponential map and the dual exponential map. Finally, we prove a theorem concerning to the determinant of our "big…
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.
We establish a purely algebraic tool for studying the Iwasawa adjoints of some natural Iwasawa modules for $p$-adic Lie group extensions of number fields, by relating them to certain continuous Galois cohomology groups via a spectral sequence.
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 classical Iwasawa theory, we mainly study codimension one behavior of arithmetic modules. Relatively recently, F. M. Bleher, T. Chinburg, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi, and M. J. Taylor started studying higher codimension behavior of unramified Iwasawa modules which are conjectured to be pseudo-null. In this paper, by developing a general algebraic theory on perfect complexes, we obtain a new perspective of their work. That allows us to extend the results to equivariant settings and, even in non-equivariant settings, to obtain more refined results concerning the higher codim…
We formulate a weak Gorenstein property for the Eisenstein component of the p-adic Hecke algebra associated to modular forms. We show that this weak Gorenstein property holds if and only if a weak form of Sharifi's conjecture and a weak form of Greenberg's conjecture hold.
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…
We prove a functional equation for two projective systems of finite abelian $p$-groups, $\{\fa_n\}$ and $\{\fb_n\}$, endowed with an action of $\ZZ_p^d$ such that $\fa_n$ can be identified with the Pontryagin dual of $\fb_n$ for all $n$. Let $K$ be a global field. Let $L$ be a $\ZZ_p^d$-extension of $K$ ($d\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$.
In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa modules over an infinite Galois extension K of a number field k whose Galois group G=G(K/k) is isomorphic to the semidirect product of two copies of the p-adic numbers. After first analyzing some general algebraic properties of the corresponding Iwasawa algebra, we apply these results to the Galois group of the p-Hilbert class field over K. As a main tool we prove a Weierstrass preparation theorem for certain skew power series rings. One striking result in our work is the discovery of the abundance of faithful torsion…
We establish the faithfulness of Verma modules for rational Iwasawa algebras of split semisimple compact $L$-analytic groups. We also prove the algebraic independence of Arens-Michael envelopes over Iwasawa algebras and compute the centre of affinoid enveloping algebras of semisimple $p$-adic Lie algebras.
In this paper we study the corresponding categories and the corresponding cohomologies of the Hodge-Iwasawa modules we developed in our series papers on Hodge-Iwasawa theory. The corresponding cohomologies will be essential in the corresponding development of the contact with the corresponding Iwasawa theoretic consideration, while they are as well very crucial in the corresponding study of the corresponding deformations of local systems over general analytic spaces. We contact with some applications in analytic geometry and arithmetic geometry which all have their own interests and deserve fu…
We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type $A_{2}$. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.
For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). Applying these maps to the Kato zeta elements …
In \citenospec{MR1097615} several spectral sequences for (global and local) Iwasawa modules over (not necessarily commutative) Iwasawa algebras (mainly of $p$-adic Lie groups) over $\Z_p$ are established, which are very useful for determining certain properties of such modules in arithmetic applications. Slight generalizations of said results can be found in \citenospec{MR2333680} (for abelian groups and more general coefficient rings), \citenospec{MR1924402} (for products of not necessarily abelian groups, but with $\Z_p$-coefficients), and \citenospec{MR3084561}. Unfortunately, some of Janns…
In this article, we study a relation between certain quotients of ideal class groups and the cyclotomic Iwasawa module $X_\infty$ of the Pontrjagin dual of the fine Selmer group of an elliptic curve $E$ defined over $\mathbb{Q}$. We consider the Galois extension field $K^E_n$ of $\mathbb{Q}$ generated by coordinates of all $p^n$-torsion points of $E$, and introduce a quotient $A^E_n$ of the $p$-sylow subgroup of the ideal class group of $K^E_n$ cut out by the modulo $p^n$ Galois representation $E[p^n]$. We describe the asymptotic behavior of $A^E_n$ by using the Iwasawa module $X_\infty$. In p…
A fundamental observation of Iwasawa gives a criterion for a module over the classical Iwasawa algebra to be torsion. In this paper, we study a certain extension of this criterion. We will then apply this to study the structure of the homology of the "Iwasawa cohomology groups" and dual fine Selmer groups.
It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$, extending the proof given by Clozel for the group $Γ_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.
This text is a mostly self-contained textbook-like introduction to classical and modern Iwasawa Theory in Spanish language. It grew out of notes for a course that the authors gave at the annual conference of the Mexican Mathematical Society (SMM) in 2018. It covers some preliminaries from algebraic number theory and profinite algebra, the theory of the Iwasawa algebra and its modules, results about class groups in $\mathbb{Z}_p$-extensions, $p$-adic $L$-functions for Dirichlet characters and their various constructions (along with some complex theory of $L$-functions), the cyclotomic main conj…
We study the behavior of Iwasawa invariants among ordinary deformations of a fixed residual Galois representation taking values in a reductive algebraic group G. In particular, under the assumption that these Selmer groups are cotorsion modules over the Iwasawa algebra, we show that the vanishing of the mu invariant is independent of the deformation, while the lambda invariant depends only on the ramification. This generalizes work of Greenberg-Vatsal and Emerton-Pollack and the author in the case G = GL(2).
