Marc Perret, Nicolas Saby
In this paper we give bounds for the Iwasawa invariants of the Igusa tower of curves investigated by Mazur and Wiles. We give an upper bound for the mu invariants and a lower bound for the sum of the lambda invariants, in terms of the genus and number of supersingular points of the base curve.
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We discuss the triviality of some classical Iwasawa modules in connection with the notion of $\ell$-rationality for totally $\ell$-adic number fields.
We develop a formalism for studying descent and codescent in the context of Iwasawa theory. The main result essentially states that to control descent or codescent amounts to the same. Arithmetic applications are given.
The article is withdrawn.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.
The article revisits a result of Antonio Lei, David Loeffler, and Sarah Livia Zerbes concerning the structure of the image by the Iwasawa regulator map of the Iwasawa module associated with a semi-stable $p$-adic representation on an unramified finite extension of $Q_p$ and give a direct proof based on the results of "Représentations $p$-adiques et normes universelles: I. le cas cristallin" in the crystalline case and of "Théorie d'Iwasawa des représentations $p$-adiques semi-stables" in the semi-stable case.
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.
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.
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.
We study equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using $p$-adic $L$-functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.
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…
We give a survey on noncommutative main conjectures of Iwasawa theory in a geometric setting, i.e. for separated schemes of finite type over a finite field, as stated and proved by Burns and the author. We will also comment briefly on versions of the main conjecture for function fields.
Let $K$ be a finite unramified extension of $\Qp$ and let $V$ be a crystalline representation of $\mathrm{Gal}(\Qpbar/K)$. In this article, we give a proof of the $C_{\mathrm{EP}}(L,V)$ conjecture for $L \subset \Qp^{\mathrm{ab}}$ as well as a proof of its equivariant version $C_{\mathrm{EP}}(L/K,V)$ for $L \subset \cup_{n=1}^\infty K(ζ_{p^n})$. The main ingredients are the $δ_{\Zp}(V)$ conjecture about the integrality of Perrin-Riou's exponential, which we prove using the theory of $(ϕ,Γ)$-modules, and Iwasawa-theoretic descent techniques used to show that $δ_{\Zp}(V)$ implies $C_{\mathrm{EP}…
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…
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.
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.
Théorie d'Iwasawa des motifs d'Artin et des formes modulaires de poids 1
本文研究 Artin 模式与权 1 模形式的 Iwasawa 理论，证明 cyclotomic Greenberg-Selmer 组的 torsion 性，并提出 Iwasawa 主猜想。
We study this subject by first proving that the p-primary subgroup of the classical Selmer group for an elliptic curve with good, ordinary reduction at a prime p has a very simple and elegant description which involves just the Galois module of p-power torsion points. We then prove theorems of Mazur, Schneider, and Perrin-Riou on the basis of this description. The final section, which is half of this long paper, contains a number of results and examples including a thorough study of the mu-invariant.
In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Z_p-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi and Perrin-Riou, we define restricted Selmer groups and λ^\pm, μ^\pm-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms of these invariants. To be able to work with non-cyclotomic Z_p-extensions, a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime …
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.
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 …
Iwasawa invariants and exceptional pairs of certain abelian fields
本文针对Q(√d,ζₚ)研究岩泽不变量与例外对，确定岩泽模的Λ同构类及例外对结构差异，在指定范围内发现15个新例外对，包括λ不变量为2的特殊情形。
Sur la fidélité de certaines représentations de GL_2(F) sous une algèbre d'Iwasawa
本文证明 Iwasawa 代数 E[[O_F]] 中 O_F*-稳定非零理想均为开理想，并据此推出上单位幂幺子群的 Iwasawa 代数对 GL2(F) 不可约容许光滑表示作用的忠实性。
We study Galois descent of K_1 of group algebras with coefficients in certain subrings of the ring of integers of C_p, the completion of an algebraic closure of Q_p.
Construction of graph coverings with prescribed Iwasawa invariants
本文研究了图覆盖的Iwasawa不变量，证明在满足条件的情况下，任何(λ, μ)对均可作为无分支覆盖的不变量，且允许分支覆盖时无需奇偶条件。
Iwasawa invariants of finite spectra
本文研究有限谱 p 进拓扑 K 理论对应的 Iwasawa 模，计算其 λ 与 μ 不变量，并证明弱形式 Iwasawa 主猜想的谱类比。
Let $p$ be an odd prime, $F$ be a number field and consider a uniform infinite pro-$p$ extension $F_\infty$ of $F$ with Galois group $G=Gal(F_\infty/F)$. Let \[G=G_0\supset G_1\supset\dots \supset G_n\supset G_{n+1}\supset \dots\] be the descending $p$ central series of $G$ and set $F_n:=F_\infty^{G_n}$. Assume that $G$ is uniform and that $F_\infty$ contains the cyclotomic $\mathbb{Z}_p$-extension of $F$. Denote by $A_n$ the $p$-primary part of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $F_n$. The $λ$-invariant of $F_n$ coincides with the corank of $A_n$ as a $\mathbb{Z}_p$…
The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves
本文利用弱于 Iwasawa 主猜想的结论刻画任意秩椭圆曲线的 Selmer 群结构，并推出 Kato Kolyvagin 系统非平凡、BSD 结构精细化、秩上界及 Kurihara 猜想等结果。
Iwasawa invariants of modular forms with $a_p=0$
本文针对满足 $a_p=0$ 的尖点新形式，建立了 Mazur-Tate 元素与带符号 $p$-进 $L$-函数的岩泽不变量之间的关系，并由此确定临界值的 $p$-进赋值及同余模形式不变量的联系，提供了计算带符号岩泽不变量的渐近方法。
For the $p$-cyclotomic tower of $\mathbb{Q}_p$ Fontaine established a description of local Iwasawa cohomology with coefficients in a local Galois representation $V$ in terms of the $ψ$-operator acting on the attached etale $(φ,Γ)$-module $D(V)$. In this article we generalize Fontaine's result to the case of arbitratry Lubin-Tate towers $L_\infty$ over finite extensions $L$ of $\mathbb{Q}_p$ by using the Kisin-Ren/Fontaine equivalence of categories between Galois representations and $(φ_L,Γ_L)$-module and extending parts of [Herr L.: Sur la cohomologie galoisienne des corps $p$-adiques. Bull. S…
We give a survey of a couple known constructions of $p$-adic $L$-functions including Iwasawa's construction from classical Stickelberger elements. We then construct "real" Stickelberger elements, i.e., explicit elements in the Galois group ring with $\mathbb{Z}_p$ coefficients that annihilate the Sylow $p$-subgroup of the ideal class group of $\mathbb{Q}(ζ_{p^{n+1}})^+$. In analogy with Iwasawa's work, we show that these elements are coherent in $\mathbb{Z}_p$-towers and give rise to twisted $p$-adic $L$-functions.
Unramified Geometric Iwasawa theory of function fields
本文研究特征$p>0$代数闭域上曲线非分歧$\mathbf{Z}_p$-塔的雅可比簇Newton多边形，证明其新部分斜率最终为$p^{-n}$的有理系数仿射函数。
Kida's formula via Selmer complexes
本文针对 Iwasawa 理论中 Kida 公式的多种类似物，提出以 Selmer 复形为视角的统一新研究方法。
Iwasawa Invariants for Symmetric Square Representations
本文研究满足剩余表示绝对不可约且同构的两个$\mathfrak{p}$-寻常尖点新形式，证明其对称平方表示的非本原$p$-进L函数模$\mathfrak{p}$同余，解析与代数岩泽不变量相关，补全了$\mathfrak{p}$-进L函数整性的证明，验证了岩泽主猜想与同余的相容性。
On the variation of the Frobenius in a non abelian Iwasawa tower
本文研究有限域上多自同构代数簇Frobenius算子特征值的ℓ进性质，证明一类岩泽塔中étale上同调Frobenius特征多项式的ℓ进收敛性，给出相关序列收敛速率，并提出多自同构曲线超奇异性的精确判据，推广统一了过往结果。
In this paper, we relate three objects. The first is a particular value of a cup product in the cohomology of the Galois group of the maximal unramified outside p extension of a cyclotomic field containing the pth roots of unity. The second is an Iwasawa module over a nonabelian extension of the rationals, a subquotient of the maximal pro-p abelian unramified completely split at p extension of a certain pro-p Kummer extension of a cyclotomic field that contains all p-power roots of unity. The third is the quotient of an Eisenstein ideal in an ordinary Hecke algebra of Hida by the square of the…
On the Wedderburn decomposition of the total ring of quotients of certain Iwasawa algebras II
本文研究了某些Iwasawa代数的总商环的Wedderburn分解，通过群环的分解来全面确定其结构。
In this paper we clarify some asymptotic formulas given by Jaulent-Maire, which relate orders of finite quotients of S-infinitesimal T-classes l-groups $Cl^S_T(K_n)$ associated to finite layers $K_n$ of a Zl-extension $K_\infty/K$ over a number field to the structural invariants of the Iwasawa module $X^S_T:=\varprojlim \Cl^S_T(K_n)$. We especially show that the lambda invariant $\tildeλ^S_T$ of those quotients sensibly differs from the structural invariant $λ^S_T$, and we illustrate this fact with explicit examples, where it can be made as large as desired, positive or negative.
$p$-adic $L$-functions for elliptic curves over global function fields
介绍与特征\(p\)的整体函数域\(K\)上的普通椭圆曲线\(A\)及\(\mathbb{Z}_{p}^{d}\)-扩张\(L/K\)相关的\(p\)-adic \(L\)函数\(\mathscr L_{A/L}\)，证明其满足函数方程等，在多种情形下证明了岩泽主猜想。
Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over $\mathbb{Z}_p^2$-extensions of an imaginary quadratic field where $p$ splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the $μ$-invariant unde…
Higher Fitting ideals and the structure of anticyclotomic Shafarevich-Tate groups
本文研究了反圆环Z_p扩展上椭圆曲线塞尔默和谢夫列夫-塔特群的结构，通过双分欧拉系统描述了更高Fitting理想，并提供了相关的新结果。
We establish a measure which describes in a precise way the local asymptotic distribution of rational points outside the locally accumulating subvarieties around a general rational point on a del Pezzo surface of degree 6 in the sense of diophantine approximation.
We consider the cohomology of local systems on the moduli space of curves of genus 2 and the moduli space of abelian surfaces. We give an explicit formula for the Eisenstein cohomology and a conjectural formula for the endoscopic contribution. We show how counting curves over finite fields provides us with detailed information about Siegel modular forms.
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