A Note on a result of Iwasawa
Bruno Angles, Thomas Herreng
利用p-adic L-函数在1处的值恢复了Iwasawa关于主单位的p-adic对数的一个结果。我们推导出一个Iwasawa样的结果在主单位的奇数部分。
We recover a result of Iwasawa on the p-adic logarithm of principal units with the use of the value at 1 of p-adic L-functions. We deduce an Iwasawa-like result in the odd part of principal units.
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The integral logarithm in Iwasawa theory: an exercise
设$l$是奇素数，$H$是有限交换$l$ -群.我们确定了$Λ_\wedge[H]$的单位群（$\Bbb{Z}_l[[T]][H]$在$l$上的局部化的完成）以及非对易岩泽理论中出现的整数对数$L:Λ_\wedge[H]^\times\to Λ_\wedge[H]$的核和上核.
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.
Seminar Notes on Open Questions in Iwasawa Theory - SNOQIT I: The $Λ[ G ]$-modules of Iwasawa theory II: Units and Kummer theory in Iwasawa extensions
在第一部分中，我们回顾了Iwasawa理论中$Λ$ -模的一些特殊性质，这些性质为Noether $Λ$ -扭模的一般性质增加了结构。第二部分涉及库默尔理论，并给出了岩泽线性空间的详细结构。这为Leopoldt和Gross关于CM扩张的证明提供了一个新的、更简单的证明。使用建设的塞恩，我们然后证明，$λ^+ = 0$在这些领域，从而证明了格林伯格的猜想的一部分-事实$μ^+ = 0$仍然要显示。在附录中，我们用岩泽的线性空间给出了主要事实的一些初等的部分证明。这两篇论文没有给出非CM域的证明，对处理这种情况的方法感兴趣的读者可以参考“$T and T^*$“关于这方面的论文。这些方法将被集成到Snoqit系列中。
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.
On the Iwasawa theory of elliptic curves at Eisenstein primes
本文为作者在2022年ICTS研讨会上关于椭圆曲线在Eisenstein素数处Iwasawa理论的迷你课程扩展笔记。
In this paper, we study the module of Euler systems. We determine the ideal of an Iwasawa algebra associated with Euler systems of rank $0$. We also show that the module of higher rank Euler systems for $\mathbb{G}_{m}$ over a totally real field is free of rank $1$ under the assumptions that Greenberg conjecture holds true and that the $μ$-invariant of a certain Iwasawa module vanishes.
This article extends our study of the geometry of the $p$-adic eigencurve at a point defined by a weight $1$ cuspform $f$ irregular at $p$ and having complex multiplication, and the implications in Iwasawa and in Hida theories. The novel results include the determination of the Fourier coefficients of certain non-classical $p$-adic modular forms belonging to the generalized eigenspace of $f$, in terms of $p$-adic logarithms of algebraic numbers. We also compute the "mysterious" cross-ratios of the $p$-ordinary filtrations of the Hida families containing $f$.
A note on the distribution of Iwasawa invariants of imaginary quadratic fields
本文建立了虚二次域类群 $p$-秩与其分圆 $\mathbb{Z}_p$-扩张 $λ$-不变量之间的关系，并据此证明了 $λ$-不变量分布的统计结果，包含依赖 Cohen--Lenstra 启发式的条件性结果和利用已有定理的无条件结果。
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…
This article is the first of a pair of articles dealing with the Iwasawa theory of modular forms of weight 1 and, more generally, of Artin representations satisfying certain conditions. The main results in this part analyze the structure of certain Selmer groups for the Artin representation. In particular, it is shown that the Selmer groups are co-torsion as $Λ$-modules. For each Selmer group, we consider a generator of the characteristic ideal of its Pontrjagin dual. We call that the algebraic p-adic L-function. We also construct an analytic $p$-adic L-function via deformation to higher weigh…
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.
Congruences between abelian pseudomeasures
继Deligne和Ribet（“全实数域上交换$L$ -函数在负整数上的值.“发明。Math.59（1980），227-286）我们证明了“扭同余”（如在我们的论文“非交换伪测度和交换岩泽$L$ -函数之间的同余”中所介绍的.“出现在纯粹和应用数学季刊）举行，因此减少'主要猜想'的等变岩泽理论的完整性对数pseudomesasure。
Iwasawa theory and the representations of finite groups
本文通过表示论精细化有限凯莱图的岩泽理论，基于图zeta函数与数论L函数的类比研究凯莱图ℤ_ℓ-塔及雅可比行列式渐近增长，证明岩泽多项式可通过基础群不可约表示典范分解并定义表示论岩泽多项式。
In this note we show how the main conjecture of the Iwasawa theory over Q has a natural place in the context of the Galois representation of the Galois group $Gal(\bar Q/Q)$ on the etale pro-p fundamental group of the projective line minus three points. However we still need to assume the Vandiver conjecture to get a proof of the main conjecture in this context.
An introduction to $p$-adic $L$-functions
本文以三种等价方式介绍p进L-函数与Iwasawa理论，并陈述Iwasawa主猜想及Vandiver素数情形下的证明。
Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes
该研究针对CM域中分歧素数处的CM椭圆曲线，构建反分圆岩泽理论的整框架，定义带符号塞尔默群，证明相关岩泽主猜想，填补了无非三角几何形变的对应理论空白。
On higher Fitting ideals of Iwasawa modules of ideal class groups over imaginary quadratic fields and Euler systems of elliptic units II
在我们以前的工作中，利用椭圆单位的Kolyvagin导数构造了Iwasawa代数的理想C_i，并证明了理想C_i成为虚二次域K的阿贝尔扩域K_0上的一元和二元p-adic非分歧Iwasawa模X的Fitting高理想的“上界“.本文利用“非算术的”特殊化论证，证明了理想C_i也是X的Fitting高理想的“下界“.特别地，我们证明了理想C_i决定X的伪同构类。请注意，在这篇文章中，我们还处理的情况下，当p-部分的等变玉川数猜想（ETNC）_p尚未证明。在证明了（ETNC）_p的情况下，Burns，Kurihara和Sano已经得到了更强的结果：在（ETNC）_p的假设下，在K_0/K的伽罗瓦群上的特征标$ψ$的一定条件下，他们用Rubin-Stark元给出了X的π-分支的高Fitting理想的完整描述。本文还证明了当（ETNC）_p被证明时，C_i的$ψ$ -部分与Burns，Kurihara和Sano所构造的理想在某些情况下重合.作为这个比较结果的推论，我们还推导出X的极大伪空子模的$ψ$ -部分的零化子理想在某些情况下与初始Fitting理想重合.
On Weil Numbers in Cyclotomic Fields
本文研究了$\mathbb Q(ζ\_p).$的分圆$\mathbb Z\_p$ -扩张中Weil数的p-adic行为.利用Iwasawa理论中出现的一些经典模的特征理想，确定了半局部单位的商与Weil数的特征理想.
A note on linear forms in two logarithms: the argument of an algebraic power
本文给出了给定代数数的幂的辐角的一个改进下界，该代数数绝对值为1但非单位根。
In these notes, we explore possible stable properties for the zeta function of a geometric Zp-tower of curves over a finite field of characteristic p, in the spirit of Iwasawa theory. A number of fundamental questions and conjectures are proposed for those Zp towers coming from algebraic geometry.
A canonical construction of signed $p$-adic $L$-functions for non-ordinary modular forms of weight $\leq p+1$
本文针对权重≤p+1的p-非普通模形式，通过无需p进霍奇理论的显式对数型矩阵分解无界p进L函数，构造出一对有界p进L函数，还计算了相关Iwasawa不变量的渐近公式并推广了已有结果。
A note on logarithmic growth Newton polygons of $p$-adic differential equations
本文回答了Y.安德烈与B有关。Dwork关于$p$ -adic线性微分方程解的对数增长的特化猜想.精确地说，我们明确地构造秩为2的$\mathbb{Q}_p[[X]]_0$上的$\nabla$ -模块$M$，使得$M$的特殊对数增长牛顿多边形的左端点严格地在$M$的一般对数增长牛顿多边形的左端点之上。
Universal norms and de Rham representations
本文证明Iwasawa上同调中通用范数计算从分圆扩张可推广至任意包含它的p-adic扩张，并给出适用于Lubin-Tate扩张的扭曲变体。
For a $p$-adic differential equation solvable in an open disc (in a $p$-adic sense), around 1970, Dwork proves that the solutions satisfy a certain growth condition on the boundary. Dwork also conjectures that a similar phenomenon should be observed without assuming the solvability. In this paper, we verify Dwork's conjecture in the rank two case, which is the first non-trivial result on the conjecture. The proof is an application of Kedlaya's decomposition theorem of $p$-adic differential equations defined over annulus.
A note on logarithmic equidistribution
本文针对单位圆上非单位根的代数数 $κ$，构造高度趋于零的严格代数数序列，证明 $f_κ$ 的共轭平均值上界，补全相关刻画并推广至 $p$-进情形。
Series representations for the logarithm of the Glaisher-Kinkelin constant
本文利用 Riemann zeta 函数导数表达式，给出了 Glaisher-Kinkelin 常数对数的两个级数展开式，其中一个涉及超几何函数。
A note on the paper by Bugeaud and Laurent "Minoration effective de la distance p-adique entre puissances de nombres algébriques"
我们将对p-adic图的一个结果作一点改进，它给出了p除以费马商x^{p-1}-1的指数的一个非平凡的上界。
A note on étale $(φ,Γ)$-modules in families
本文利用形式概形棱镜位点上结构层的形变，建立棱镜 $(\Lambda,F)$-晶体与 $\Lambda$-étale 局部系统及其上同调的等价关系，并由此推出岩泽上同调同构、重证 Dee 的族等价定理。
Averages of Ramanujan sums: Note on two papers by E. Alkan
本文给出了E.阿尔坎关于加权平均的拉马努金总和。我们推导出其他加权平均数的Ramanujan和与权重有关的矩阵，值的算术函数的GCD的，Γ函数，伯努利多项式，和二项式系数的身份。
Notes on $\log(ζ(s))^{\prime\prime}$
出于对黎曼零点相关的连接，我们调查的二阶导数的对数的黎曼ζ函数，特别是该函数的零点。定理1给出了一个零自由区域。定理2给出了高度T的非平凡零点个数的渐近估计。定理3是零密度估计。
On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms
我们通过Jacquet-Langlands对应构造了p-adic L函数，推广了Bertolini-Darmon，Vatsal等人的工作，并在某些二面角或反分圆扩张中构造了与平行权为2的尖点Hilbert模特征形相关联的L函数.这里给出的结构是adelic的，它允许我们从Waldspurger型公式推导出精确的插值公式，以及二面角μ-不变量的公式。我们还注意到这些p-adic L函数的Howard非零准则，它可以用来减少相关的Iwasawa主要猜想到一定的非平凡性准则的p-adic L函数的家庭。
The discrete logarithm problem over prime fields: the safe prime case. The Smart attack, non-canonical lifts and logarithmic derivatives
在这个简短的说明中，我们将安全素数情形下的素域上的离散对数问题与对数导数联系起来。
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。
Computation of the Galois groups occuring in M. Papanikolas's study of Carlitz logarithms
本文给出了一个关于定义在正特征域上的Tannakian范畴的对象$U$的伽罗瓦群的幂幺根的计算定理，该对象是单位对象的半单扩张.然后，我们给出了一个标准的代数解的对象的线性相关的扩展组。我们将这些结果应用于M. Papanikolas;特别是，我们给出了另一种证明的代数独立的Carlitz矩阵。
A note on the standard zero-free region for $L$-functions
本文为一类对数系数实部非负的一般 $L$-函数（含酉尖点自守表示的 Rankin--Selberg $L$-函数）建立了标准无零点区域。
Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p
构造了全实数域的一维非交换p-扩张的p-adic zeta函数，使得其伽罗瓦群的有限部分是指数为p的p-群.首先利用积分算子的Oliver-Taylor理论计算了Iwasawa代数的Whitehead群及其典型Ore局部化.这种计算减少了存在的非交换p-adic zeta函数之间的阿贝尔p-adic zeta伪测度的某些同余条件。最后利用Deligne-Ribet理论和一定的归纳技巧对这些同余式进行了验证。作为应用，我们将证明一个特殊情况下（的p-部分）的非交换等变玉川数猜想的关键泰特动机。本文的主要结果给出了作者的前一份文件的概括。
On a function of Ramanujan twisted by a logarithm
本文利用拉马努金《遗失笔记》中的模关系和德宁格函数导数的近期结果，推导出涉及双伽马函数与对数因子的无穷级数函数方程，并研究了其中起关键作用的积分 $\mathscr{H}(x)$ 的多种表示。
The Ramanujan Machine project detects new expressions related to constants of interest, such as $ζ$ function values, $γ$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures involving even and odd $ζ$ function values, logarithms etc. We show that many relations detected by the Ramanujan Machine Project stem from a specific algebraic observation and show how to generate infinitely many. This provides an automated proof and/or an explanation of many of the relations listed as conjectures by the project (although not all of them).
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