Actions of Galois groups on invariants of number fields
Aristides Kontogeorgis
本文研究了作用于$L$上的子群$H$对应的数域$L^H$的各种不变量之间的关系和范数幂等元之间的线性关系之间的联系。
In this paper we investigate the connection between relations among various invariants of number field s $L^H$ coresponding to subgroups $H$ acting on $L$ and of linear relations among norm idempotents.
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We define degree two cohomological invariants for G-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self--dual normal basis. In some cases, we show that these conditions are also sufficient.
Galois action on class groups
众所周知，扩张的伽罗瓦群对相对理想类群的结构施加了约束。只使用基本部分的理论群表示，我们给这样的结果的统一方法。
We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.
Finding Galois splitting models to compute local invariants
本研究利用伽罗瓦分裂模型思想计算合次数分歧扩张的相关伽罗瓦不变量，补充了Jones和Roberts建立的p进扩张不变量数据库，并提出了含泛多项式与Panayi根查找算法的多种找伽罗瓦分裂模型的策略。
Iwasawa invariants of Galois deformations
研究了在约化代数群G中取值的固定剩余伽罗瓦表示的普通变形中岩泽不变量的行为。特别是，假设这些Selmer群是余挠模的岩泽代数，我们表明，消失的亩不变是独立的变形，而lambda不变只依赖于分歧。这推广了Greenberg-Vatsal和Emerton-Pollack以及作者在G = GL（2）情况下的工作。
A control theorem for the images of Galois actions on certain infinite families of modular forms
在这封信中，我们表明，对于某些无限家庭的模形式的增长水平是可能有一个控制结果的例外素数所附的伽罗瓦表示。作为应用，证明了任意大指数的有限域上线性群的WieseonGalois实现的一个结果的统一形式.
Elliptic curves with maximal Galois action on their torsion points
给定一条数域k上的椭圆曲线E，在E的扭点上的伽罗瓦作用导出一个伽罗瓦表示ρ_E：Gal（\bar{k} /k）\to GL_2（\hat{Z}）.对于一个定数域k，我们描述了ρ_E对k上“随机”椭圆曲线E的象.特别地，如果k \neq Q与Q的分圆扩张线性无缘，则ρ_E对k上的“大多数”椭圆曲线是满射的.
On the $μ$-invariants of residually reducible Galois representations
研究剩余可约伽罗瓦表示塞尔梅群的岩泽$μ$-不变量，定义细化该不变量的岩泽不变量，证明存在模伽罗瓦表示提升可约模$p$伽罗瓦表示且其塞尔梅群$μ$-不变量≠指定整数，细化剩余可约情形下的塞尔猜想。
Galois action on the $\bar\mathbb{Q}$-isogeny classes of abelian surfaces with quaternionic multiplication
数域上的阿贝尔簇称为L-阿贝尔簇，如果对于数域L的绝对伽罗瓦群的任何元素，共轭阿贝尔簇通过保持自同态环上伽罗瓦作用的同构与给定的阿贝尔簇同构。我们可以认为它们是定义在L上的阿贝尔簇的推广，其自同态也定义在L上。在一维情况下，定义在L上的椭圆曲线产生由其Tate模上的伽罗瓦作用提供的伽罗瓦表示。这种经典的伽罗瓦表示在过去的几十年里一直是数论研究的中心对象。此外，给定椭圆L-曲线，可以构造以前的伽罗瓦表示的投影模拟。在这项工作中，我们在二维情况下构建了类似的射影表示，即通过四元数乘法或假椭圆曲线附着到阿贝尔L曲面。此外，我们证明了这样的射影表示也描述了Galois作用的椭圆L-曲线的Isonite类，在经典的情况下，或阿贝尔L-曲面与四元数乘法，在一般情况下。
Galois invariants of finite abelian descent and Brauer sets
该文研究全局域上代数簇的有限阿贝尔下降、Brauer-Manin障碍对应的阿德尔点子集的伽罗瓦不变量还原基域障碍集的条件，并将常曲线的相关结果推广到函数域上的等平凡曲线。
Using Shimura's reciprocity law, we calculate the Galois action on the special values of theta functions associated to primitive Dirichlet characters of odd conductor, normalised by the Dedekind eta function, at the point i. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime conductor.
Drinfeld modules with maximal Galois action on their torsion points
对于具有类属特征的广义生成域上的每个Drinfeld模，可以对应一个由其扭点上的伽罗瓦作用产生的伽罗瓦表示。Pink和Rütsche最近的工作描述了这种表征的图像，直到可解释性。他们的定理是定性的，本文的目的是补充这一理论与一个工作的例子。特别是，我们给出的例子Drinfeld模块的秩2的伽罗瓦行动，其扭点是尽可能大的。我们将遵循的方法，塞尔用来给明确的例子，他的开放性定理的椭圆曲线。使用我们的具体例子，我们将在数值上测试一些著名的椭圆曲线曲线的类似物。
We gives an explicit genus 3 curve over Q such that the Galois action on the torsion points of its Jacobian is a large as possible. That such curves exist is a consequence of a theorem of D. Zureick-Brown and the author; however, those methods do not produce explicit examples. We shall apply the general strategies of Hall and Serre in their open image theorems. We also make use of Serre's conjecture to show that the modulo ell Galois actions are irreducible. While we computationally focus on a single curve, the methods of this paper can be applied to a large family of genus 3 curves.
$λ$-invariant stability in families of modular Galois representations
本文从定量角度研究权为 2 的模形式族中 Iwasawa $λ$-不变量的变化，提供了 $λ$-不变量增长或保持稳定的频率下界。
Counting Polynomials via Galois Actions on Root Subsets
本文研究了高度不超过H的单项整系数多项式数量，其伽罗华群在根上的作用同构于给定的置换群(G,Ω)。针对多种群族获得了新的上界。
A simplified algorithmic realization of Galois actions on special values of modular functions
本研究基于扩展形式类群理论，提出一种显式实用的算法，用于计算虚二次序相关CM点处模函数特殊值的Galois共轭元与不可约多项式，改进了Stevenhagen和Cho的早期方法。
Computation of Galois groups of rational polynomials
计算伽罗瓦论，特别是计算给定多项式的伽罗瓦群的问题是一个非常古老的问题。目前，最好的算法解决方案是Stauduhar的方法。在计算上，应用Stauduhar方法的关键挑战之一是对于给定的一对群H<G，找到一个G-相对H-不变量，即H-不变量但不是G-不变量的多元多项式F。虽然通用的理论方法已知可以找到这样的F，但通常它们会产生不切实际的答案。我们给出了一个计算大次数不变量的一般方法，它改进了以前已知的方法，以及各种特殊的不变量，来自组的结构。然后，我们将我们的新的不变量的任务计算的伽罗瓦群的多项式的有理数，从而在第一个实用的度独立算法。
Given two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's u…
A del Pezzo surface of degree one defined over the rationals has 240 exceptional curves. These curves are permuted by the action of the absolute Galois group. We show how a solution to the classical inverse Galois problem for a subgroup of the Weyl group of type $D_8$ gives rise to a solution of the inverse Galois problem for the action of this subgroup on the 240 exceptional curves. A del Pezzo surface of degree one with such a Galois action contains a Galois invariant sublattice of type $D_8$ within its Picard lattice; this can be characterized in terms of a certain set of sixteen points in …
At present, the inverse Galois problem over $\mathbb{Q}$ is unsolved for the Mathieu group $M_{23}$. Here an overview of the current state in realizing $M_{23}$ as Galois group using the rigidity method and the action of braids is given. Computing braid orbits for $M_{23}$ revealed new invariants of the action of braids in addition to Fried's lifting invariant. These invariants can be used to construct generic braid orbits and more Galois realizations over $\mathbb{Q}$ for the Mathieu group $M_{24}$, but until now did not lead to success for realising $M_{23}$ as Galois group over $\mathbb{Q}$…
In this paper we present a description of the Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and $SL_2(\mathbb{F}_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over $\mathbb{Q}_2$ is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.
Galois invariant smoothness basis
本文回答了Joux和第二作者提出的关于有限域的乘法群中离散元的计算问题。给定一个有限剩余域$\bK$，人们寻找$\bK^*$的光滑基，该光滑基通过$\bK$的自同构左不变。对于广泛的一类有限域，我们设法构建模型，允许这样的平滑的基础。这项工作的目的是加速离散对数计算在这些领域。我们处理余维1（线性筛）和余维2（函数域筛）的情况。
We study the Galois action attached to the Dwrok surfaces $X_λ:X_0^4+X_1^4+X_2^4+X_3^4-4λX_0X_1X_2X_3=0$ with parameter $λ$ in a number field $F$. We show that when $X_λ$ has geometric Picard number $19$, its Néron-Severi group $NS(\overline{X}_λ)\otimes \mathbb{Q}$ is a direct sum of quadratic characters. We provide two proofs to this conclusion in our article. In particular, the geometrically proof determines the conductor of each of quadratic characters. Our result matches the one in \cite{Voight2}. With this decomposition, we give another proof to a result of Wan.
In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.
Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be an odd prime such that the residual representation $E[p]$ is reducible. The $μ$-invariant of the fine Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ is studied. We do not impose the Heegner hypothesis on $E$, thus allowing certain primes of bad reduction to decompose infinitely in the anticyclotomic $\mathbb{Z}_p$-extension. It is shown that the fine $μ$-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven.
Construction and classification of some Galois modules
在我们以前的文章中，我们描述了$p$次幂类群$K^\times/{K^{\times p}}$的伽罗瓦模结构，其中$K/F$是$p$在包含本原$p$次单位根的域$F$上的循环扩张。我们的描述依赖于与$K/F$相关的算术不变量。在这里，我们构造域扩张$K/F$与规定的算术不变量，从而完成我们的分类伽罗瓦模块$K^{\times}/K^{\times p}$。
On Kummer characters arising from Galois actions on the pro-$p$ fundamental groups of once-punctured CM elliptic curves
本文研究了来自CM椭圆曲线一次punctured时的pro-$p$基本群的Kummer字符，探讨了其非平凡性和满射性条件。
We make some observations concerning the Galois actions on Hurwitz curves and on the closely related but lesser-known Hurwitz origamis.
We define invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$ of Galois extensions of number fields with a fixed Galois group. Then, we propose a heuristic in the spirit of Malle's conjecture which asymptotically predicts the number of extensions that satisfy $\operatorname{inv}_i\leq X_i$ for all $X_i$. The resulting conjecture is proved for abelian Galois groups. We also describe refined Artin conductors that carry essentially the same information as the invariants $\operatorname{inv}_1,\dots,\operatorname{inv}_m$.
In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all automorphisms of the de Rham fundamental groupoid of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we also discuss other applications of the confluence relations such as an explicit $\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis…
Galois invariants of K_1-groups of Iwasawa algebras
本文研究了系数在整数环的某些子环上的群代数K_1的Galois下降，它是Q_p的一个代数闭包的完备化.
The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.
Information about the absolute Galois group $G_K$ of a number field $K$ is encoded in how it acts on the étale fundamental group $π$ of a curve $X$ defined over $K$. In the case that $K=\mathbb{Q}(ζ_n)$ is the cyclotomic field and $X$ is the Fermat curve of degree $n \geq 3$, Anderson determined the action of $G_K$ on the étale homology with coefficients in $\mathbb{Z}/n \mathbb{Z}$.The étale homology is the first quotient in the lower central series of the étale fundamental group.In this paper, we determine the structure of the graded Lie algebra for $π$. As a consequence, this determines the…
Constructing Galois representations with large Iwasawa $λ$-Invariant
本文利用模形式同余、Greenberg–Vatsal 结果及 Fakruddin–Khare–Patrikis 提升定理，构造了在分圆 $\mathbb{Z}_p$-扩张上 Selmer 群 $λ$-不变量任意大的模 Galois 表示。
This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and Z…
Arboreal Galois representations and uniformization of polynomial dynamics
给定一个定义在完备局部域上的d次多项式f，我们构造一个定义在无穷邻域内的双全纯变量变量变换，它将作用z->f（z）变换为乘法作用z->z^d。这种构造与复多项式动力学中的Bottcher坐标之间的关系类似于椭圆曲线的复一致化与Tate的p-adic一致化之间的关系。具体地说，这个双全纯是伽罗瓦等变的，将关于f的原像的伽罗瓦论的某些问题减少到关于乘法库默尔理论的问题。
Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on profinite knots is rigorously established, which is attained by our extending the action of Drinfeld's Grothendieck-Teichmüller group on profinite braid groups into on profinite knots.
We define and study a collection of special cycles on certain non-PEL Shimura varieties for $U(2,1) \times U(1,1)$ that appear naturally in the context of the recent conjectures of Gan, Gross and Prasad on restrictions of automorphic forms for unitary groups and conjectural generalizations of the Gross--Zagier formula. We prove a combinatorial formula expressing the Galois action in terms of the distance function on the Bruhat--Tits buildings for these groups. In addition, we calculate explicitly the Hecke polynomial appearing in the congruence relation conjectured by Blasius and Rogawski. Usi…
On symmetric power L-invariants of Iwahori level Hilbert modular forms
我们计算的算术L-不变量（Greenberg-Benois）的扭曲的对称功率的p-adic伽罗瓦表示附加到Iwahori水平希尔伯特模形式（在某些技术条件下）。我们的方法使用的对称权力的自同构和研究的分析伽罗瓦表示的p-adic家庭的自守形式辛和酉群。将这些族与GL（2）表示论中的一些显式体积函数相结合，构造了系数为对称幂的整体Galois上同调类，并给出了用Hecke特征值的对数导数表示的L-不变量的公式.
We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group $π_0 \mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that $π_0 \mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of $π_* \mathcal{R…
Characteristic p Galois representations that are produced by Drinfeld
本文研究了有限特征域的绝对伽罗瓦群在同一有限域域上的象的表示，可以由伽罗瓦群在适当的Drinfeld模的分点上的作用来构造.
We compare the Pontryagin duals of fine Selmer groups of two congruent $p$-adic Galois representations over admissible pro-$p$, $p$-adic Lie extensions $K_\infty$ of number fields $K$. We prove that in several natural settings the $π$-primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $μ$-invariants. In the special case of a $\mathbb{Z}_p$-extension $K_\infty/K$, we also compare the Iwasawa $λ$-invariants of the fine Selmer groups, even in situations …
Composantes irréductibles de lieux spéciaux d'espaces de modules de courbes, action galoisienne en genre quelconque
本文证明了绝对伽罗瓦群在标记曲线的模空间的无限几何基本群的无栈惯性的代数分解的几何有限循环群上的作用。作为一个补充结果，我们对亏格2中的素数阶profinite元给出了相同的作用。
The inverse Galois problem for PSL_2(F_p)
证明了对p>3的素数，单群PSL_2（F_p）是有理数扩张的伽罗瓦群.通过研究椭圆曲面的二阶上同调群上的Galois作用，得到了我们的Galois扩张。
Based on the analogies between mapping class groups and absolute Galois groups, we introduce an arithmetic pro-$\ell$ analogue of Orr invariants for a Galois element associated with Galois action on étale fundamental groups of punctured projective lines. At the same time, we also introduce pro-$\ell$ Orr space as an arithmetic analogue of Orr space whose third homotopy group is a target group of Orr invariant. We then determine its rank as $\mathbb{Z}_{\ell}$-module following Igusa-Orr's computation. Moreover, we investigate its relation with Ellenberg's obstruction to $π_1$-sections associate…
Weighted Completion of Galois Groups and Galois Actions on the Fundamental Group of P^1 - {0,1,infinity}
这是以前题为“伽罗瓦群的加权完备化和Deligne的一些猜想”的论文的修订版。修复一个素数$ł$。本文证明了Ihara提出的一个猜想，他认为这个猜想是Deligne提出的，关于绝对伽罗瓦群对三次穿孔射影线的基本群的$ł$完备化的作用。类似的技术也被用来证明贡恰洛夫猜想的一部分，也是关于绝对伽罗瓦群对三次穿孔射影线的基本群的作用。主要的技术工具是一个投射有限群关于一个约化表示（和其他适当的数据）的加权完备化。这一理论是在本文中发展。
Linking Invariants for Valuations and Orderings on Fields
该论文将Morishita的算术Milnor不变量理论从数域推广到一般域，引入离散赋值与序的链环不变量，并揭示其与Magnus同态系数及Galois上同调中Massey积的联系。
Arithmetic E_8 lattices with maximal Galois action
我们构造了在算术中出现的E_8格的显式例子，其中自然Galois作用等于格的自同构的完全群，即，E_8的外尔群。特别地，我们给出了Q（t）上Mordell-Weil格同构于E_8且具有极大Galois作用的显式椭圆曲线.我们的主要研究对象是数域上的1次del Pezzo曲面。几何Picard群通过交对的负性被看作格，它包含一个同构于E_8的子格。我们构造这样的表面的例子，伽罗瓦的几何皮卡德群的行动是最大的。
Galois actions on the pro-l-unipotent fundamental group
本文建立了在局部域K上好约化的光滑簇X的pro-l-幂幺基本群上可能的Galois作用的限制。特别地，如果X是真的，且l不等于K的特征残数p，则这个作用量唯一地由上同调作用量决定。一个类似的结果，l=p，然后证明通过比较定理之间的结晶和pro-p同伦类型。
Galois actions on complex braid groups
我们建立了可以定义在与有限复反射群相关的判别簇上的有理数的绝对伽罗瓦群的几何作用的忠实性，并回顾了与profinite Grothendieck-Teichmüller群的一些可能的联系。
We expand on Nekovář's construction of the plectic half transfer to define a plectic Galois action on Hilbert modular varieties. More precisely, we study in a unifying fashion Shimura varieties associated to groups that differ only in the centre from $R_{F/\mathbb{Q}}{\rm GL}_2$. We define plectic Galois actions on the CM points and on the set of connected components of these Shimura varieties, and show that these two actions are compatible. This extends the plectic conjecture of Nekovář--Scholl.
For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.
Belyi-extending maps and the Galois action on dessins d'enfants
通过寻找绝对伽罗瓦群对Grothendieck的dessins d 'enfants的忠实作用的不变量和轨道来研究绝对伽罗瓦群。我们定义了一类函数称为贝利扩展映射，我们用它来构建新的伽罗瓦不变量的dessins从以前已知的不变量。Belyi扩张映射是绝对伽罗瓦群注入Grothendieck-Teichmuller群的“新型”关系的来源。我们明确了如何从一般的Belyi扩张映射得到其相关不变量的公式，该公式可以在计算机代数软件包中实现。我们给出了一个例子，一个新的不变量不同的两个dessins具有相同的价值，其他容易计算的不变量。
We prove in some cases a formula for the Greenberg-Benois $\mathcal{L}$-invariant of the spin, standard and adjoint Galois representations associated with Siegel-Hilbert modular forms. In order to simplify the calculation, we give a new definition of the $\mathcal{L}$-invariant for a Galois representation $V$ of a number field $F\neq \mathbb{Q}$; we also check that it is compatible with Benois' definition for $\mathrm{Ind}_F^{\mathbb{Q}}(V)$.
$\mathscr{L}$-invariants and logarithm derivatives of eigenvalues of Frobenius
令$K$为$p$ -adic局部域。本文研究了它的一类特殊的$p$ -adic伽罗瓦表示，这些表示类似于模形式的例外零猜想中的伽罗瓦表示.特别是，我们验证了Colmez公式可以推广到我们的情况。我们还包括Colmez公式的退化版本。
Constructing Galois representations with prescribed Iwasawa $λ$-invariant
本文在分圆Zp-扩张下研究模Bloch-Kato Selmer群的Iwasawa λ-不变量，证明对任意不小于给定p-普通新形式g的λ-不变量的整数n，存在无穷多个与g模p同余且λ-不变量恰为n的新形式，并给出其水平的定量结果。
In the present paper, we study the $p$-adic $L$-functions and the (strict) Selmer groups over $\mathbb{Q}_{\infty}$, the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, of the $p$-adic weight one cusp forms $f$, obtained via the $p$-stabilization of weight one Eisenstein series, under the assumption that a certain Eisenstein component of the $p$-ordinary universal cuspidal Hecke algebra is Gorenstein. As an application, we compute the Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with the same residual Galois representations as the one of $f$, which in our setting, is r…
On l-adic Galois L-functions
我们正在研究基本群上的伽罗瓦作用。利用复盖塔构造了Z_p的有限个复积上的测度，利用这些测度可以计算伽罗瓦表示的系数.在Z_p上测度的最简单情形下，得到了Kubota_Leopoldt p-adic L函数和p-adic Hurwitz zeta函数.
Galois Action on Diameter Four Trees
本文的研究对象是一类被称为直径四树的Dessins d'enfants。这些对象首先由G. Shabat，可以被认为是最简单的非平凡的例子，etale覆盖的投影线减去三点。它们的算术性质仍然是神秘的，它们的研究可以启发对更一般情况的理解。在这里，主要的兴趣是致力于行动的绝对伽罗瓦群对这些（同构类）覆盖。特别是，在许多情况下，我们能够区分伽罗瓦轨道和描述的分解群的行动。另一个中心结果是关于野生分歧的研究，我们展示了如何减少到驯服的情况下，然后推导出一些详细的算术信息。
This paper studies the Galois action on a special lattice of geometric origin, which is related to mod-$\ell$ abelian-by-central quotients of geometric fundamental groups of varieties. As a consequence, we formulate and prove the mod-$\ell$ abelian-by-central variant/strengthening of a conjecture due to Ihara/Oda-Matsumoto.
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