Galois groups over function fields of positive characteristic

Galois scaffolds and Galois module structure in extensions of characteristic $p$ local fields of degree $p^2$

A Galois scaffold, in a Galois extension of local fields with perfect residue fields, is an adaptation of the normal basis to the valuation of the extension field, and thus can be applied to answer questions of Galois module structure. Here we give a sufficient condition for a Galois scaffold to exist in fully ramified Galois extensions of degree $p^2$ of characteristic $p$ local fields. This condition becomes necessary when we restrict to $p=3$. For extensions $L/K$ of degree $p^2$ that satisfy this condition, we determine the Galois module structure of the ring of integers by finding necessa…

正特征局部域上最大幂零类的某些有限伽罗瓦扩张的分歧群

Ramification groups of some finite Galois extensions of maximal nilpotency class over local fields of positive characteristic

本文将 Brylinski 关于等特征局部域上阿贝尔伽罗瓦扩张分歧群的结果，推广到阶不超过 $p^{p+1}$ 且具有最大幂零类的非阿贝尔伽罗瓦扩张情形。

秩2德林费尔德模的弗罗贝尼乌斯迹，高维伽罗华表示，以及正特征中的强多重性一定理

Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic

本文证明了在有限多个地方外弗罗贝尼乌斯迹相等时，两类2阶非CM德林费尔德模的l-adic伽罗华表示同构。进一步，作者在弱维度假设下，展示在正特征局部域上，伽罗华表示在有限多个地方外弗罗贝尼乌斯迹相等强制同构仅在绝对不可约性假设下成立。

特征为正的局部域上以幺三角矩阵群为伽罗瓦群的伽罗瓦扩张的分歧群

Ramification groups of Galois extensions over local fields of positive characteristic with Galois group isomorphic to the group of unitriangular matrices

本文研究特征p>0、完美剩余域的局部域上，伽罗瓦群为UT_n(F_p)的有限伽罗瓦扩张的分歧群，推导上分歧断点的线性表达式，实现无需L-K元素的分歧群计算。

混合特征局部域的绝对伽罗瓦群的外自同构群的有限子群

Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields

本文从单阿贝尔几何角度研究混合特征局部域绝对伽罗瓦群外自同构群的有限子群，证明了存在非拟几何的有限子群，从而表明拟几何子群不能等同于所有有限子群。

特征零局部域的循环扩张的Galois模结构

Galois module structure of cyclic extensions of local Fields of characteristic zero

本文描述了特征零局部域循环扩张的精确Galois模结构及通常作为其直接因子的参数化空间Wa,b,m,n。

Galois scaffolds for cyclic $p^n$-extensions in characteristic $p$

Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to sufficient conditions for the ring of integers $O_L$ to be free of rank 1 over its associated order $A_0$, and to stricter conditions which imply that $A_0$ is a Hopf order in the group ring $K[C_{p^n}]$.

Akashi series, characteristic elements and congruence of Galois representations

In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible p-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain similar results for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.

Equality of two non-logarithmic ramification filtrations of abelianized Galois group in positive characteristic

We prove the equality of two non-logarithmic ramification filtrations defined by Matsuda and Abbes-Saito for the abelianized absolute Galois group of a complete discrete valuation field in positive characteristic. We also compute the refined Swan conductor and the characteristic form of a character of the fundamental group of a smooth separated scheme over a perfect field of positive characteristic by using sheaves of Witt vectors.

关于某些正特征多项式动力学伽罗瓦群的动态伽罗瓦群

On the dynamical Galois group of certain affine polynomials in positive characteristic

本文在正特征下研究了特定多项式动力学伽罗瓦群的判据，提出了一种在特征2情况下与Stoll判据相关的动态判据方法。

Compatible systems of symplectic Galois representations and the inverse Galois problem III. Automorphic construction of compatible systems with suitable local properties

This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with l…

Integral Galois Module Structure for Elementary Abelian Extensions with a Galois Scaffold

This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata, con…

树形Galois表示与有理函数：固定点比例与扩展问题

Arboreal Galois representations of rational functions: fixed-point proportion and the extension problem

本文研究了有理函数的树形Galois表示，解决了扩展问题，并给出了具有正固定点比例的群族，扩展了多项式结果并提供了反例。

通过伽罗瓦理论与Brauer群刻画正特征局部域

Characterising local fields of positive characteristic by Galois theory and the Brauer group

本文通过绝对伽罗瓦群的小商及Brauer群的p-挠公理，在亏指数至多1的域类中同构刻画了正特征局部域，补充了Efrat-Fesenko的工作。

An explicit integral polynomial whose splitting field has Galois group W(E_8)

Using the principle that characteristic polynomials of matrices obtained from elements of a reductive group over a number field typically have splitting field with Galois group isomorphic to its Weyl group, we construct an explicit monic integral polynomial of degree 240 whose splitting field has Galois group the Weyl group of the exceptional group of type E_8.

Lifting $N$-dimensional Galois representations to characteristic zero

Let $F$ be a number field, let $N\geq 3$ be an integer, and let $k$ be a finite field of characteristic $\ell$. We show that if $\rb:G_F\longrightarrow GL_N(k)$ is a continuous representation with image of $\rb$ containing $SL_N(k)$ then, under moderate conditions at primes dividing $\ell\infty$, there is a continuous representation $ρ:G_F\longrightarrow GL_N(W(k))$ unramified outside finitely many primes with $\rb\simρ\mod{\ell}$.

Explicit Galois representations of automorphisms on holomorphic differentials in characteristic $p$

We determine the representation of the group of automorphisms for cyclotomic function fields in characteristic $p > 0$ induced by the natural action on the space of holomorphic differentials via construction of an explicit basis of differentials. This includes those cases which present wild ramification and automorphism groups with non-cyclic $p$-part, which have remained elusive. We also obtain information on the gap sequences of ramified primes. Finally, we extend these results to rank one Drinfel'd modules.

椭圆曲线在特定伽罗瓦扩张中具有正密度的丢番图稳定性

A positive density of elliptic curves are diophantine stable in certain Galois extensions

本文证明，对于循环p-扩张L/Q（p=3或5），存在正密度的椭圆曲线E（按高度排序）在L中具有丢番图稳定性。

Galois cohomology of $p$-adic fields and $(φ, τ)$-modules

We construct various explicit Herr complexes that compute the Galois cohomology of a $p$-adic representation of the absolute Galois group of a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, using the associated $(φ,τ)$-modules (defined by Xavier Caruso), instead of $(φ,Γ)$-modules. We also give an application to $p$-divisible groups.

绝对伽罗华群的外自同构群的anabelian方面

Anabelian aspects of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields

本文研究混合特征局部域绝对伽罗华群的外自同构群，通过拓扑流形映射类群的类比，扩展了Hoshi和Nishio之前的工作，揭示了相关表示的不可约性和结晶性特性。

The Galois group of random elements of linear groups

Let F be a finitely generated field of characteristic zero and Γ<GL_n(F) a finitely generated subgroup. For an element g in Γ, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over F. We show that the structure of Gal(F(g)/ F) has a typical behaviour depending on F, and on the geometry of the Zariski closure of Γ(but not on Γ).

Sufficient Conditions for Large Galois Scaffolds

Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and for $K$ of either possible characteristic (0 or $p$), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper. But since our conditions can be difficult to check, we specialize to elementary abelian extensions …

A Valuation Criterion for Normal Basis Generators of Hopf-Galois Extensions in Characteristic p

Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let D_{S/R} be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element $ρ$ of L with $v_L(ρ)$ congruent to $-v_L(D_{S/R})-1$ mod n generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. Elder for Galois extensions.

A survey of Galois theory of curves in characteristic p

This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open questions. We illustrate important phenomena which occur for covers of curves in characteristic p. We explain key results on the structure of fundamental groups. We end by describing areas of active research and giving two new results about the genus and p-rank of certain covers of the affine line.

特征0中特殊p-扩张的Galois支架

Galois scaffolds for extraspecial p-extensions in characteristic 0

本文研究了特征0局部域上特殊p-群的全分歧扩张，找到了其容许Galois支架的充分条件，并由此导出了整数环在关联序上秩为1自由及关联序为Hopf序的条件。

The Hasse principle for finite Galois modules allowing exceptional sets of positive density

We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets of positive density. For a Galois module whose underlying abelian group is isomorphic to $\mathbb{F}_p^{\oplus r}$ ($r \leq 2$), we show that the product of the restriction maps for places in a set of places $S$ is injective if the Dirichlet density of $S$ is strictly larger than $1 - p^{-r}$. We give applications to the local-global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.

Galois module structure of Galois cohomology and partial Euler-Poincare characteristics

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group G_F of F. Using the Bloch-Kato Conjecture we determine the structure of the cohomology group H^n(U,Fp) as an Fp[G_F/U]-module for all n in N. Previously this structure was known only for n=1, and until recently the structure even of H^1(U,Fp) was determined only for F a local field, a case settled by Borevic and Faddeev in the 1960s. We apply these results to study partial Euler-Poincare characteristics of open subgroups N of the maximal pro-p quotient T of G_…

小特征下中等维数某些Galois表示的不存在性

The non-existence of some Galois representations of moderate dimension in small characteristic

本文在广义 Riemann 假设下证明了维数 n ≤ 4 时不存在不可约 mod 2 Galois 表示，并无条件证明了 n ≤ 3 时不存在此类表示。

Lifting Galois representations over arbitrary number fields

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero $p$-adic representation, if local lifting problems at places above $p$ are unobstructed.

正常域和B-扩展的Galois理论类比（无特征方法）

Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach)

本文引入B-扩展作为最对称的有限域扩展，并获得其Galois理论类比。B-扩展与正常域扩展一致，从而为正常域扩展提供Galois理论类比，涵盖Galois和纯 inseparable 扩张。

On the Jacobi sums over Galois rings

In this paper, we study the Jacobi sums over Galois rings of arbitrary characteristics and completely determine their absolute values, which extends the work in \cite{feng1}, where the Jacobi sums over Galois rings with characteristics a square of a prime number were discussed.

数论中绝对伽罗瓦群的Ab-忠实性

Ab-faithfulness of absolute Galois groups in number theory

本文证明了混合特征离散赋值域及正维数高维局部域的绝对伽罗瓦群是ab-忠实的，并给出形式Hilbert~90判据以推导其瘦性。

具有完美剩余域的混合特征完备离散赋值域的绝对伽罗瓦群之间同态的几何性质

Geometric properties of homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with perfect residue fields

本文将 Mochizuki 与 Hoshi 关于 $p$-进局部域绝对伽罗瓦群几何同态的判别条件推广到具有完美剩余域的一般混合特征完备离散赋值域，并得到弱 Isom 远阿贝尔结果。

关于伽罗瓦上同调的第二部分全局欧拉-庞加莱特征

On the second partial Global Euler-Poincare characteristics for Galois cohomology

本文研究有限伽罗瓦模M的第二部分欧拉-庞加莱特征χ2(G_{K,S},M)，通过添加素数集得到显式公式，并应用于伽罗瓦群表示和维数猜想的反例构造。

Abhyankar's conjectures in Galois theory: Current status and future directions

In this paper, we survey the major contributions of Abhyankar to the development of the theory of fundamental groups and Galois covers in positive characteristic. We first discuss the current status of four conjectures of Abhyankar about Galois covers in positive characteristic. Then we discuss research directions inspired by Abhyankar's work, including many open problems.

Adelic Openness for Drinfeld Modules in Special Characteristic

For any Drinfeld module of special characteristic p0 over a finitely generated field, we study the associated adelic Galois representation at all places different from p0 and \infty, and determine the image of the geometric Galois group up to commensurability.

On the Galois group over Q of a truncated binomial expansion

For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $\le r$ where $1 \le r \le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. We show here that for a fixed positive integer $r \ne 6$ and $n$ sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group $S_{r}$. For $r = 6$, we show the number of exceptional $n \le N$ for which the Galois group of this polynomial is not $S_r$ is at most $O(\log N)$.

Abelian Varieties and Finitely Generated Galois Groups

This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.

偶特征下 Dwork family 的剩余单值性及其在 Galois 表示中的应用

The residual monodromy for the Dwork family in even characteristic and its applications to Galois representations

本文研究了特征二中 Dwork family 的剩余单值性表示及其在 2-adic 和模 2 Galois 表示中的应用，并结合已有成果证明了源自 Dwork 五次超曲面族的秩 4 辛动机在特定条件下的自守性。

Galois module structure of Milnor K-theory mod p^s in characteristic p

Let E be a cyclic extension of pth-power degree of a field F of characteristic p. For all m, s in N, we determine K_mE/p^sK_mE as a (Z/p^sZ)[Gal(E/F)]-module. We also provide examples of extensions for which all of the possible nonzero summands in the decomposition are indeed nonzero.

特征2下整体伽罗瓦变形环的维数猜想

On the dimension conjecture for global Galois deformation rings in characteristic two

该研究针对特征2下整体伽罗瓦变形环的维数猜想，证明了相关下界并构造了维数猜想的反例，还提出了修正的维数猜想。

特征 $p$ 中 $p$-扩张的 Galois 脚手架

Galois scaffolds for $p$-extensions in characteristic $p$

本文利用 Saltman 的 generic $G$-扩张构造，在特征 $p$ 的局部域上构造了具有 Galois scaffolds 的全分歧 $G$-扩张，并得到了整数环在关联阶上为秩 1 自由模及关联阶为 Hopf order 的特例。

Canonical Nonclassical Hopf-Galois Module Structure of Nonabelian Galois Extensions

Let $L/K$ be a finite Galois extension of local or global fields in characteristic $0$ or $p$ with nonabelian Galois group $G$, and let ${\mathfrak B}$ be a $G$-stable fractional ideal of $L$. We show that ${\mathfrak B}$ is free over its associated order in $K[G]$ if and only if it is free over its associated order in the Hopf algebra giving the canonical nonclassical Hopf-Galois structure on the extension.

Corestriction Principle in non-abelian Galois cohomology

This is a revision of a McMaster University preprint, with extension. In this paper we prove that over local or global fields of characteristic 0, the Corestriction Principle holds for kernel and image of all maps which are connecting maps in group cohomology and the groups of $R$-equivalences. Some related questions over arbitrary fields of characteristic 0 are also discussed. AMS Mathematics Subject Classification (1991): Primary 11E72, Secondary 18G50, 20G10

On Galois cohomology of unipotent algebraic groups over local fields

In this paper, we give a necessary and sufficient condition for the finiteness of Galois cohomology of unipotent groups over local fields of positive characteristic

Galois covers of the open p-adic disc

This paper investigates Galois branched covers of the open $p$-adic disc and their reductions to characteristic $p$. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves.

关于函数域上的逆Galois问题的注记

Remarks on the inverse Galois problem over function fields

本文通过Galois理论和自守方法构造取值于半单群的$\ell$-adic Galois表示相容系统，并利用Larsen的经典结果证明Galois像在正密度素数集上极大，从而证明了Lie型有限群在全局函数域上逆Galois问题的新实例。

Dynamical Galois groups of trinomials and Odoni's conjecture

We prove Odoni's conjecture in all prime degrees; namely, we prove that for every positive prime $p$, there exists a degree $p$ polynomial $φ\in\mathbb{Z}[x]$ with surjective arboreal Galois representation. We also show that Vojta's conjecture implies the existence of such a polynomial in many degrees $d$ which are not prime.

Producing Geometric Deformations of Orthogonal and Symplectic Galois Representations

For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine-Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at $p$ arising from Fontaine-Laffaille theory.

Le problème inverse de Galois sur les corps des fractions tordus à indéterminée centrale

In this article, we show that the Inverse Galois Problem over a skew field $H$ of finite dimension over its center $k$ is equivalent to a variant of the Inverse Galois Problem over $k$ involving a polynomial constraint. As an application, we show that if $k$ contains an ample field, then the Inverse Galois Problem has a positive answer over the skew field $H(t)$ of rational fractions with central indeterminate.

Scaffolds and Generalized Integral Galois Module Structure

Let $L/K$ be a finite, totally ramified $p$-extension of complete local fields with residue fields of characteristic $p > 0$, and let $A$ be a $K$-algebra acting on $L$. We define the concept of an $A$-scaffold on $L$, thereby extending and refining the notion of a Galois scaffold considered in several previous papers, where $L/K$ was Galois and $A=K[G]$ for $G=\mathrm{Gal}(L/K)$. When a suitable $A$-scaffold exists, we show how to answer questions generalizing those of classical integral Galois module theory. We give a necessary and sufficient condition, involving only numerical parameters, f…

Galois Module Structure of $p$th-Power Classes of Extensions of Degree p

For fields F of characteristic not p containing a primitive $p$th root of unity, we determine the Galois module structure of the group of $p$th-power classes of K for all cyclic extensions K/F of degree p.

An Application of Maeda's Conjecture to the Inverse Galois Problem

It is shown that Maeda's conjecture on eigenforms of level 1 implies that for every positive even d and every p in a density-one set of primes, the simple group PSL_2(F_{p^d}) occurs as the Galois group of a number field ramifying only at p.

与加法多项式相关的局部伽罗瓦表示

Local Galois representations associated to additive polynomials

本文针对加法多项式构造了非阿基米德局部域Weil群的不可约光滑表示，研究其不变量并给出原始性的充要条件。