Ramification in C-extensions of local field of characteristic 0

Massey积与限制分歧的单幂扩张

Massey products and unipotent extensions with restricted ramification

本文构造了具有限制分歧和分裂的数域pro-p扩张新例子，其伽罗瓦群分解为局部域pro-p绝对伽罗瓦群的余积，并证明这些扩张满足强Massey消失性质，从而具有大的单幂商。

Ramification groups and Artin conductors of radical extensions of the rationals

We compute the higher ramification groups and the Artin conductors of radical extensions of the rationals. As an application, we give formulas for their discriminant (using the conductor-discriminant formula). The interest in such number fields is due to the fact that they are among the simplest non-abelian extensions of the rationals (and so not classified by Class Field Theory). We show that this extensions have non integer jumps in the superior ramification groups, contrarily to the case of abelian extensions (as prescribed by Hasse-Arf theorem).

The ramification filtration in certain $p$-extensions

We show that the recent result of Castañeda and Wu about the ramification filtration in certain $p$-extensions of function fields of prime characteristic $p$ is equally valid over local fields of mixed characteristic $(0,p)$. Apart from being applicable to both equicharacteristic and mixed characteristic cases, our method has the advantage of being purely local, purely conceptual, more natural, and much shorter.

Calcul des groupes de ramification de certaines extensions radicales

For a rational prime $p\neq 2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L/F$ where $F/\mathbb{Q}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of Gal$(L/F)$ is a power of $p$, by a Hecke's theorem and some inductions,. Finally, we deal with the general case. For the case $p=2$, the same method can work, provided that we make an additional hypothesis.

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。

Refined ramification breaks in characteristic $p$

Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified elementary abelian $p$-extension with a single ramification break $b$. Byott and Elder defined the refined ramification breaks of $L/K$, an extension of the usual ramification data. In this paper we give an alternative definition for the refined ramification breaks, and we use Artin-Schreier theory to compute both versions of the breaks in some special cases.

Wild ramification in a family of low-degree extensions arising from iteration

This article gives a first look at wild ramification in a family of iterated extensions. For integer values of c, we consider the splitting field of $(x^2 + c)^2 + c$, the second iterate of $x^2 + c$. We give complete information on the factorization of the ideal (2) as c varies, and find a surprisingly complicated dependence of this factorization on the parameter c. We show that 2 ramifies (necessarily wildly) in all these extensions except when c = 0, and we describe the higher ramification groups in some totally ramified cases.

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。

On Abhyankar's lemma about ramification indices

We provide a simple proof of the fact that the ramification index of the compositum of two finite extensions of local fields is equal to the least common multiple of the ramification indices when at least one of the extensions is tamely ramified.

Enumerating Extensions of $(π)$-Adic Fields with Given Invariants

We give an algorithm that constructs a minimal set of polynomials defining all extension of a $(π)$-adic field with given, inertia degree, ramification index, discriminant, ramification polygon, and residual polynomials of the segments of the ramification polygon.

On the necessity of new ramification breaks

Ramification invariants are necessary, but not in general sufficient, to determine the Galois module structure of ideals in local number field extensions. This insufficiency is associated with elementary abelian extensions, where one can define a refined ramification filtration -- one with more ramification breaks [JNTB 17 (2005)]. The first refined break number comes from the usual ramification filtration and is therefore necessary. Here we study the second refined break number.

关于限制分歧的 $\mathbb{Z}_p$ 扩张的非分歧岩泽模的有限性性质

On finiteness properties of the unramified Iwasawa module of a $\mathbb{Z}_p$-extension with restricted ramification

本文研究由 $p$ 进素数处分歧限制定义的 $\mathbb{Z}_p$ 扩张，探讨其非分歧岩泽模的有限性、非平凡有限子模的存在性及平凡性，并给出格林伯格广义猜想的一个新充分条件及非平凡实例。

关于具有单个剧烈分支跳跃的局部域的伽罗瓦扩张

On Galois Extensions of Local Fields with a Single Wild Ramification Jump

研究了具有单个剧烈分支跳跃的局部域伽罗瓦扩张的计数方法，结合卢宾-塔特扩张和规范子群的性质分析。

Minimal Ramification in Nilpotent Extensions

Let $G$ be a finite nilpotent group and $K$ a number field with torsion relatively prime to the order of $G$. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of $K$ ramified in a Galois extension of $K$ with Galois group isomorphic to $G$. This sharpens and extends results of Geyer and Jarden and of Plans. Also we confirm Boston's conjecture on the minimum number of ramified primes for a family of central extensions by the Schur multiplicator.

Extensions of number fields with wild ramification of bounded depth

We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend well-known properties of tame extensions to this more general setting; for instance, we show that these towers, when infinite, are ``asymptotically good'' (an explicit bound for the root discriminant is given). We study the difficult problem of bounding the relation-rank of the Galois groups in question. Results of Gordeev and Wingberg imply that the relation-rank can tend to infinity when the set of ramified primes is fixed but the leng…

On the numerology of ramification data for power series in characteristic $p$

Let $k$ be a perfect field of characteristic $p$ and set $K=k((t))$. In this paper we study the ramification properties of elements of Aut$_k(K)$. By choosing a uniformizer for $K$ we may interpret our theorems in terms of power series over $k$. The most important tool that we use is the field of norms construction of Fontaine and Wintenberger.

Indices of inseparability and refined ramification breaks

Let K be a finite extension of Q_p and let L/K be a totally ramified (Z/pZ)^2-extension which has a single ramification break b. Byott and Elder defined a "refined ramification break" b_* for L/K. In this paper we prove that if p>2 and the index of inseparability i_1 of L/K is not equal to p^2b-pb then b_*=i_1-p^2b+pb+b.

On Ramification Filtrations and p-adic Differential Equations, II: mixed characteristic case

Let K be a complete discretely valued field of mixed characteristic (0, p) with possibly imperfect residue field. We prove a Hasse-Arf theorem for the arithmetic ramification filtrations on G_K, except possibly in the absolutely unramified and non-logarithmic case, or p=2 and logarithmic case. As an application, we obtain a Hasse-Arf theorem for filtrations on finite flat group schemes over O_K.

Ramification theory for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(0,p)$

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more general valuations and residue fields. As seen in VT16, the "defect" case gives rise to many interesting complications. In this paper, we present analogous results for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(0,p)$ in a more general setting. More specifically, the only assumption here is that the base field $K$ is henselian. In…

On tame ${\mathbb Z}/p{\mathbb Z}$ extensions with prescribed ramification

The tame Gras-Munnier Theorem gives a criterion for the existence of a ${\mathbb Z}/{\mathbb Z}$-extension of a number field $K$ ramified at exactly a set $S$ of places of $K$ prime to $p$ (allowing real Archimedean places when $p=2$) in terms of the existence of a dependence relation on the Frobenius elements of these places in a certain governing extension. We give a new and simpler proof of this theorem that also relates the set of such extensions of $K$ to the set of these dependence relations. After presenting this proof, we then reprove the key Proposition 3 using the more sophisticated …

On ramification filtrations and $p$-adic differential modules, I: equal characteristic case

Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors a…

On enumerating extensions of p-adic fields with given invariants

We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given ramification polygon. We supplement this with an algorithm to produce all ramification polygons of a given degree, and hence we can produce all totally ramified extensions of a given degree.

Purely inseparable extensions and ramification filtrations

In this article, we investigate the shift of Abbes and Saito's ramification filtrations of the absolute Galois group of a complete discrete valuation field of positive characteristic under a purely inseparable extension. We also study a functoriality property for characteristic forms.

Ramification Filtrations of Certain Abelian Lie Extensions of Local Fields

Let $G\subset x{\mathbb F}_q[\![x]\!]$ ($q$ is a power of the prime $p$) be a subset of formal power series over a finite field such that it forms a compact abelian $p$-adic Lie group of dimension $d\ge 1$. We establish a necessary and sufficient condition for the APF extension of local field corresponding to $\left({\mathbb F}_q(\!(x)\!), G\right)$ under the field of norms functor to be an extension of $p$-adic fields. We then apply this result to study family of invertible power series with coefficients in a $p$-adic integers ring and commute with a fixed noninvertible power series under the…

单位根与迭代扩张中的高分歧

Roots of unity and higher ramification in iterated extensions

本文研究迭代扩张中单位根的出现条件，证明若干有理函数族（如具有周期临界点的双临界函数）的高分歧群无限，并给出满足算术有限性的新例子，提出PCF映射的Sen定理动力类比猜想。

Invitation to higher local fields, Part I, section 18: On ramification theory of monogenic extensions

Ramification theory of monogenic extensions of complete discrete valuation fields is presented. Relations to Kato's conductor are discussed.

Counting Extensions of $\mathfrak{p}$-Adic Fields with Given Invariants

We give two specializations of Krasner's mass formula. The first formula yields the number of extensions of a $\mathfrak{p}$-adic field with given, inertia degree, ramification index, discriminant, and ramification polygon. We then refine this formula further to the case where an additional invariant related to the residual polynomials of the segments of the ramification polygon is given.

Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields

Let p>2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fractional field of the Witt ring of k. Let G and H be finite flat commutative group schemes killed by p over O_K and k[[u]], respectively. In this paper, we show the upper and the lower ramification subgroups of G and H in the sense of Abbes-Saito are naturally isomorphic to each other when they are associated to the same Kisin module.

An ultrametric space of Eisenstein polynomials and ramification theory

We give a criterion whether given Eisenstein polynomials over a local field K define the same extension over K in terms of a certain non-Archimedean metric on the set of polynomials. The criterion and its proof depend on ramification theory.

On the structure of the Galois group of the maximal pro-$p$ extension with restricted ramification over the cyclotomic $\mathbb{Z}_p$-extension

Let $k_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_\infty)$ the set of primes of $k_\infty$ lying above $S$. In the present paper, we will study the structure of the Galois group $\mathcal{X}_S (k_\infty)$ of the maximal pro-$p$ extension unramified outside $S (k_\infty)$ over $k_\infty$. We mainly consider the question whether $\mathcal{X}_S (k_\infty)$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic f…

Tame ramification and group cohomology

We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class of a tamely ramified finite galoisian extension coincide, and can be recovered from the parameter. We also give an elementary proof of Serre's mass formula in the tame case and in the simplest wild case, and we classify tame galoisian extensions of degree the cube of a prime.

Frobenius and non logarithmic ramification

A $\ell$-extension is said logarithmically unramified if it is locally cyclotomic. The purpose of this article is to explain the construction of the logarithmic Frobenius, which plays the role usually played by the classical Frobenius, but in the context of the logarithmic ramification.The interesting point is that usual and logarithmic Frobenius coincide when usual and logarithmic ramification are the same.

Characteristic cycle of a rank one sheaf and ramification theory

We compute the characteristic cycle of a rank one sheaf on a smooth surface over a perfect field of positive characteristic. We construct a canonical lifting on the cotangent bundle of Kato's logarithmic characteristic cycle using ramification theory and prove the equality of the characteristic cycle and the canonical lifting. As corollaries, we obtain a computation of the singular support in terms of ramification theory.

Specialization results and ramification conditions

Given a hilbertian field $k$ of characteristic zero and a finite Galois extension $E/k(T)$ with group $G$ such that $E/k$ is regular, we produce some specializations of $E/k(T)$ at points $t_0 \in \mathbb{P}^1(k)$ which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of $\mathbb{Q}$ of various finite groups with specified local behavior - ramified or unramified - at finitely many given primes. Secondly, in the case $k$ is a number field, we pr…

Minimal ramification and the inverse Galois problem over the rational function field $\mathbb{F}_p(t)$

The inverse Galois problem is concerned with finding a Galois extension of a field $K$ with given Galois group. In this paper we consider the particular case where the base field is $K=\F_p(t)$. We give a conjectural formula for the minimal number of primes, both finite and infinite, ramified in $G$-extensions of $K$, and give theoretical and computational proofs for many cases of this conjecture.

Analytic lie extensions of number fields with cyclic fixed points and tame ramification

- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Γ$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $σ$ be a generator of Gal(K/k), we study the constraints posed on the …

剩余域不完备的局部域的分歧群的刻画

A characterization of ramification groups of local fields with imperfect residue field

该研究借助剩余域完备的局部域的分歧群，刻画了剩余域不完备的局部域的分歧群，并以此重新证明了不同定义下阿贝尔扩张分歧群的一个等式。

Existence of covers with fixed ramification in positive characteristic

We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain information on the structure of the Hurwitz curve parametrizing these covers.

Characteristic cycle and wild ramification for nearby cycles of étale sheaves

In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito's logarithmic ramification filtration. This provides a positive answer to the main conjecture in Isabel Leal's article "On the ramification of étale cohomology groups" for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.

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.

Practice of incomplete p-ramification over a number field -- History of abelian p-ramification

The theory of p-ramification, regarding the Galois group of the maximal pro-p-extension of a number field K, unramified outside p and $\infty$, is well known including numerical experiments with PARI/GP programs. The case of ``incomplete p-ramification'' (i.e., when the set S of ramified places is a strict subset of the set P of the p-places) is, on the contrary, mostly unknown in a theoretical point of view. We give, in a first part, a way to compute, for any S $\le$ P, the structure of the Galois group of the maximal S-ramified abelian pro-p-extension H\_(K,S) of any field K given by means o…

Artin-Schreier-Witt 扩张与分歧断点

Artin-Schreier-Witt extensions and ramification breaks

本研究针对特征p>0的局部域上的Artin-Schreier-Witt循环扩张，采用直接方法证明：既约Witt向量对应的全分歧扩张的上分歧断点计算公式，在剩余域为任意完满域时仍成立，推广了有限域情形下的已有结果。

亏格理论、控制域、分歧与弗罗贝尼乌斯

Genus theory, governing field, ramification and Frobenius

本文通过控制域方法推广亏格理论，将循环扩张的亏格数表示为分歧素数弗罗贝尼乌斯元素所构矩阵的秩，并在二次扩张情形下用勒让德符号刻画。

On the Shafarevich Group of Restricted Ramification Extensions of Number Fields in the Tame Case

Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X} \stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X} \stackrel{\simeq}{\t…

Relative Bogomolov extensions

An algebraic extension K of the rationals has the Bogomolov property if the absolute logarithmic height of non-torsion points of K* is bounded away from 0. We define a relative extension L/K to be Bogomolov if this holds for points of L\K. We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K, when K is Galois over Q.

The minimal ramification problem for rational function fields over finite fields

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases.

秩2 Drinfeld 模的 Tate 模的分歧

Ramification of Tate modules for rank $2$ Drinfeld modules

本文研究秩2 Drinfeld模分点生成的函数域扩张的分歧，定义其导子作为椭圆曲线导子的类似物，通过导子计算在特定受限情形下证明了Szpiro猜想的类似结论。

The v-function in the wild McKay correspondence is not determined by the ramification filtration

The \(v\)-function appears in the wild McKay correspondence and it is an invariant of a Galois extension over a local field which measures about the ramification of extension. The ramification filtration is also such invariant. Yasuda presented the problem whether the \(v\)-function is determined by ramification filtration. In this paper, we give a formula for the \(v\)-function for 2-dimensional representations of the bicyclic group of order \(p^2\) with \(p\) the characteristic of our base field. Using this formula, we construct an example of \(v\)-function not determined by the ramification…

Fontaine's property (Pm) at the maximal ramification break

We completely determine which extension of local fields satisfies Fontaine's property (Pm) for a given real number m. A key ingredient of the proof is the local class field theory of Serre and Hazewinkel.

On ramification theory in the imperfect residue field case

We consider the class of complete discretely valued fields such that the residue field is of prime characteristic p and the cardinality of a $p$-base is 1. This class includes two-dimensional local and local-global fields. A new definition of ramification filtration for such fields is given. It appears that a Hasse-Herbrand type functions can be defined with all the usual properties. Therefore, a theory of upper ramification groups, as well as the ramification theory of infinite extensions, can be developed. Next, we consider an equal characteristic two-dimensional local field $K$. We introduc…

On the $p$-rank of class groups of $p$-extensions

We prove a local-global principle for the embedding problems of global fields with restricted ramification. By this local-global principle, for a global field $k$, we use only the local information to give a presentation of the maximal pro-$p$ Galois group of $k$ with restricted ramification, when some Galois cohomological conditions are satisfied. For a Galois $p$-extension $K/k$, we use our presentation result for $k$ to study the structure of pro-$p$ Galois groups of $K$. Then for $k=\mathbb{Q}$ and $k=\mathbb{F}_q(t)$ with $p\nmid q$, we give upper and lower bounds for the rank of $p$-tors…

具有受限分歧的数域的单远阿贝尔重构

Mono-anabelian Reconstruction of Number Fields with Restricted Ramification

本文利用Hoshi的单远阿贝尔重构方法，从射有限群G_{K,S}出发，对密度为1的素数子集重构数域K及其在S外的最大非分歧扩张K_S。

迭代扩张与分歧二分法

Iterated extensions and the ramification dichotomy

研究局部域上多项式迭代生成塔的分歧性质，证明塔要么非分歧要么深度分歧，并应用于Frobenius型情形得到完美域。

任意赋值环的上分歧群

Upper Ramification Groups for Arbitrary Valuation Rings

本文通过将 Hensel 赋值环的整闭包表示为完全交子环的滤过并，把 Saito 分歧理论推广到任意 Hensel 赋值环，并研究了此前未处理的亏差扩张。