David S. Dummit, Hershy Kisilevsky
We examine whether it is possible to realize finite groups $G$ as Galois groups of minimally tamely ramified extensions of $\mathbb{Q}$ and also specify both the inertia groups and the further decomposition of the ramified primes.
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Let $K/F$ be a quadratic tamely ramified extension of a non-Archimedean local field $F$ of characteristic zero. In this paper, we give an explicit formula for Langlands' lambda function $λ_{K/F}$.
In this note, we treat two dimensional complete local rings which are called "semi-stable local rings" and discuss the tame class filed theory. In 1987, Professor S. Saito completed the unramified class field theory of the general two dimensional excellent local rings. On the other hand, we discuss the tame class field theory for a specific local rings.
Pro-$\ell$-by-cyclotomic and tamely ramified variants of the Neukirch-Uchida Theorem
本文证明了通过 pro-$\ell$-by-cyclotomic 和 tame 分支商可以恢复数域的同构类型，扩展了 Neukirch-Uchida 定理。
Let K be a discretly henselian field whose residue field is separably closed. Answering a question raised by G. Prasad, we show that a semisimple K-- group G is quasi-split if and only if it quasi--splits after a finite tamely ramified extension of K.
For a prime number p, we denote by K the cyclotomic Z_p-extension of a number field k. For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension of K unramified outside S. This paper treats the case where S does not contain p and k is the rational number field or an imaginary quadratic field. In this case, we prove the explicit formulae for the free ranks of the S-ramified Iwasawa modules as abelian pro-p groups, by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of a…
We prove a large finite field version of the Boston--Markin conjecture on counting Galois extensions of the rational function field with a given Galois group and the smallest possible number of ramified primes. Our proof involves a study of structure groups of (direct products of) racks.
Let p be an odd prime, and k_\infty the cyclotomic Z_p-extension of an abelian field k. For a finite set S of rational primes which does not include p, we will consider the maximal S-ramified abelian pro-p extension M_S(k_\infty) over k_\infty. We shall give a formula of the Z_p-rank of Gal(M_S(k_\infty)/k_\infty). In the proof of this formula, we also show that M_{q}(k_\infty)/L(k_\infty) is a finite extension for every real abelian field k and every rational prime q distinct from p, where L(k_\infty) is the maximal unramified abelian pro-p extension over k_\infty.
The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O_k of k, which, together with the degree [K:k] of the extension determines the O_k-module structure of O_K. We call R_t(k,G) the classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. In this paper we will develop some of the ideas contained in arXiv:0910.5080 to study some l-groups, where l is an odd prime number. In particular, together with [1] we…
We will give an explicit construction of irreducible suparcuspidal representations of the special linear group over a non-archimedean local field and will speculate its Langlands parameter by means of verifying the Hiraga-Ichino-Ikeda formula of the formal degree of the supercuspidal representations.
On the Finiteness and Structure of Galois Groups of Tamely ramified pro-p Extensions of Imaginary Quadratic Fields
本文研究虚二次域在有限集合外非分歧的最大 pro-p 扩张的 Galois 群，给出 p 为奇数时的显式表示，并确定 Q(i) 的最大 pro-3 扩张的结构。
The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O_k of k, which, together with the degree [K:k] of the extension determines the O_k-module structure of O_K. We call rt(k,G) the classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A'-groups inductively, starting by abelian groups and then considering semidirect products of A'-groups with abelian groups of relatively prime order and direc…
Let $ K $ be a number field and let $ L/K $ be a tamely ramified radical extension of prime degree $ p $. If $ K $ contains a primitive $ p^{th} $ root of unity then $ L/K $ is a cyclic Kummer extension; in this case the group algebra $ K[G] $ (with $ G=\mbox{Gal}(L/K) $) gives the unique Hopf-Galois structure on $ L/K $, the ring of algebraic integers $ \mathfrak{O}_L $ is locally free over $ \mathfrak{O}_{K}[G] $ by Noether's theorem, and Gómez Ayala has determined a criterion for $ \mathfrak{O}_L $ to be a free $ \mathfrak{O}_{K}[G] $-module. If $ K $ does not contain a primitive $ p^{th} $…
Hecke action on tamely ramified Eisenstein series over $\mathbb{P}^1$
本文研究有理函数域在三个位点驯分歧的自守函数空间，提出 Eisenstein 级数生成的仿射 Hecke 代数三模的生成元与关系猜想，并在 PGL(2) 和 SL(3) 情形给出证明。
Assuming a certain form of resolution of singularities, we prove a general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics. This specializes to well-known results in residue characteristic $0$ and unramified mixed characteristic. It also encompasses the conditional existential decidability results known for $\mathbb{F}_p(\!(t)\!)$ and its finite extensions, due to Denef-Schoutens. On the other hand, it also applies to the setting of infinite ramification, providing us with an abundance of infinitely ramified extensions of $\mathbb{Q}_p$ and $\mathbb{F}_…
Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli…
We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Saïdi, Sugiyama-Yasuda, and Anbar-Tutdere.
The formal degree conjecture and the root number conjecture are verified with respect to supercuspidal representations of $Sp_{2n}(F)$ and $L$-parameters associated with tamely ramified extension $K/F$ of degree $2n$. The supercuspidal representation is constructed as a compact induction from an irreducible unitary representation of the hyper special compact group $Sp_{2n}(O_F)$, which is explicitly constructed, based upon the general theory developed by the author, by $K$ and certain character $θ$ of the multiplicative group $K^{\times}$. $L$-parameter is constructed by the data $\{K,θ\}$ by …
On a tamely ramified local relative Langlands conjecture via categorical representations
该研究在非分歧相对局部朗兰兹猜想等假设成立的前提下，用相对朗兰兹对偶给出环路空间上伊瓦霍里等变D-模满子范畴的谱描述，证实了驯分歧局部相对朗兰兹猜想的一个轻微变体。
On tamely ramified infinite Galois extensions
本文通过引入稳定惯性生成pro-p群的概念，刻画并构造了数域最大驯顺分歧扩张的伽罗瓦群的有限生成pro-p商群，并给出了可实现与不可实现商群的分类结果。
For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conj…
We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.
The finitude of tamely ramified pro-$p$ extensions of number fields with cyclic $p$-class groups
本文研究了 $p$-类群循环的数域在单个非 $p$-进素理想外不分歧的极大 pro-$p$ 扩张的 Galois 群的有限性，并证明了在生成元秩为 2 时该群对大多数素理想有限。
The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(ϕ, Γ_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.
Based upon the general theory, developed by the author, on the parametrization of the irreducible representations of the hyper special compact groups corresponding to the regular adjoint orbit, supercuspidal representations of $SL_n(F)$ are explicitly constructed for which the formal degree conjecture and the root number conjecture are verified with respect to certain $L$-parameter defined, by means of Kaletha, that is, the local Langlands correspondence of tori and the Langlands-Schelstad procedure, by the data parametrizing the irreducible representations of the hyper special compact subgrou…
Shalika germs for tamely ramified elements in $GL_n$
该研究证明了GLₙ中稳分歧元轨道积分的显式组合公式，涵盖仿射Springer纤维权多项式等副产品，为相关猜想提供了证据。
This paper gives some restrictions on finite groups that can occur as Galois groups of extensions over $\Q$ and over $\F_q(t)$ ramified only at one finite prime. Over $\Q$, we strengthen results of Jensen and Yui about dihedral extensions and rule out some non-solvable groups. Over $\F_q(t)$ restrictions are given for symmetric groups and dihedral groups to occur as tamely ramified extension over $\F_q(t)$ ramified only at one prime.
We give all normal integral bases for the simplest cubic field $L_n$ generated by the roots of Shanks' cubic polynomial when these bases exist, that is, $L_n/\mathbb Q$ is tamely ramified. Furthermore, as an application of the result, we give an explicit relation between the roots of Shanks' cubic polynomial and the Gaussian periods of $L_n$ in the case $L_n/\mathbb Q$ is tamely ramified, which is a generalization of the work of Lehmer, Châtelet and Lazarus in the case that the conductor of $L_n$ is equal to $n^2+3n+9$.
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.
Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly $…
We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.
On S-ramified T-split Iwasawa modules
本文修正了S-分歧T-分裂Iwasawa模的公式，计算了λ不变量的缺陷群，并推广了相关结果。
Let $p$ and $\ell$ be distinct primes, and $ρ$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $ρ$ "ramified no worse than $ρ$", generalizing the minimally ramified deformation condition for $GL_n$ studied in \cite{cht08}. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the…
On tamely ramified $\mathcal G$-bundles on curves
本文证明光滑射影曲线上的仿射 Bruhat-Tits 群概形可表示为驯顺覆盖上约化群概形的不变量，并建立两类扭子及其模叠之间的联系，同时发展等变扭子的局部类型理论。
We prove the geometrical Satake isomorphism for a reductive group defined over F=k((t)), and split over a tamely ramified extension. As an application, we give a description of the nearby cycles on certain Shimura varieties via the Rapoport-Zink-Pappas local models.
Logarithmic Geometry and Geometric Class Field Theory
利用对数几何升级Deligne的几何类域论，通过定义带框架的对数Picard空间并证明除子到Picard映射的对数紧化具有对数单连通纤维，建立了曲线上的局部系统与乘法局部系统之间的典范双射，从而重新推导了函数域的适度分歧整体Artin互反律。
We consider a tamely ramified abelian extension of local fields of degree n, without assuming the presence of the nth roots of unity in the base field. We give an explicit formula which computes the local reciprocity map in this situation.
We compute equivariant Euler characteristics of locally free sheaves on curves, thereby generalizing several results of Kani and Nakajima. For instance, we extend Kani's computation of the Galois module structure of the space of global meromorphic differentials which are logarithmic along the ramification locus from the tamely ramified to the weakly ramified case.
We provide an explicit construction of the local Langlands correspondence for general tamely-ramified reductive p-adic groups and a class of wildly ramified Langlands parameters. Furthermore, we verify that our construction satisfies the expected properties of such a correspondence. More precisely, we show that each L-packet we construct admits a parameterization in terms of the Langlands dual group, contains a unique generic element for a fixed Whittaker datum, satisfies the formal degree conjecture, is compatible with twists by central and cocentral characters, provides a stable virtual char…
Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I
本文证明了离散赋值域上Z/n-覆盖的P^1的导子-判别式不等式，推广了超椭圆曲线情形，并加强了Kohls关于Jacobian导子指数的结果。
Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say that L/K is weakly ramified when G_2 is trivial. Let O_L be the valuation ring of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring O_K[G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every…
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。
Let $K$ be a finite tamely ramified extension of $\Q_p$ and let $L/K$ be a totally ramified $(\Z/p^n\Z)$-extension. Let $π_L$ be a uniformizer for $L$, let $σ$ be a generator for $\Gal(L/K)$, and let $f(X)$ be an element of $Ø_K[X]$ such that $σ(π_L)=f(π_L)$. We show that the reduction of $f(X)$ modulo the maximal ideal of $Ø_K$ determines a certain subextension of $L/K$ up to isomorphism. We use this result to study the field extensions generated by periodic points of a $p$-adic dynamical system.
Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module.
As an analogue of a link group, we consider the Galois group of the maximal pro-$p$-extension of a number field with restricted ramification which is cyclotomically ramified at $p$, i.e, tamely ramified over the intermediate cyclotomic $\mathbb Z_p$-extension of the number field. In some basic cases, such a pro-$p$ Galois group also has a Koch type presentation described by linking numbers and mod $2$ Milnor numbers (Rédei symbols) of primes. Then the pro-$2$ Fox derivative yields a calculation of Iwasawa polynomials analogous to Alexander polynomials.
Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a `weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of $p$-adic valuations of epsilon constants.
We study the nonclassical Hopf-Galois module structure of rings of algebraic integers in some extensions of $ p $-adic fields and number fields which are at most tamely ramified. We show that if $ L/K $ is an unramified extension of $ p $-adic fields which is $ H $-Galois for some Hopf algebra $ H $ then $ \OL $ is free over its associated order $ \AH $ in $ H $. If $ H $ is commutative, we show that this conclusion remains valid in ramified extensions of $ p $-adic fields if $ p $ does not divide the degree of the extension. By combining these results we prove a generalisation of Noether's th…
N. Katz has shown that any irreducible representation of the Galois group of F_q((t)) has unique extension to a special representation of the Galois group of k(t) unramified outside 0 and infinity and tamely ramified at infinity. In this paper we analyze the number of not necessarily special such extensions and relate this question to a description of bases in irreducible representations of multiplicative groups of division algebras.
We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a can…
We study ultrametric germs in one variable having an irrationally indifferent fixed point at the origin with a prescribed multiplier. We show that for many values of the multiplier, the cycles in the unit disk of the corresponding monic quadratic polynomial are "optimal" in the following sense: They minimize the distance to the origin among cycles of the same minimal period of normalized germs having an irrationally indifferent fixed point at the origin with the same multiplier. We also give examples of multipliers for which the corresponding quadratic polynomial does not have optimal cycles. …
On trace-one generators of abelian cubic fields
该研究针对驯分歧阿贝尔三次数域，证明了特定首一整系数迹一多项式的数量与对应二次域中特定范数的理想数量相等，为数域生成元相关问题提供了数论结论。
Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers II
本文为温和分歧循环覆盖建立 conductor-discriminant 不等式，证明其贡献量非负性并刻画负值情形为有理双重点，推广了 Ogg、Saito 等人的工作。
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…
The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes.
Local models are schemes which are intended to model the étale-local structure of p-adic integral models of Shimura varieties. Pappas and Zhu have recently given a general group-theoretic construction of flat local models with parahoric level structure for any tamely ramified group, but it remains an interesting problem to characterize the local models, when possible, in terms of an explicit moduli problem. In the setting of local models for ramified, quasi-split GU_n, work towards an explicit moduli description was initiated in the general framework of Rapoport and Zink's book and was subsequ…
On Classifying Extensions of $p$-adic Fields
针对 $\mathbb{Q}_p$ 的三次域扩张，提出基于不可约多项式系数确定同构类的闭式方法，并讨论了野分歧情形及任意素数次顺分歧扩张的分类方法。
Let $K$ be a finite extension of $\mathbb{Q}_p$. We study the geometry and cohomology of the tamely ramified cover of Drinfeld's symmetric space over $K.$ We prove, in a purely local way, that the cohomology in middle degree realises the local Langlands correspondence and local Jacquet-Langlands correspondence for the depth zero supercuspidal representations.
We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters.
Normal integral bases of Lehmer's cyclic quintic fields
本研究针对Emma Lehmer参数多项式生成的温和分歧循环五次域$K_n$，仅用多项式根给出其所有正规整基，推广了Lehmer和Spearman-Williams在特定参数条件下的相关研究结果。
- 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 …
We prove three theorems concerning the Hopf-Galois module structure of fractional ideals in a finite tamely ramified extension of $ p $-adic fields or number fields which is $ H $-Galois for a commutative Hopf algebra $ H $. Firstly, we show that if $ L/K $ is a tame Galois extension of $ p $-adic fields then each fractional ideal of $ L $ is free over its associated order in $ H $. We also show that this conclusion remains valid if $ L/K $ is merely almost classically Galois. Finally, we show that if $ L/K $ is an abelian extension of number fields then every ambiguous fractional ideal of $ L…
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