Group schemes and local densities of ramified hermitian lattices in residue characteristic 2 Part I

关于签名 $(n-1,1)$ 的分歧酉局部模型的模描述

On the moduli description of ramified unitary local models of signature $(n-1,1)$

本文在特征非2条件下显式刻画签名 $(n-1,1)$ 分歧酉局部模型的特殊纤维，证实 Smithling 猜想，并推广至 Pappas–Zhu 模型与 Shimura 簇整模型。

生成全分歧扩张的艾森斯坦多项式族、扩张的识别及类域的构造

A family of Eisenstein polynomials generating totally ramified extensions, identification of extensions and construction of class fields

设$K$是一个具有有限剩余域的局部域，我们定义了艾森斯坦多项式的一个标准形，它依赖于均匀化子$π_K$和剩余表示的选择.族中多项式生成的扩张的同构类穷尽了所有的全分歧扩张，并且每个同构类$L/K$出现的重数总是小于$L$在$K$上的共轭数.本文提出了一种由一般艾森斯坦多项式确定扩张的特殊多项式集合的恢复算法。我们还给出了一个快速判定一个多项式生成的扩张不同于一组特殊多项式生成的扩张的判据，这个判据不仅依赖于Krasner等人所考虑的艾森斯坦多项式集合上的通常距离.最后给出了一个构造确定一般度上完全分歧类域的唯一特殊方程的算法，给出了一组范数的适当表示。

分歧素数上某些酉 Shimura 簇的半稳定模型

Semi-stable models for some unitary Shimura varieties over ramified primes

本文针对签名 $(2,n-2)$ 的酉群相关的 Shimura 簇，通过显式分解局部模型，在虚二次域分歧的奇素数 $p$ 上构建了正则 $p$-adic 整模型。

具有分歧自同态结构的 $p$-可分群的 Hodge-Newton 滤波

Hodge-Newton filtration for $p$-divisible groups with ramified endomorphism structure

本文基于有限平坦群概型的 Harder-Narasimhan 理论，证明了具有分歧自同态结构的 $p$-可分群上 Hodge-Newton 滤波的存在性，并给出了其存在的充分条件。

特殊 parahoric 的野分歧酉局部模型。奇数维情形

Wildly ramified unitary local models for special parahorics. The odd dimensional case

本文为奇数维野分歧酉相似群构造了特殊 parahoric 级局部模型，利用 Bruhat-Tits 理论分析了两类共轭子群，证明了其平滑性或正规且 Cohen-Macaulay 性质，并建立了与 Scholze-Weinstein v-层模型的联系及显式模空间解释。

Transfers of some Hecke elements for possibly ramified base change in GL_n

In this paper we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residue characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy-Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semi-simple descend in the twisted space according to Waldspurger's fundamental work "L'endoscopie tordue n'est pas si tordue".

k-Isomorphism Classes of Local Field Extensions

Let K be a local field of characteristic p with perfect residue field k. In this paper we find a set of representatives for the k-isomorphism classes of totally ramified separable extensions L/K of degree p. This extends work of Klopsch, who found representatives for the k-isomorphism classes of totally ramified Galois extensions L/K of degree p.

On the structure of certain valued fields

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th residue rings are isomorphic for each $n\ge 1$, then $K_1$ and $K_2$ are isometric and isomorphic. More generally, for $n_1\ge 1$, there is $n_2$ depending only on the ramification indices of $K_1$ and $K_2$ such that any homomorphism from the $n_1$-th residue ring of $K_1$ to the $n_2$-th residue ring of $K_2$ can be lifted to a homomorphism between the valuat…

Explicit integral Galois module structure of weakly ramified extensions of local fields

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…

抛物群方案的简化

Reductification of parahoric group schemes

本文研究抛物群方案在有限伽罗华扩张下的可约性，并证明在良好余特征下，一般平凡抛物 torsors 的平凡性。

局部模型的奇点

Singularities of local models

本文构造了Shimura簇的局部模型，证明了除剩余特征2下的奇酉群外其特殊纤维的既约性、Cohen-Macaulay性质及等特征下的（伪）有理性，并基于parahoric群概形与完美几何分析了野分歧情形下的奇点。

On ramification in transcendental extensions of local fields

Let $L/K$ be an extension of complete discrete valuation fields, and assume that the residue field of $K$ is perfect and of positive characteristic. The residue field of $L$ is not assumed to be perfect. In this paper, we prove a formula for the Swan conductor of the image of a character $χ\in H^1(K, \mathbb{Q}/\mathbb{Z})$ in $H^1(L, \mathbb{Q}/\mathbb{Z})$ for $χ$ sufficiently ramified. Further, we define generalizations $ψ_{L/K}^{\mathrm{ab}}$ and $ψ_{L/K}^{\mathrm{AS}}$ of the classical Hasse-Herbrand $ψ$-function and prove a formula for $ψ_{L/K}^{\mathrm{ab}}(t)$ for sufficiently large $t…

邀请到更高的局部域，第一部分，第15节：关于完备离散赋值域的Milnor K-群的结构

Invitation to higher local fields, Part I, section 15: On the structure of the Milnor K-groups of complete discrete valuation fields

这项工作包含了一个列表的所有已知结果的商过滤的Milnor K-群的一个完整的离散赋值领域的微分模的剩余域。利用指数映射和同伦复形研究了特征为0的弱分歧域与特征为p的剩余域的情形。

The density of ramified primes

Let $F$ be a number field, $\mathcal{O}$ be a domain with fraction field $\mathcal{K}$ of characteristic zero and $ρ: \mathrm{Gal}(\overline F/F) \to \mathrm{GL}_n(\mathcal{O})$ be a representation such that $ρ\otimes\overline{\mathcal{K}}$ is semisimple. If $\mathcal{O}$ admits a finite monomorphism from a power series ring with coefficients in a $p$-adic integer ring (resp. $\mathcal{O}$ is an affinoid algebra over a $p$-adic number field) and $ρ$ is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes o…

关于 Krämer 模型的 Kudla-Rapoport 猜想的证明

A proof of the Kudla-Rapoport conjecture for Krämer models

本文证明了分歧处酉 Rapoport-Zink 空间 Krämer 模型的 Kudla-Rapoport 猜想，确立了特殊闭链算术相交数与修正导出局部密度的精确等式，并据此放宽了酉 Shimura 簇算术 Siegel-Weil 公式的局部假设。

关于某些值域域的结构 II

On the structure of certain valued fields II

本文研究有限分歧亨泽尔值域域的结构，通过高长度剩余环证明近似提升定理并确定最优误差界，进而得到若干 Ax-Kochen-Ershov 原理及公式等价性结果。

高局部域的邀请，第一部分，第16节：不使用K-群的高类域理论

Invitation to higher local fields, Part I, section 16: Higher class field theory without using K-groups

本文将一维类域理论推广到具有任意不可分p-闭剩余域的完备离散赋值域的交换全分歧p-扩张理论及其应用。

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.

具有非完美剩余域的局部域的Hasse-Arf性质与阿贝尔扩张

Hasse-Arf property and abelian extensions for local fields with imperfect residue fields

本文在剩余域非完美的完备离散赋值域中，证明了类似完美剩余域情形下全分歧扩张为阿贝尔扩张的条件，并给出Hasse-Arf定理的逆定理，构造了具有非整数上分歧断裂的Galois扩张。

Ramification of Wild Automorphisms of Laurent Series Fields

Let $K$ be a complete discrete valuation field with residue class field $k$, where both are of positive characteristic $p$. Then the group of wild automorphisms of $K$ can be identified with the group under composition of formal power series over $k$ with no constant term and $X$-coefficient $1$. Under the hypothesis that $p > b^2$, we compute the first nontrivial coefficient of the $p$th iterate of a power series over $k$ of the form $f = X + \sum_{i \geq 1} a_iX^{b+i}$. As a result, we obtain a necessary and sufficient condition for an automorphism to be ``$b$-ramified,'' having lower ramifi…

The maximal discrete extension of the Hermitian modular group

Let $Γ_n(\mathcal{\scriptstyle{O}}_\mathbb{K})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $SU(n,n;\mathbb{C})$, which coincides with the normalizer of $Γ_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$. The description involves the $n$-torsion subgroup of the ideal class group of $\mathbb{K}$. This group is defined over a particular number field $\widehat{\mathbb{K}}_n$ and we can describe the ramified primes in it. In the case $n=2$ we give an explicit description, which invol…

任意剩余特征下驯顺超尖表示（supercuspidal representations）的构造

Construction of tame supercuspidal representations in arbitrary residue characteristic

本文在任意剩余特征下，通过紧致诱导构造了非阿基米德局部域上连通约化群的超尖表示，涵盖 Yu 的构造并适用于剩余特征 2，且无需满足条件 GE2。

The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its étale version is …

关于几何 $C_1$ 域

On geometrically $C_1$ fields

本文定义了几何 $C_1$ 域，并证明在等特征为 0 且值群可除的 Henselian 赋值域中，该性质可从剩余域提升至赋值域，同时证明具有可除值群与有限剩余域的代数极大赋值域及局部域的极大全分歧扩张均为几何 $C_1$ 域。

Witt向量仿射Springer纤维

Witt vector affine Springer fibers

本文在混合特征局部域上，对本质驯分歧且剩余特征非坏的重约群，建立了Witt向量仿射Springer纤维的维数公式，涉及判别式赋值、Artin导子和Kottwitz不变量。

Generation of Local Unitary Groups

Let $E$ be a two-dimensional étale algebra over a non-Archimedean local field $K$ of characteristic zero. We show that the unitary group of a non-degenerate hermitian lattice over $E$ is generated by symmetries and rescaled Eichler isometries. In the appendix we show that unless $E/K$ is a ramified dyadic field extension and the residue field has two elements, symmetries suffice.

On the ramification of étale cohomology groups

Let $K$ be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let $X$ be a connected, proper scheme over $\mathcal{O}_K$, and let $U$ be the complement in $X$ of a divisor with simple normal crossings. Assume that the pair $(X,U)$ is strictly semi-stable over $\mathcal{O}_K$ of relative dimension one and $K$ is of equal characteristic. We prove that, for any smooth $\ell$-adic sheaf $\mathscr{G}$ on $U$ of rank one, at most tamely ramified on the generic fiber, if the ramification of $\mathscr{G}$ is bounded by $t+$ for the logarithmic upper rami…

Local-global principles for hermitian spaces over semi-global fields

Let $K$ be a complete discrete valued field with residue field $k$ and $F$ the function field of a curve over $K$. Let $A \in {}_2Br(F)$ be a central simple algebra with an involution $σ$ of any kind and $F_0 =F^σ$. Let $h$ be an hermitian space over $(A, σ)$ and $G = SU(A, σ, h)$ if $σ$ is of first kind and $G = U(A, σ, h)$ if $σ$ is of second kind. Suppose that $\text{char}(k) \neq 2$ and ind$(A)\leq 4$. Then we prove that projective homogeneous spaces under $G$ over $F_0$ satisfy a local-global principle for rational points with respect to discrete valuations of $F$.

指数幂级数、伽罗瓦模结构和微分模

Exponential power series, Galois module structure and differential modules

利用Pulita工作的启发，利用新的过收敛p-adic指数幂级数，构造了Qp的某个非分歧扩张K的交换弱分歧扩张的逆差的平方根的自对偶正规基生成元.这些扩张，其集合我们表示为M，是Mp，2，K的最大交换完全，广泛和弱分歧的扩张，其范数群包含p的K上的p次子扩张。我们的结构遵循Pickett的，谁处理了相同的集合M的K的扩张，但不依赖于一个基础的选择的剩余域k的K。相反，它从k的射影空间到M上建立了一对一的对应，与K/Qp的伽罗瓦群的作用交换。我们非常精确地描述了M中扩张的范数群。当K不等于Qp时，它们的合成Mp，2给出了Qp的非交换弱分歧扩张的一个有趣例子，其中伽罗瓦群同构于一个圈积.最后，我们表明，稍加修改，我们的过收敛指数幂级数赋予某些微分模与Frobenius结构，推广Pulita的结果。不幸的是，它们失去了我们需要建立自对偶正规基生成元的性质，因此伽罗瓦模结构和微分模之间的联系还没有得到。

局部特征$p$域上正规基生成元的一个赋值准则

A valuation criterion for normal basis generators in local fields of characteristic $p$

设$K$是特征$p$的完备局部域，具有完美剩余域。设$L/K$是有限的、完全分歧的Galois $p$ -扩张.若$π_L\in L$是素元，$p'(x)$是$π_L$的极小多项式在$K$上的导数，则相对不同的$\euD_{L/K}$由$p'(π_L)\in L$生成.令$v_L$为使用$v_L(L)=\mathbb{Z}$标准化的标准化估值。我们发现，任何元素$ρ\in L$与$v_L(ρ)\equiv -v_L(p'(π_L))-1\bmod[L:K]$生成一个正规基，$K[{Gal}(L/K)]\cdotρ=L$。这个准则是严格的：给定任何整数$i$使得$i\not\equiv -v_L(p'(π_L))-1\bmod[L:K]$，存在具有$v_L(ρ_i)=i$的$ρ_i\in L$使得$K[{Gal}(L/K)]\cdotρ_i\subsetneq L$。

关于$(φ,Γ)$-模不完全系数环积分结构的注记

A remark on an integral structure of the imperfect coefficient ring of $(φ,Γ)$-modules

本文证明了在$p$进数情况下，$(φ,Γ)$-模的不完全系数环与$A_\mathrm{inf}$的交集与$W(k_{K_\infty})[[μ]]$存在同构关系，扩展了Nathalie Wach的结论。

单位根的群模式

Models of the group schemes of roots of unity

设O_K是具有混合特征（0，p）的离散赋值环，其剩余域为k.利用Sekiguchi和Suwa的工作，我们构造了p^n次根的群格式μ_{p^n，K}的有限平坦O_K-模型，我们称之为Kummer群格式.我们仔细设置的一般框架和代数性质的建设。当k是完备的，O_K是Witt向量环W（k）的完全全分歧扩张时，我们对μ_{p^n，K}的有限平坦模型的Breuil-Kisin模进行了平行的研究，从而比较了Kummer群和Breuil-Kisin模的构造.我们计算n < 4的这些对象。这使我们猜想μ_{p^n，K}的所有有限平坦模型都是Kummer群格式。

强可除格与不完备剩余域情形下的晶体上同调

Strongly divisible lattices and crystalline cohomology in the imperfect residue field case

本文在不完备剩余域情形下，用强可除格分类半稳定表示的$\mathbf{Z}_p$-格，并给出相关晶体上同调的上同调描述，推广了Liu及Cais-Liu的结果。

关于不完全剩余域情形的Milnor $K$-理论及其在周期-指数问题中的应用

On Milnor $K$-theory in the imperfect residue case and applications to period-index problems

本文通过定义伪完美扩张，证明其可分裂混合特征完备离散赋值域的高阶上同调类，从而给出Brauer p-维数上界，并改进半全局域上的周期-指数界。

$p^2$次特征PDMATHAAA局部域扩张的Galois支架和Galois模结构

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

在具有完美剩余域的局部域的伽罗瓦扩张中，伽罗瓦支架是正规基对扩张域赋值的一种适应，因此可以用来回答伽罗瓦模结构问题.本文给出了特征$p$局部域的度$p^2$的全分歧伽罗瓦扩张中存在伽罗瓦支架的一个充分条件。当我们局限于$p=3$时，这个条件就变得必要了。对于扩展$L/K$度$p^2$，满足这个条件，我们确定的伽罗瓦模结构的整数环的$L$的整数环是免费的充要条件，其相关的顺序在$K[Gal(L/K)]$。

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。

关于局部域的正无穷pro-p-群和算术profinite扩张

On just infinite pro-p-groups and arithmetically profinite extensions of local fields

野群是特征p的局部域的野自同构的群。本文应用Fontaine-Wintenberger的范数域场论研究了野群的结构。特别地，我们给出了R的一个新的简短证明.卡米纳定理，它说，每一个亲p-群与可数个开放子群是同构的一个闭子群的野生集团。研究了换位子群异常小的野群的闭子群T.将群T看作是p进域的算术有限扩张的伽罗瓦群，我们肯定地回答了关于局部域的深分歧扩张的Coates-Greenberg问题。最后利用子群T证明了在具有特征p的有限剩余域的交换完备局部Noether整环上的野群不是解析的。

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。

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 …

局部域上的 Chebotarev 密度定理

A Chebotarev Density Theorem over Local Fields

本文在局部域上建立 Chebotarev 型密度定理，利用可容许对和 p-进积分计算分裂密度，并由此证明 Bhargava 等人关于 p-进多项式分解密度的猜想。

关于二元Hermitian型对空间的Shintani zeta函数

On the Shintani zeta function for the space of pairs of binary Hermitian forms

在本文中，k是数域。我们固定k的二次扩张k_1 = k（a_0），其中a_0 = sqrt（B_0），对于k ^\times/（k ^\times）^2中的某个B_0。在本文中，我们考虑zeta函数定义的二元厄米特形式对的空间。这是我们在第2节[1]中讨论的预齐次向量空间，并且是[4]中D_4情形的非分裂形式。本文的目的是确定调整zeta函数的主体部分。我们的主要结果是定理（8.15）。我们的情况类似于我们在第5章[5]中讨论的二元二次型对空间。然而，问题的意义，调整zeta函数的极点结构，以及调整项与上述情况不同。我们情形的zeta函数是类数乘以形式为k（sqrt（B_0），sqrt（B））的域的调节器的计数函数，其中B_0固定，B在k中，而上述情形的zeta函数本质上与二元二次型空间相同。我们的情况是第一个例子，其中我们需要两个调整项，而上述情况的调整项基本上与二元二次型空间的调整项相同。本文不再证明调节项的亚纯延拓。为了密度定理，我们仍然得到zeta函数最右边极点的留数（没有调整）。如果这个空间出现在一个更大的预齐次向量空间中。因此，调整zeta函数的主部公式应足以满足大多数应用。由于我们没有进行局部理论，所以还没有得到相应的密度定理。