Ahmed Abbes, Takeshi Saito
Classically the ramification filtration of the Galois group of a complete discrete valuation field is defined in the case where the residue field is perfect. In this paper, we define without any assumption on the residue field, two ramification filtrations and study some of their properties.
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The graded quotients of the logarithmic ramification groups of a local field of mixed characteristic is killed by the residue characteristic. Its characters are described by differential forms.
We study the ramification of fierce cyclic Galois extensions of a local field $K$ of characteristic zero with a one-dimensional residue field of characteristic $p>0$. Using Kato's theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs $(δ_i,ω_i)$, where $δ_i$ is a positive rational number and $ω_i$ a differential form on the residue field of $K$. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of $K$.
Graded quotients of ramification groups of local fields with imperfect residue fields
该研究针对剩余特征为$p>0$的亨泽尔离散赋值域，证明其分歧群滤过的分次商为$F_p$-向量空间，并构造了分次商特征群到基于余切复形的扭曲余切空间的单射。
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…
Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that the modulo s^m of the extension O_L/O_K and modulo t^m of O_E/O_F are isomorphic. Let j=<m be a positive rational number. In this paper, we prove that the ramification of L/K is bounded by j if and only if the ramification of E/F is bounded by j. As an application, we prove that the categories of finite separable extensions of K and F whose ramifications ar…
Faltings extension and Hodge-Tate filtration for abelian varieties over $p$-adic local fields with imperfect residue fields
针对剩余域不完备的特征0p进局部域，定义Z_p上O_K的Faltings扩张，推广Fontaine1981年的构造，为该类域上的阿贝尔簇构建Hodge-Tate滤过。
Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$, whose residue field may not be perfect, and $G_K$ the absolute Galois group of $K$. In the first part of this paper, we prove that Scholl's generalization of fields of norms over $K$ is compatible with Abbes-Saito's ramification theory. In the second part, we construct a functor $\mathbb{N}_{\mathrm{dR}}(V)$ associating a de Rham representation $V$ with a $(φ,\nabla)$-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of $\mathbb{N}_{\mathrm{dR}}(V)$ and …
This is an introduction to author's ramification theory of a complete discrete valuation field with residue field whose p-basis consists of at most one element. New lower and upper filtrations are defined; cyclic extensions of degree p may have non-integer ramification breaks. A refinement of the filtration for two-dimensional local fields which is compatible with the reciprocity map is discussed.
We develop a new approach to construction of numerical invariants for ramified coverings of algebraic surfaces of prime characteristic. Let A be a two-dimensional regular local ring of prime characteristic p with algebraically closed residue field. Let L/K be a solvable finite Galois extension of its fraction field. Let C be a germ of a regular curve on the spectrum of A which is not a component of the ramification divisor R. Then L/K determines (up to conjugation) an extension of the function filed of C. This is an extension of discrete valuation fields with algebaically closed residue fields…
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…
Let $C$ be a curve of genus $g\geqslant 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $\char(K)=0$ and that the characteristic of the residue field is not 2. Suppose that the Jacobian $\Jac(C)$ has semi-stable reduction over $R$. Embed $C$ in $\Jac(C)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique moderately ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on $\Jac(C)$.
M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.
We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a finite last residue field the constructed local central extension is isomorphic to a central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by M. Kapranov.
We develop a theory of residues for arithmetic surfaces, establish the reciprocity law around a point, and use the residue maps to explicitly construct the dualizing sheaf of the surface. These are generalisations of known results for surfaces over a perfect field. In an appendix, explicit local ramification theory is used to recover the fact that in the case of a local complete intersection the dualizing and canonical sheaves coincide.
We propose a geometric method to measure the wild ramification of a smooth etale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of positive characteristic with arbitrary residue field. We also define the characteristic cycle of an l-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number. Definition 2.1.1 is corrected in v2.
A characterization of ramification groups via Taylor morphism
本文提出一种函子方法，通过将分歧群识别为复合jet代数上的不变群来刻画其结构，为定义涉及群概型的高阶分歧群奠定基础，并在交换含幺环范畴内引入Taylor映射计算其核。
For a simple, normal and finite extension of a valued field, we prove that we can related the order of the ramification group of the field extension and the set of key polynomials associated to the extension of the valuation. More precisely, the order of this group can be expressed in terms of a product of a power of the characteristic of the residue field of the valuation and the effective degrees of the key polynomials. We also give a condition on the order of the ramification group so that there is no limit key polynomials for a valuation of rank one. This condition also allow us to have a …
Let $A$ be a regular 2-dimensional local ring of characteristic $p>0$, and let $L/K$ be a cyclic extension of degree $p$ of its field of fractions such that the corresponding branch divisor is normal crossing. For each $\gp\in\Spec A$ of height 1 such that $A/\gp$ is regular, consider the ramification jump $h_\gp$ of the extension of the residue field at $\gp$. In this paper the semi-continuity of $h_\gp$ with respect to Zariski topology of suitable jet spaces is proved. The asymptotic of $h_\gp$ with respect to intersection multiplicity of the prime divisor defined by $\gp$ and the branch div…
We define a motivic conductor for any presheaf with transfers $F$ using the categorical framework developed for the theory of motives with modulus by Kahn-Miyazaki-Saito-Yamazaki. If $F$ is a reciprocity sheaf this conductor yields an increasing and exhaustive filtration on $F(L)$, where $L$ is any henselian discrete valuation field of geometric type over the perfect ground field. We show if $F$ is a smooth group scheme, then the motivic conductor extends the Rosenlicht-Serre conductor; if $F$ assigns to $X$ the group of finite characters on the abelianized étale fundamental group of $X$, then…
Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $Γ_{L}^{(v)}$ the ramification subgroups of $Γ_{L}=\operatorname{Gal}(L^{sep}/L)$. We consider the category $\operatorname{MΓ}_{L}^{Lie}$ of finite $\mathbb{Z}_p[Γ_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $Γ_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the ramification subgroups $Γ_L^{(v)}$ can be explicitly extracted from some differential forms $Ω[N]$ on the Fontaine etale $ϕ$-mod…
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。
We extend the formalism and results on motivic integration from ["Constructible motivic functions and motivic integration", Invent. Math., Volume 173, (2008) 23-121] to mixed characteristic discretely valued Henselian fields with bounded ramification. We also generalize the equicharacteristic zero case of loc. cit. by giving, in all residue characteristics, an axiomatic approach (instead of only using Denef-Pas languages) and by using richer angular component maps. In this setting we prove a general change of variables formula and a general Fubini Theorem. Our set-up can be specialized to prev…
Ramification theory of reciprocity sheaves, II, Higher local symbols
本文构建互反层沿 Parsin 链的高阶局部符号理论，统一导出剩余映射与 Kato 互反映射，并据此刻画互反层截面的模。
On Abelian extensions in mixed characteristic and ramification in codimension one
针对Paul Roberts关于正则局部环整闭包性质的定理在模情形下不成立的问题，通过修改余维一p上分歧的相关表述，将该定理推广至混合特征下非分歧正则局部环的模情形（伽罗瓦群p挠元被p零化）。
Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic
本文证明了在特征不为2的奇剩余特征下，离散赋值域上超椭圆曲线的导子与判别式之间的不等式，将Artin导子的负值与特殊纤维不可约分支数量由判别式赋值上界控制，推广了前人工作。
This article studies the variation of the Swan conductor of a lisse étale sheaf of $\mathbb{F}_{\ell}$-modules $\mathcal{F}$ on the rigid unit disc $D$ over a complete discrete valuation field $K$ with algebraically closed residue field of characteristic $p\neq \ell$. We associate to $\mathcal{F}$ a function ${\rm sw}_{\rm AS}(\mathcal{F}, \cdot): \mathbb{Q}_{\geq 0}\to \mathbb{Q}$, defined with the Abbes-Saito logarithmic ramification filtration, which measures, at each $t\in \mathbb{Q}_{\geq 0}$, the ramification of the restriction of $\mathcal{F}$ to the subdisc of radius $t$ along the spec…
For a prime $p$, a cyclic-by-$p$ group $G$ and a $G$-extension $L|K$ of complete discrete valuation fields of characteristic $p$ with algebraically closed residue field, the local lifting problem asks whether the extension $L|K$ lifts to characteristic zero. In this paper, we characterize $D_4$-extensions of fields of characteristic two, determine the ramification breaks of (suitable) $D_4$-extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every $D_4$-extension of complete discrete valuation fields of characterist…
We develop, for an l-adic etale sheaf on a complete trait of characteristic p>0, the notion of characteristic variety. Our approach, inspired by the microlocal analysis of Kashiwara and Schapira, is a complement to our ramification theory for local fields with general residue fields. We formulate the main property that should be satisfied by the characteristic variety (the isogeny conjecture), and prove it for rank one sheaves unconditionally on the trait, or unconditionally on the sheaf if the residue field of the trait is perfect.
Existence of global Néron models beyond semi-abelian varieties
本文证明了完美剩余域情形下非半阿贝尔代数群的Bosch-Lütkebohmert-Raynaud猜想，给出了非完美剩余域的反例，并分类了完美剩余域上的单有理有界幂幺群。
The relatively perfect Greenberg transform and cycle class maps
本文将Greenberg变换推广至非完美剩余域情形，并在半阿贝尔簇Néron模型的广义Greenberg变换上构造取值于相对完美邻近闭链函子的闭链类映射。
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。
Mori fibre spaces for $3$-folds over imperfect fields
研究特征\(p>5\)的函数域上射影三维空间，证明典范除子非伪有效时LMMP有效，过程中表明此情形下三维空间对数典范阈值的ACC完全成立。
Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two general results: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any information about a Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space.
Unirational algebraic groups and tame ramification
研究无理代数群在基变换成整环时Néron lft模型的行为，揭示跳跃现象与动机ζ函数的有理性，回答Halle-Nicaise和Edixhoven的问题。
Triangulations of non-archimedean curves, semi-stable reduction, and ramification
本文利用 Berkovich 曲线的最小三角剖分研究非阿基米德曲线的半稳定约化，证明最小三角剖分重数的最小公倍数整除 Galois 扩张次数，并在驯顺情形下给出 Saito 定理的新证明及椭圆曲线和野分歧情形的推广分析。
On Kawamata-Viehweg Vanishing for Surfaces of del Pezzo type over imperfect fields
本文在特征 p>5 的非完美域上证明了 del Pezzo 型曲面的 Kawamata-Viehweg 消失定理，并由此推出优秀除子对数终端三维簇的 Grauert-Riemenschneider 消失定理，进而证明三维 klt 奇点是有理的。
We define a numerical invariant, the differential Swan conductor, for certain differential modules on a rigid analytic annulus over a p-adic field. This gives a definition of a conductor for p-adic Galois representations with finite local monodromy over an equal characteristic discretely valued field, which agrees with the usual Swan conductor when the residue field is perfect. We also establish analogues of some key properties of the usual Swan conductor, such as integrality (the Hasse-Arf theorem), and the fact that the graded pieces of the associated ramification filtration on Galois groups…
We consider non-linear changes of variables and Fubini's theorem for certain integrals over a two-dimensional local field. An interesting example is presented in which imperfectness of a finite characteristic local field causes Fubini's theorem to unexpectedly fail. The relationship to ramification theory is discussed.
Let k be a perfect field of characteristic p>0. When p>2, Fontaine and Laffaille have classified p-divisibles groups and finite flat p-groups over the Witt vectors W(k) in terms of filtered modules. Still assuming p>2, we extend these classifications over an arbitrary complete discrete valuation ring A with unequal characteristic (0,p) and residue field k by using "generalized" filtered modules. In particular, there is no restriction on the ramification index. In the case k is included in \bar{F}_p (and p>2), we then use this new classification to prove that any crystalline representation of t…
Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue field characteristics. We define a class of functions, called functions of motivic exponential class, which we show to be stable under integration and under Fourier transformation, extending results and definitions from previous papers. We prove uniform results related to rationality and to various kinds of loci. A key ingredient is a refined form of Denef-…
Let R be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K' of K. We prove that there exists a constant c which depends on the absolute ramification index e(K'/Q_p) and the height of G such that G has good reduction over K if and only if G[p^c] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's p-adic Ner…
Brauer group of varieties over local fields of finite characteristic
本文在正特征局部域上研究了代数簇的Brauer群，证明了Kato挠 filtration 与评估 filtration 的等价性，并扩展了相关结果到正特征情况。
Let $p$ be a prime. Let $R$ be a regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ of $k[[x_1,\ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $\text{Frac}(R)$ which extends to every local ring of $\text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group ove…
Bases of associated Galois modules in general wildly ramified extensions and in elementary abelian extensions of degree $p^2$
本文研究了在剧烈反常扩张中如何构造各种关联伽罗瓦模和阶的基底，特别在G=(Z/pZ)^2的情况下，计算了元素(σ_1-1)^i(σ_2-1)^j在估值过滤中的作用，并证明当反常跳跃在模p^2下不同则这些元素能构成合适的基底。
Formal groups and $(φ,Γ)$-modules
本文通过引入指数周期映射构造环$R_{H,K}$，证明有限生成$\mathcal{O}_E$-模上连续Galois作用与étale $(\varphi_E,\Gamma)$-模的范畴等价，推广了经典分圆与Lubin-Tate情形，并利用$F$-动力系统建立一般等价。
We classify group schemes in terms of their Cartier modules. We also prove the equivalence of different definitions of the tangent space and the dimension for these group schemes; in particular, the minimal dimension of a formal group law that contains $S$ as a closed subgroup is equal to the minimal number of generators for the affine algebra of $S$. As an application the following reduction criteria for Abelian varieties are proved. Let $K$ be a mixed characteristic local field, let its residue field have characteristic $p$, $L$ be a finite extension of $K$, let $\mathfrak{O}_K\subset\mathfr…
Sen Operators and Lie Algebras arising from Galois Representations over $p$-adic Varieties
本研究将p-进局部域绝对伽罗瓦群表示的Sen算子构造推广到带半稳定图的p-进仿射簇基本群，证明其模的典范性，结合p-进辛普森对应描述李代数作用并延拓到无限维表示，还证明几何Sen算子零化局部解析向量。
Let $K$ be the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field $k$. When the characteristic of $k$ is not 2, we prove that every quadratic form of rank $\ge 9$ is isotropic over $K$ using methods of Parimala and Suresh, and we obtain the local-global principle for isotropy of quadratic forms of rank 5 with respect to discrete valuations of $K$. The latter result is proved by making a careful study of ramification and cyclicity of division algebras over the field $K$, following Saltman's methods. A key step is the proof of the following result, whi…
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