Igor Zhukov
This work presents author's explicit methods of constructing abelian extensions of complete discrete valuation fields. His approach to explicit equations of a cyclic extension of degree p^n which contains a given cyclic extension of degree p is explained.
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This is an introduction to the author theory of cyclic p-extensions of an absolutely unramified complete discrete valuation field K with arbitrary residue field of characteristic p. In this theory a homomorphism is constructed from the p-part of the group of characters of K to Witt vectors over its residue field.
This is a sketch of main steps of the proof of Bloch--Kato's theorem which states that the norm residue homomorphism K_q(K)/m\to H^q(K,\Bbb Z/m(q)) is an isomorphism for a henselian discrete valuation field K of characteristic 0 with residue field of positive characteristic.
This work contains a list of all known results on the quotient filtration on the Milnor K-groups of a complete discrete valuation field in terms of differential modules over the residue field . Author's recent study of the case of a tamely ramified field of characteristic 0 with residue field of characteristic p by using an exponential map and a syntomic complex is explained.
This work sketches the author classification of complete discrete valuation fields K of characteristic 0 with residue field of characteristic p into two classes depending on the behaviour of the torsion part of a differential module. For each of these classes, the quotient filtration of the Milnor K-groups of K is characterized for all sufficiently large members of the filtration, as a quotient of differential modules. For a higher local field the previous result and higher local class field theory imply certain restrictions on types of cyclic extensions of the field of sufficiently large degr…
Ramification theory of monogenic extensions of complete discrete valuation fields is presented. Relations to Kato's conductor are discussed.
Author's generalization of one-dimensional class field theory to theory of abelian totally ramified p-extensions of a complete discrete valuation field with arbitrary non-separably p-closed residue field and its applications are described.
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.
An exponential homomorphism for a complete discrete valuation field of characteristic zero which relates differential forms and the Milnor K-groups of the field is studied. An application to explicit formulas is included.
A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a second application (which is a joint work with Fesenko) we analyze the arithmetic structure of fields with the same absolute …
This paper is a summary of author's results on finite flat commutative group schemes. The properties of the generic fibre functor are discussed. A complete classification of finite local flat commutative group schemes over mixed characteristic complete discrete valuation rings in terms of their Cartier modules (defined by Oort) is given. We also state several properties of tangent space of these schemes. These results are applied to the study of reduction of Abelian varieties. A finite $p$-adic semistable reduction criterion is formulated. It looks especially nice in the ordinary reduction cas…
The monograph "Invitation to higher local fields" is the result of the conference on higher local fields held in Muenster, August 29 to September 5, 1999. The aim is to provide an introduction to higher local fields (more generally complete discrete valuation fields with arbitrary residue field) and render the main ideas of this theory (Part I), as well as to discuss several applications and connections to other areas (Part II). The volume grew as an extended version of talks given at the conference. The two parts are separated by a paper of K. Kato, an IHES preprint from 1980 which has never …
In this note we investigate the problem of existence of a torsor structure for Galois covers of (formal) schemes over a complete discrete valuation ring of residue characteristic $p>0$ in the case of abelian Galois groups of type (p,p,...,p).
Let $p$ be a prime number and let $K$ be a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. Let $N$ be a fully ramified, elementary abelian extension of $K$. Under a mild hypothesis on the extension $N/K$, we show that every element of $N$ with valuation congruent mod $[N:K]$ to the largest lower ramification number of $N/K$ generates a normal basis for $N$ over $K$.
Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that n_R is the greatest positive integer which divides the order of (R mod p) for all but finitely many primes p of K. Furthermore, let m>0 be a multiple of n_R and let S be a finite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every l in S the l-adic valuation of the order of (R mod p) e…
We study how the $\ell$-adic valuation of the number of spanning trees varies in regular abelian $\ell$-towers of multigraphs. We show that for an infinite family of regular abelian $\ell$-towers of bouquets, the behavior of the $\ell$-adic valuation of the number of spanning trees behave similarly to the $\ell$-adic valuation of the class numbers in $\mathbb{Z}_{\ell}$-extensions of number fields.
Cohomological flatness over discrete valuation rings: numerical and logarithmic criteria
本文提出在离散赋值环上实现上同调平坦性的数值和对数准则，应用于曲线和具有良好减少的阿贝尔簇torsors，给出正则模型的对数光滑性条件。
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。
Let $\mathcal{O}_K$ be a discrete valuation ring with fraction field $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $p > 0$. Let $A/K$ be an abelian variety of dimension $g$ with a $K$-rational point of order $p$. In this article, we are interested in the reduction properties that $A/K$ can have. After discussing the general case, we specialize to $g=1$, and we study the possible Kodaira types that can occur.
Given an abelian surface $A$, defined over a discrete valuation field and having good reduction, does the attached Kummer surface $\mathrm{Km}(A)$ also have good reduction? In this paper we give an affirmative answer in the extreme case, that is, when the abelian surface has supersingular reduction in characteristic $2$.
Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the Néron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the logarithmic setting, i.e. there exists a projective, log smooth model of $A$ over $\mathcal{O}_K$. This implies in particular the existence of a projective, regular model of $A$, generalizing a result of Künnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings-Chai et al.
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。
Finite extensions of abelian schemes and duality for commutative group stacks
该研究在离散赋值环及任意基上，证明交换群概形与群层的结构定理，推导其对偶的性质，应用Fourier-Mukai论证得到导出范畴的等价，证明了Brochard对偶猜想。
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 A_K be an abelian variety over a discrete valuation field K. Let A be the Neron model of A_K over the ring of integers O_K of K and A_k its special fibre. We study the set of rational points of the group of components ϕ_A of A_k. In particular, we look at the cases where A_K is a Jacobian or where A_K is an abelian variety having semi-stable reduction. We also consider algebraic tori over K instead of abelian varieties.
This paper explores some first-order properties of commuting-liftable pairs in pro-$\ell$ abelian-by-central Galois groups of fields. The main focus of the paper is to prove that minimized inertia and decomposition groups of many valuations are first-order definable using a predicate for the collection of commuting-liftable pairs.
Let $K$ be a complete discrete valuation field. Let $\mathcal{O}_K$ be its ring of integers. Let $k$ be its residue field which we assume to be algebraically closed of characteristic exponent $p\geq1$. Let $G/K$ be a semi-abelian variety. Let $\mathcal{G}/\mathcal{O}_K$ be its Néron model. The special fiber $\mathcal{G}_k/k$ is an extension of the identity component $\mathcal{G}_k^0/k$ by the group of components $Φ(G)$. We say that $G/K$ has split reduction if this extension is split. Whereas $G/K$ has always split reduction if $p=1$ we prove that it is no longer the case if $p>1$ even if $G/K…
Integer group determinants for the elementary abelian group of order 49
本文通过分类能被7整除的取值，完整确定了$C_7\times C_7$的整数群行列式集合，给出赋值10和11的余因子条件，并求得最小正取值。
We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local function fields of characteristic $p$, called {\em one-dimensional}, that, in a particular sense, are as simple as cyclic degree $p$ extensions, and prove the statement in the title above.
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…
Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation of the number of spanning trees in an abelian $\ell$-tower of connected multigraphs. This is analogous to the classical theorem of Washington--Sinnott on the growth of the $p$-part of the class group in a cyclotomic $\mathbb{Z}_\ell$-extension of abelian extensions of $\mathbb{Q}$. Furthermore, we show that under certain hypotheses, the number of primes dividing the number of spanning trees is unbounded in such a tower.
If $L/K$ is a finite Galois extension of local fields, we say that the valuation criterion $VC(L/K)$ holds if there is an integer $d$ such that every element $x \in L$ with valuation $d$ generates a normal basis for $L/K$. Answering a question of Byott and Elder, we first prove that $VC(L/K)$ holds if and only if the tamely ramified part of the extension $L/K$ is trivial and every non-zero $K[G]$-submodule of $L$ contains a unit. Moreover, the integer $d$ can take one value modulo $[L:K]$ only, namely $-d_{L/K}-1$, where $d_{L/K}$ is the valuation of the different of $L/K$. When $K$ has positi…
Let K be a field and \ell be a prime such that char K \neq \ell. In the presence of sufficiently many roots of unity in K, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal (\Z/\ell)-abelian resp. pro-\ell-abelian Galois group of K using its (Z/\ell)-central resp. pro-\ell-central extensions.
In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This in particular implies Grothendieck's conjecture on the perfectness of his pairing between the Neron component groups of an abelian variety and its dual. The point is that our formulation is well-suited with Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.
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。
Given a Hilbertian field $k$ and a finite set $\mathcal{S}$ of Krull valuations of $k$, we show that every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/k)$ over $k$ with abelian kernel has a solu\-tion ${\rm{Gal}}(F/k) \rightarrow G$ such that every $v \in \mathcal{S}$ is totally split in $F/L$. Two applications are then given. Firstly, we solve a non-constant variant of the Beckmann--Black problem for solvable groups: given a field $k$ and a non-trivial finite solvable group $G$, every Galois field extension $F/k$ of group $G$ is shown to occur as the specialization at some $t_0…
It is proved that, if $K$ is a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field satisfying a mild condition, then any abelian variety over $K$ with potentially good reduction has finite $K(K^{1/p^\infty})$-rational torsion subgroup. This can be used to remove certain conditions assumed in some theorems in Iwasawa theory.
Let $F/E$ be a finite Galois extension of fields with abelian Galois group $Γ$. A self-dual normal basis for $F/E$ is a normal basis with the additional property that $Tr_{F/E}(g(x),h(x))=δ_{g,h}$ for $g,h\inΓ$. Bayer-Fluckiger and Lenstra have shown that when $char(E)\neq 2$, then $F$ admits a self-dual normal basis if and only if $[F:E]$ is odd. If $F/E$ is an extension of finite fields and $char(E)=2$, then $F$ admits a self-dual normal basis if and only if the exponent of $Γ$ is not divisible by $4$. In this paper we construct self-dual normal basis generators for finite extensions of fini…
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 …
Colmez conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions. In a series of articles we investigate the analog of Colmez's theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher dimensional generalizations, so-called $A$-motives. In the present article we prove the product formul…
Let $K$ be a global or local field, $E/K$ a Galois extension, and Br$(E)$ the Brauer group of $E$. This paper shows that if $K$ is a local field, $v$ is its natural discrete valuation, $v'$ is the valuation of $E$ extending $v$, and $q$ is the characteristic of the residue field $\widehat E$ of $(E, v')$, then Br$(E) = \{0\}$ if and only if the following conditions hold: $\widehat E$ contains as a subfield the maximal $p$-extension of $\widehat K$, for each prime $p \neq q$; $\widehat E$ is an algebraically closed field in case the value group $v'(E)$ is $q$-indivisible. When $K$ is a global f…
Let $K$ be a finite extension of $\Q_p$, let $L/K$ be a finite abelian Galois extension of odd degree and let $\bo_L$ be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional $\bo_L$-ideal with square equal to the inverse different of $L/K$. For $p$ an odd prime and $L/\Q_p$ contained in certain cyclotomic extensions, Erez has described integral normal bases for $A_{L/\Q_p}$ that are self-dual with respect to the trace form. Assuming $K/\Q_p$ to be unramified we generate odd abelian weakly ramified extensions of $K$ using Lubin-Tate formal groups. We then use Dwork's expon…
Let $(R, \mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/\mathfrak{m} = \mathbb{F}_{q}$. Given a monic polynomial $P(t) \in R[t]$ whose reduction modulo $\mathfrak{m}$ gives an irreducible polynomial $\bar{P}(t) \in \mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $\mathrm{coker}(P(A))$, where $A \in \mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $\mathrm{Mat}_{n}(R)$ of $n \times n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. O…
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…
Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. Let $\mathbb{W}_n(\cdot)$ denote the ring of $p$-typical Witt vectors of length $n$. Hesselholt conjectured that the pro-abelian group $\{H^1(G,\mathbb{W}_n(\mathcal{O}_L))\}_{n\geq 1}$ is isomorphic to zero. Hogadi and Pisolkar have recently provided a proof of this conjecture. In this paper, we provide an alternative proof o…
Let K/k be an Abelian extension of number fields, S be a set of places of k, and p be an odd prime number. We continue an earlier investigation of the author into the values at zero of the S-imprimitive partial zeta functions of K/k. An earlier result provides, under the assumption that the p-power roots of unity in K are cohomologically trivial, a criterion for the values to have larger than expected p-valuation. The present paper provides such a criterion for a special class of degree 2p extensions for which the p-power roots of unity are not cohomologically trivial. For such extensions, new…
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…
On the Distribution of Class Groups of Abelian Extensions
本文研究有限阿贝尔群Γ下，Q或F_q(t)的伽罗瓦群同构于Γ的扩张K的类群Cl(K)的p-部分分布，构造离散赋值环分析其结构，给出分布猜想并证明函数域情形加权矩结果，补充相关核的平均大小结论。
Character Sheaves on Tori over Local Fields
本文研究了局部域上环面的局部特征sheaves，通过计算基本群并建立局部朗兰兹对应，将乘法局部系统与惯性局部朗兰兹参数联系起来。
On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-François Jaulent
本文研究了完全p-次虚二次域的Z_p扩张性质，不依赖Iwasawa理论，探讨了p-类群的滤子及p-类群的阶，提出新的视角和猜想。
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