On polynomial reciprocity law

The well-known law of quadratic reciprocity has over 150 proofs in print. We establish a relation between polynomial Jacobi symbols and resultants of polynomials over finite fields. Using this relation, we prove the polynomial reciprocity law and obtain a polynomial analogue of classical Burde's quartic reciprocity law. Under the use of our polynomial Poisson summation formula and the evaluation of polynomial exponential map, we get a reciprocity for the generalized polynomial quadratic Gauss sums.     

A note on Gersten's conjecture for logarithmic Hodge-Witt sheaves

We prove a purity theorem for logarithmic Hodge-Witt sheaves and construct a complex, which gives thep-part of the reciprocity sequence for a function field over a finite field of characteristicp.     

Non-divisible cycles on surfaces over local fields

In this paper, we are concerned with the reciprocity map of unramified class field theory for smooth projective surfaces over non-archimedean local fields which do not have potentially good reduction. We will construct two types of smooth projective surfaces whose reciprocity maps modulo positive integers are not injective. The first type is the case where the kernel of the reciprocity map is not divisible. The second is the case where the kernel of the reciprocity map is divisible, but where nevertheless the reciprocity map modulo some integer is not injective.     

Class field theory for a product of curves over a local field

We prove that the kernel of the reciprocity map for a product of curves over a p-adic field with split semi-stable reduction is divisible. We also consider the K ₁ of a product of curves over a number field.      

The Class Number of Derived Subgroups and the Structure of Camina Groups

A finite group G is called a Camina group if G has a proper normal subgroup N such that gN is precisely a conjugacy class of G for any g ∈E G - N. In this paper, the structure of a Camina group G is determined when N is a union of 2, 3 or 4 conjugacy classes of G.     

Non-abelian local reciprocity law

Following our work on the generalized Fesenko reciprocity map, we construct the non-abelian local reciprocity map ${\pmb{\Phi}_K^{(\varphi)}}$ of a local field K as a certain isomorphism from the absolute Galois group G _K of K onto a topological group ${\nabla_{K,Y}^{(\varphi)}}$ whose definition involves Fontaine–Wintenberger theory of field of norms, and build the non-abelian local class field theory over K in the sense of Fesenko and Koch.     

有限群的Deskins极大完备

对于有限群G的一个极大予群M,Deskins称子群C为M的一个完备,如果C(?)M,但C的G-不变真子群包含在M中.令I(M)表示M的所有完备之集. I(M)的一个极大元叫做M的一个极大完备.利用极大完备,本文获得关于群的可解性和超可解性某些新的刻划.     

Class field theory for open curves over p-adic fields

We introduce the idèle class group for quasi-projective curves over p-adic fields and show that the kernel of the reciprocity map is divisible. This extends Saito’s class field theory for projective curves (Saito in J Number Theory 21:44–80, 1985).     

On the Connectivity of Proper Power Graphs of Finite Groups

We study the connectivity of proper power graphs of some family of finite groups including nilpotent groups, groups with a nontrivial partition, and symmetric and alternating groups. Also, for such a group, the corresponding proper power graph has diameter at most 26 whenever it is connected.     

A geometric approach to the quadratic reciprocity law

A geometric viewpoint of much broader potential yields a unified proof of the quadratic reciprocity law based on counting points on quadratic surfaces over finite prime fields.     

有限域上型$B_n$的Chevalley群之间的同态

研究了特征为$p$的有限域上型$B_{n}$的Chevalley群的结构,并确定了特征为$p \ (p\neq 2)$的有限域上型$B_{n}$的Chevalley群 之间的非平凡同态.     

The Influence of Primitive Subgroups on the Structure of Finite Groups

A subgroup $H$ of a group $G$ is said to be primitive if it is a proper subgroup of the intersection of all subgroups of $G$ containing $H$ as its proper subgroup. The purpose of this note is to go further into the influence of primitive subgroups on the structure of finite groups. Some new results are obtained.     

Measurable groups of low dimension

We consider low‐dimensional groups and group‐actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is (finite‐by‐Abelian)‐by‐finite, and that any 2‐dimensional asymptotic group is soluble‐by‐finite. We obtain a field‐interpretation theorem for certain measurable groups, and give an analysis of minimal normal subgroups and socles in groups definable in a supersimple theory of finite rank where infinity is definable. We prove a primitivity theorem for measurable group actions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Density of vector bundles periodic under the action of Frobenius

Let X be a proper and smooth curve of genus g?2 over an algebraically closed field k of positive characteristic. If , it follows from Hrushovski's work on the geometry of difference schemes that the set of rank r vector bundles with trivial determinant over X that are periodic under the action of Frobenius is dense in the corresponding moduli space. Using the equivalence between Frobenius periodicity of a stable vector bundle and its triviality after pull-back by some finite étale cover of X (due to Lange and Stuhler) on the one hand, and specialization of the fundamental group on the other hand, we prove that the same result holds for any algebraically closed field of positive characteristic.     

On Weakly Factorizable Groups

Groups with complemented subgroups, which are also called completely factorizable groups, were studied by P. Hall, S. N. Chernikov, and N. V. Chernikova (Baeva). For complete factorizability, it is sufficient (Theorem 1) that each proper subgroup have a normal complement in some larger subgroup. A group is said to be weakly factorizable if each of its proper subgroups is complemented in some larger subgroup; the problem of describing finite groups with this property is posed (Question 8.31) in the Kourovka Notebook. Some properties of these groups are considered. The question is studied for Sylow p-subgroups of Chevalley-type groups of characteristic p. The main theorem, Theorem 2, establishes the weak factorizability of the Sylow p-subgroups in the symmetric and alternative groups and in the classical linear groups over fields of characteristic p> 0, excluding the unitary groups of odd dimension > p.     

A new trichotomy theorem for groups of finite Morley rank

There is a longstanding conjecture, of Gregory Cherlin and BorisZilber, that all simple groups of finite Morley rank are simplealgebraic groups. Here we will conclude that a simple K*-groupof finite {M}orley rank and odd type either has normal rankof at most 2, or else is an algebraic group over an algebraicallyclosed field of characteristic not 2. To this end, it sufficesto produce a proper 2-generated core in groups with \Pruferrank 2 and normal rank at least 3, which is what is proved here.Our final conclusion constrains the Sylow 2-subgroups availableto a minimal counterexample and, finally, proves the trichotomytheorem in the nontame context.     

Sur l'' application de réciprocité pour une surface fibrée en coniques définie sur un corps local

For a proper smooth variety X defined over a local field k, unramified class field theory investigates the reciprocity map _X: SK₁(X) ^ab ₁(X) as introduced by S. Saito. We study this map in the case when X is a surface admitting a proper surjection onto a smooth geometrically connected curve C with a smooth conic as generic fibre. Without any assumption on the reduction of C, we prove that _X is injective modulo n for all n invertible in k and its cokernel is the same as that of _C.     

Deciding finiteness of matrix groups in positive characteristic

We present a new algorithm to decide finiteness of matrix groups defined over a field of positive characteristic. Together with previous work for groups in zero characteristic, this provides the first complete solution of the finiteness problem for finitely generated matrix groups over a field. We also give an algorithm to compute the order of a finite matrix group over a function field of positive characteristic by constructing an isomorphic copy of the group over a finite field. Our implementations of these algorithms are publicly available in Magma.     

Tannaka duality for proper Lie groupoids

By replacing the category of smooth vector bundles of finite rank over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.     

A note on adjacency preservers on hermitian matrices over finite fields

It is shown that any map which preserves adjacency on hermitian matrices over a finite field is necessary bijective and hence of the standard form.