On Supersingular Abelian Varieties of Dimension Two over Finite Fields

In this paper we study the characteristic polynomials and the rational point group structure of supersingular varieties of dimension two over finite fields. Meanwhile, we also list allL-functions of supersingular curves of genus two over ₂and determine the group structure of their divisor class groups over all finite algebraic extension of ₂.     

The Rank of Abelian Varieties over Infinite Galois Extensions

G. Frey and M. Jarden (1974, Proc. London Math. Soc.28, 112-128) asked if every Abelian variety A defined over a number field k with dim A>0 has infinite rank over the maximal Abelian extension k^ab of k. We verify this for the Jacobians of cyclic covers of P¹, with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d -dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2).     

Galois Theory for the Selmer Group of an Abelian Variety

This paper concerns the Galois theoretic behavior of the p-primary subgroup Sel _A (F) _p of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group G(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map s _K/F Sel _A (F) _p Sel _A (K) _p ^G(K/F) where F is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B.Mazurand M. Harris.     

有限域上单变量代数函数域中的素数定理

设q是素数的幂次，Fq为一有限域；F为Fq上的单变量代数函数域．在这篇文章中我们证明了下面的素数定理，πF（x）=1/（q-1）.x/logqx＋O（x/log^2qx）.x=q^n→∞其中logqx以q为底的对数，这一结果改进了M．Kruse，H．Stichtenoth的结果．     

On Solvable Congruences in Finitely Decidable Varieties

In this paper we establish the (1, 2)- and (2, 1)-transfer principles for finitely decidable locally finite varieties, where a class of structures is finitely decidable if the first order theory of its finite members is recursive. The transfer principles deal with the local structure of finite algebras and have strong global consequences. Mathematics Subject Classification : 08A05, 03C13, 08B05, 08B10.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

On the Existence of Simple BIBDs with Number of Elements a Prime Power

We show that, when the number of elements is a prime power q, in many situations the necessary conditions

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The Moduli Space of Real Abelian Varieties with Level Structure

The moduli space of principally polarized Abelian varieties with real structure and with level N = 4m structure (with m1) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over , and to consist of finitely many copies of the quotient of the space GL(n, )/O(N) (of positive definite symmetric matrices) by the principal congruence subgroup of level N in GL(n, ).     

On a cardinal group invariant related to decompositions of Abelian groups

For each Abelian groupG, a cardinal invariant χ(G) is introduced and its properties are studied. In the special caseG = ℤ ⁿ , the cardinalχ(ℤ ⁿ ) is equal to the minimal cardinality of an essential subset of ℤ ⁿ , i.e., a of a subsetA ⊂ ℤ ⁿ such that, for any coloring of the group ℤ ⁿ inn colors, there exists an infinite one-color subset that is symmetric with respect to some pointα ofA. The estimaten( ^{n +} l)/2 ≤χ(ℤ ⁿ ) < 2n is proved for alln and the relationχ(ℤ ⁿ ) =n(n + 1)/2 forn ≤ 3. The structure of essential subsets of cardinalityχ(ℤ ⁿ ) in ℤ ⁿ is completely described forn ≤ 3. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 341–350, September, 1998.     

On a System of Martix Equations over an Arbitrary Skew Field

ＯｎａＳｙｓｔｅｍｏｆＭａｒｔｉｘＥｑｕａｔｉｏｎｓｏｖｅｒａｎＡｒｂｉｔｒａｒｙＳｋｅｗＦｉｅｌｄ￥ＷａｎｇＱｉｎｇｗｅｎ（Ｄｅｐｔ．ｏｆＭａｔｈ．，ＣｈａｎｇｗｅｉＴｅａｃｈｅｒｓＣｏｌｌｅｙｅ，Ｗｅｉｆａｎｇ，Ｗｅｉｆａｎｇ，２６１０４３）Ａｂ...     

A Note on the Global Generation of Primitive Line Bundles on Abelian Varieties

Let (X,L) be a generic polarized abelian variety of dimension g and type (d ₁,...,d _g ). We show that L is globally generated when d ₁+...+d _g ≥ 2g. Mathematics Subject Classification: Primary 14K05; secondary 14C20     

任意体上的双矩阵分解与矩阵方程

本文给出了任意体上具有相同行数或相同列数的双矩阵分解定理；利用此定理，给出了任意体上的矩阵方程ＡＸＢ＋ＣＹＤ＝Ｅ及［Ａ１ＸＢ１，Ａ２ＸＢ２］＝［Ｅ１；Ｅ２］有解的充要条件及其一般解的表达式．     

Abelian varieties over finitely generated fields and the conjecture of Geyer and Jarden on torsion

In this paper we prove the Geyer‐Jarden conjecture on the torsion part of the Mordell‐Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on ?‐torsion points, for almost all primes ?, contains the full symplectic group.     

有限域上遍历矩阵的特性研究

对有限域上遍历矩阵的性质进行了分析,给出了有限域上遍历矩阵的计数定理,并对遍历矩阵序对(A,B)关于矩阵M的双侧幂乘集〈A〉M〈B〉的秩及基数进行了全面分析.给出了R_k(A,B)集的构成及其基数的有关定理,所得到的结论对利用遍历矩阵实现有关的公钥密码具有理论上的指导意义.     

On Recognition of the Finite Simple Orthogonal Groups of Dimension 2m, 2m+1, and 2m+2 over a Field of Characteristic 2

The spectrum (G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if HG for every finite group H such that (H)=(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum.