F_3上代数曲线的有理点个数

如果用Ｎｑ（ｇ）表示Ｆｑ上亏格为ｇ的所有光滑不可约代数曲线的Ｆｑ－有理点个数最大值，则确定Ｎｑ（ｇ）的值是一个困难问题．Ｓｅｒｒｅ确定了当ｑ＝２，１≤ｇ≤１０及ｑ＝３，１≤ｇ≤３时的全部Ｎｑ（ｇ），本文确定了ｑ＝３，４≤ｇ≤６的Ｎｑ（ｇ）值．     

代数等距曲线、曲面的有理性质

A simple method is developed to identify the rationality of offsets to algebraic curves and surfaces which are given either implicitly or parametrically and to parametrize the offsets if they are rational. In particular, we show that offsets to ellipses and hyperbolas are of nonzero genus except for the circles, and somewhat surprisingly, that offsets to paraboioids and hyperboioids with one sheet can all be rationally parametrtised.     

有限域上的高斯和与有理点的计算

设f是有限域Fq上的n元m项多项式,D_f∈Z_(≥0)~(n×m)为其次数矩阵,用N(f)表示由超曲面f=0在仿射空间An(Fq)确定的Fq-有理点的个数.若矩阵A∈Z~(n×m)在环Z/(q-1)Z中与Df行等价,则记为D_fr～A.本文利用高斯和给出了当m≤n且0D_fr～diag(λ_1,···,λ_m),其中λi∈{1,3}时N(f)的具体表达式,从而推广了已知结论.     

椭圆曲线有理点扭子群的明显分类

本文给出了有理数域Q上椭圆曲线E按其偶数阶循环扭子群Etors(Q)的分 类,并给出了Etors(Q)的生成元.这些结果,连同新近Ono在Etors(Q)非循环情形 的结果,完全解决了E含2阶有理点时的分类和参数化问题.     

一种基于有限域上有理正规曲线的密钥预分配方案

在网络通讯中,有时某些用户需要共享一个密钥,并且还要防止非授权的用户得到有关这个密钥的任何信息。Beimelt和Chor在[1]中提出了密钥预分配方案（KPS,Key Predistribution Scheme);Stinson在[2-4]中对KPS进行了总结,并且提出了一些构造方案。本文第二作者在[5]中利用有限域上的有理正规曲线构造出一类新的强部分平衡t-设计。本文将说明,利用强部分平衡t-设计可构造KPS,从而构造出一类新的KPS。     

复射影平面中全纯叶层的不变代数曲线和有理首次积分

借助代数几何作为工具给出复射影平面中一类全纯叶层具有有理首次积分的必要条件. 进一步, 得到复射影平面中一类全纯叶层的不可约不变代数曲线次数的上界. 最后, 应用所得结果证明了任何二次全纯叶层没有三次代数极限环.     

直线上整点个数及圆上有理点个数问题

2011年普通高等学校招生全国统一考试(安徽卷)中有一道这样的题目:在平面直角坐标系中,如果x和y都是整数,则称点(x,y)为整数,下列命题中正确的是:(1)存在这样的直线,既不与坐标轴平行又不经过任何整点;     

超椭圆曲线yk = x(x + 2m) 上的有理点

设k ≥ 2; m ≥ 0 是整数, 给出了超椭圆曲线y^k = x(x + 2^m) 上所有的有理点(x, y).     

On Drinfeld Modular Curves with Many Rational Points over Finite Fields

It is known that Drinfeld modular curves can be used to construct asymptotically optimal towers of curves over finite fields. Using reductions of the Drinfeld modular curves X₀( ), we try to find individual curves over finite fields with many rational points. The main idea is to divide by an Atkin–Lehner involution which has many fixed points in order to obtain a quotient with a better ratio #{rational points}/genus. In a few cases we can improve the known records of rational points.     

The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields

We show that for all finite fields F_q, there exists a curve C over F_q of genus 3 such that the number of rational points on C is within 3 of the Serre–Weil upper or lower bound. For some q, we also obtain improvements on the upper bound for the number of rational points on a genus 3 curve over F_q.with an Appendix by Jean-Pierre Serre     

Normal Rational Curves Over Prime Fields

It is shown that in the projective spaces PG(n,p),p prime, 2 n p-2, the normal rational curves are the only (p+1)-arcs fixed by a projective group G isomorphic to PSL(2,p).     

Families of Algebraic Varieties and Towers of Algebraic Curves over Finite Fields

We introduce a new construction of towers of algebraic curves over finite fields and provide a simple example of an optimal tower.     

On Rational Cuspidal Projective Plane Curves

In 2002, L. Nicolaescu and the fourth author formulated a verygeneral conjecture which relates the geometric genus of a Gorensteinsurface singularity with rational homology sphere link withthe Seiberg--Witten invariant (or one of its candidates) ofthe link. Recently, the last three authors found some counterexamplesusing superisolated singularities. The theory of superisolatedhypersurface singularities with rational homology sphere linkis equivalent with the theory of rational cuspidal projectiveplane curves. In the case when the corresponding curve has onlyone singular point one knows no counterexample. In fact, inthis case the above Seiberg--Witten conjecture led us to a veryinteresting and deep set of ‘compatibility properties’of these curves (generalising the Seiberg--Witten invariantconjecture, but sitting deeply in algebraic geometry) whichseems to generalise some other famous conjectures and propertiesas well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yauinequalities). Namely, we provide a set of ‘compatibilityconditions’ which conjecturally is satisfied by a localembedded topological type of a germ of plane curve singularityand an integer d if and only if the germ can be realized asthe unique singular point of a rational unicuspidal projectiveplane curve of degree d. The conjectured compatibility propertieshave a weaker version too, valid for any rational cuspidal curvewith more than one singular point. The goal of the present articleis to formulate these conjectured properties, and to verifythem in all the situations when the logarithmic Kodaira dimensionof the complement of the corresponding plane curves is strictlyless than 2. 2000 Mathematics Subject Classification 14B05,14J17, 32S25, 57M27, 57R57 (primary), 14E15, 32S45, 57M25 (secondary).     

On the Group of Automorphisms of Shimura Curves and Applications

Let V _D be the Shimura curve over attached to the indefinite rational quaternion algebra of discriminant D. In this note we investigate the group of automorphisms of V _D and prove that, in many cases, it is the Atkin–Lehner group. Moreover, we determine the family of bielliptic Shimura curves over and over and we use it to study the set of rational points on V _D over quadratic fields. Finally, we obtain explicit equations of elliptic Atkin–Lehner quotients of V _D .     

A Family of K3 Surfaces and Towers of Algebraic Curves over Finite Fields

Mathematical Notes -     

Legendre Elliptic Curves over Finite Fields

We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.