有限域上几类椭圆曲线簇的有理点数分布  

On the distribution of the number of rational points on several families of elliptic curves over finite field

作　　者：赵龙[1] 韩文报[1] 冀会芳[1] 

机构地区：[1]解放军信息工程大学信息研究系,郑州450002

出　　处：《四川大学学报（自然科学版）》2011年第4期771-776,共6页Journal of Sichuan University(Natural Science Edition)

基　　金：国家"973"基金项目(2007CB807902);国家"863"基金项目(2009AA01Z417);全国优秀博士学位论文作者专项基金(FANEDD-2007B74)

摘　　要：设E_(a,b)为定义在有限域F_q上的椭圆曲线y^2=x^3+ax+b,其中q=p^n,素数p≥5,t_(a,b)表示Frobenius映射的迹,于是有理点数#E_(a,b)(F_q)=q+1-t_(a,b).本文作者利用有限域上的指数和计算了t_(az,bz)、(?)t_(az^2,bz^2)、t_(az^2,b)以及∑t_(a,bz^2).基于这些结果作者给出了当a和b相等且跑遍F_q或F_q~*的二次剩余类时#E(F_q)的均值,以及a或b固定,另外一个参数跑遍F_q或F_q~*的二次剩余类时#E(F_q)的均值.Let Fq be the finite field with q = p＂ and the prime p ≥ 5, Ea,b denote the elliptic curve over Fq with equation y^2 = x^3 ＋ ax ＋b, also define to,b be the trace of Frobenius by # Ea,b （Fq） = q ＋ 1- ta,b. The authors use some elementary properties of exponential sums to evaluate ∑x∈Fq^taz,bc、∑x∈Fq^tax^2、∑x∈Fq^ta,bx^2.and ∑x∈Fq^ta,bx^2., through which the average number of the rational points can be derived when a equals b and runthrough Fq or the quadratic residue of F^＊q, also when a or b is fixed and the other parameter runs through Fq or the quadratic residue of F^＊q.

关 键 词：椭圆曲线 有理点数 Frobenius映射 指数和 

分 类 号：O156.1[理学—数学]