GENERATORS AND DEFINING EQUATIONS OF MODULAR FUNCTION FIELDS

我们的结果如下。(1)利用Weierstrass P-functuins构造了模群Γ_1(N)的模函数域K(N)的新生成器，得到了模函数域的一个简单定义方程。(2)在K(N)有属2的情况下，利用(1)中的生成子构造了K(N)的属1子域的生成子，给出了椭圆曲线正则幂级数解的几个例子。(3)给出了一种确定定义方程解对应椭圆曲线j不变量的算法。在研究有限域上定义方程性质的过程中，得到了以下两个结果：(4)利用代数数域上有理数的椭圆曲线，用复乘法构造大阶循环有理点群有限域上的椭圆曲线族。(5)确定了复乘R的椭圆曲线的Frobenius内同性的迹线，其中R是一个阶的判别式除以3,4,5，其类数为2或3。结果(4)、(5)适用于椭圆曲线密码系统。

Our results are as follows.(1) We constracted new generators of the modular function field K(N)of the modular group Γ_1(N) by Weierstrass P-functuins and obatained a plain defining equation of the modular function field.(2) We gave some examples of the canonical power series solutions of the elliptic curve by constructing the generator of the genus 1 subfields of K(N) from the generators in (1) in the cases K(N) has genus 2.(3) We obatined an algorithm to determine the j-invariant of the elliptic curve corresponding to the solution of the defining equation. But this algorithm is *neffective for large N.In the course of studing the properties of the defining equation over the finite fields, the following two results were obtained.(4) We offered a method of constracting a family of elliptic curves over finite fields of cyclic rational point group of large order using the elliptic curve,rational over an algebraic number field, with complex multiplication.(5) We determined the trace of Frobenius endomoprphism of the elliptic curves with complex multipication R, where R is an order of discriminant divided by 3, 4,5 and its class number is 2 or 3.The results (4), (5) are applicable to the elliptic curve cryptosysytems.