Study on arithmetic properties of modular function fields and elliptic curves by constructive methods.

在本研究项目中，我们研究了以下三个方面的内容：(1)研究了有限域上椭圆曲线E的有理点的群结构，以了解模函数域上定义方程的解的算术性质。如果椭圆曲线E是复乘为O的椭圆曲线的约化，则E的有理点的群结构由Frobenius自同态的迹α决定。由于α的绝对值是很容易知道的，所以问题是确定α的符号。我们给出了一种确定符号的方法，并在O是被2，3或5阶除以2，3或5的判别式的情况下确定了迹的符号。(2)设p=5，7，11.研究了模函数域的生成元将模不变函数J表示为p次多项式，并利用这种表示构造了有限域上具有循环有理点群的椭圆曲线族，确定了椭圆曲线的p-除点群上的Galois表示。(3)模函数域的定义方程的每一个解都对应于一条椭圆曲线。为了确定这种对应关系，我们研究了模函数域的生成元对模不变函数J的表示。设g是模函数域的亏格。我们得到了由g+1模函数Fj计算J的定义方程和表示的一个算法，这些模函数Fj是正则的，只有一个尖点非魏尔斯特拉斯点。实现该算法的关键是构造g+1模函数Fj。在水平N的Hecke群的情况下，我们构造了每个N&lt；53的模函数Fj。

In this research project, we studied following three subjects.(1)We studied the group structure of rational points of an elliptic curve E over finite fields in understanding arithmetic properties of solutions of a defining equation of a modular function field. In the case the elliptic curve E is a reduction of an elliptic curve with complex multiplication O, the group structure of rational points of E is determined by the trace α of the Frobenius endomorphism. Since the absolute value of α is easily known, the problem is to determine the sign of α. We showed a method to determine the sign and determined the sign of the trace in the case O is an orders of discriminant divided by 2, 3 or 5 and of class number 2 or 3.(2)Let p=5, 7, 11. We studied the representation of the modular invariant function J as a polynomial of degree p by a generator of a modular function field associated with the subgroup of SL_2(Z) of index p. We applied this representation to construct a family of elliptic curves with cyclic rational points groups over a finite field and to determine the Galois representation on the group of p-division points of elliptic curves.(3)Each solution of the defining equation of a modular function field corresponds to an elliptic curve. To determine this correspondence, we studied the representation of modular invariant function J by generators of the modular function field. Let g be the genus of the modular function field. We have obtained an algorithm to calculate the defining equation and the representation of J from g+1 modular functions fj which are regular except one cuspidal non-Weierstrass point. The essential part in practising the algorithm is to construct g+1 modular functions fj. In the case of Hecke group of level N, we constructed the modular functions fj for every N<53.