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

用构造方法研究模函数域和椭圆曲线的算术性质。

基本信息

批准号：
15540042
负责人：
ISHII Noburo
金额：
$ 1.28万
依托单位：
Osaka Prefecture University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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.

在这个研究项目中，我们研究了以下三个主题。(1)研究有限域上椭圆曲线E有理点的群结构，以理解模函数域定义方程解的算术性质。在椭圆曲线E是椭圆曲线复乘O的化简情况下，E的有理点群结构由Frobenius自同态的迹线α决定。因为α的绝对值很容易知道，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，并确定了当0是一个阶的判别式除以2、3或5，且类数为2或3时，迹的符号。(2)设p= 5,7,11。研究了模不变函数J作为p次多项式的模函数域的生成器与指标p的SL_2(Z)子群相关联的表示。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，并确定了椭圆曲线的p分点群上的Galois表示。(3)模函数场定义方程的每一个解对应一条椭圆曲线。为了确定这种对应关系，我们研究了模不变函数J用模函数域的生成器表示。设g为模函数域的格。本文给出了一种由g+1个除一个cusidal非weierstrass点以外的正则模函数fj计算定义方程和表示J的算法。练习算法的关键部分是构造g+1个模函数fj。对于水平N的Hecke群，我们构造了每个N<53的模函数fj。