Study on arithmetic properties of modular function fields and elliptic curves by constructive methods.
Project/Area Number 
15540042

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Osaka Prefecture University 
Principal Investigator 
ISHII Noburo Osaka Prefecture University, Faculty of Liberal arts and Sciences, Professor, 総合教育研究機構, 教授 (30079024)

Project Period (FY) 
2003 – 2005

Project Status 
Completed (Fiscal Year 2005)

Budget Amount *help 
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2005: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2004: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)

Keywords  modular function fields / elliptic curves / defining equation / Jinvariant function / Frobenius endomorphism / モジュラー曲線 / J不変関数 / 巡回的有理点群 / J関数 / フロベニウス凖同型 / トレース 
Research Abstract 
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 pdivision 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 nonWeierstrass 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.

Report
(4 results)
Research Products
(7 results)