Rational points on algebraic curves and its application to number theory
Project/Area Number  09640074 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  RIKKYO UNIVERSITY 
Principal Investigator 
AOKI Noboru RIKKYO UNIV.COLLEGE OF SCIENCE.ASSISTANT PROF., 理学部, 助教授 (30183130)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,000,000 (Direct Cost : ¥2,000,000)
Fiscal Year 1998 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1997 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  Gauss sum / Jacobi sum / Fermat curve / elliptic curve / congruent number / Selmer group / Cassels pairing / TateShafarevich group / ガウス和 / ヤコビ和 / フェルマー曲線 / 楕円曲線 / 合同数 / セルマー群 / カッセルズ対 / テイト・シャファレビッチ群 / テイト・シャファレヴィッチ群 / 円分体 / フェルマ曲線 
Research Abstract 
In this research, I have studied rational points on algebraic curves. The main targets are Fermat curves and elliptic curves. As for Fermat curves, I treated the purity problem on Gauss sums and Jacobi sums which appear in the zeta functions of those curves. I was able to generalize certain known results, and proposed a new conjecture on the purity of Gauss sums. I showed that the conjecture is valid in some cases. As for elliptic curves, I treated special elliptic curves'which are closely related to the congruent number problem. The main result is an explicit formula for the size of the 2Selmer groups. As far as I know, such an explicit formula has never been obtained in the literature. The key point in the proof is an explicit calculation of the Cassels pairing restricted to a subgroup of the TateShafarevich group.

Report
(4results)
Research Output
(6results)