Rational points and integer points on elliptic curves and the related Diophantine equations

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400025
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nihon University
• Principal Investigator: FUJITA Yasutsugu 日本大学, 生産工学部, 准教授 (50514163)
• Co-Investigator(Kenkyū-buntansha): TERAI Nobuhiro 大分大学, 工学部, 教授 (00236978)
• Research Collaborator: NARA Tadahisa 東北学院大学, 工学部, 非常勤講師
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 楕円曲線 / 不定方程式 / モーデルヴェイユ群 / 生成元 / 整数点 / 連立ペル方程式

Outline of Final Research Achievements

Our purpose of this research are (1) to study the generators and integer points on elliptic curves defined by equations having integral coefficients, and (2) to examine the extensibility of Diophantine pairs to Diophantine quintuples. For (1), let C_m be the elliptic curve defined by x＾3+y＾3=m, where m is cube-free. Then we determined the generators for the group C_m(Q) of rational points on C_m and integer points on C_m in the cases where C_m(Q) has rank 1 or 2. Moreover, we explicitly investigated the generators for the group E＾N(Q) of rational points on E＾N defined by y＾2=x＾3-N＾2x in the case where the rank of E＾N(Q) is 2 or 3. As for (2), we showed that any Diophantine pair {a,b} satisfying a<b<4*sqrt{a} or b<3a cannot be extended to a Diophantine quintuple.

Free Research Field

数物系科学