| Project/Area Number |
16K05079
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Nihon University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | ディオファンタスのm組 / 楕円曲線のMordell-Weil群 / 連立ペル方程式 / 楕円曲線の Mordell-Weil 群 / ディオファンタスの m 組 / 楕円曲線 / 不定方程式 |
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| Outline of Final Research Achievements |
A set of m distinct positive integers is called a Diophantine m-tuple if the product of any two elements in the set increased by 1 is a perfect square. In this study, we obtain some results supporting the open conjecture asserting that any Diophantine quadruple is regular. For example, we showed that for a Diophantine triple {a,b,c} (a<b<c) the number of d's such that {a,b,c,d} (c<d) is a irregular Diophantine quadruple is at most 7.
Diophantine quadruples are closely related to elliptic curves, which are certain curves having group structures. This study also explicitly examined integral points (that is, points with integer coordinates) and generators as groups on certain families of elliptic curves.
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| Academic Significance and Societal Importance of the Research Achievements |
任意のディオファンタスの4組は正則であると予想されており，様々な傍証はあるが，一般には正則でないディオファンタスの4組の有限性さえ示されていない．ディオファンタスの3組{a,b,c}(a<b<c)を固定してはいるが，{a,b,c,d}(c<d)が正則でないディオファンタスの4組であるようなdの個数の具体的な上限が得られた意義は大きい．一般に，与えられた１つの楕円曲線の整数点や群としての生成元を求めることは困難である．ここでは例えば u＾2-v＾4=m (mは0でない整数）で定義される楕円曲線の族について，階数が1または2の場合に整数点や生成元を具体的に決定した．
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