【摘 要】
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A positive integer n is called a congruent number if it is the area of a right triangle with rational side lengths,or equivalently the elliptic curve En : n
【机 构】
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AcademyofMathematicsandSystemsScienceandMorningsideCenterofMathematics,ChineseAcademyofSciences,Chin
【出 处】
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第二届中法算术代数几何会议(SFCAG 2016)
论文部分内容阅读
A positive integer n is called a congruent number if it is the area of a right triangle with rational side lengths,or equivalently the elliptic curve En : ny2 = x3-x has Mordell-Weil group of rank > 0.Note that for square-free positive integer n,the L-function L(En; s)has sign-1 if and only if n ≡5; 6; 7 mod 8.Goldfeld conjectured that among all positive square-free integers congruent to 5,6,7 modulo 8,those n with ords=1L(En; s)= 1 has density one.In this talk,we show this density is > 50%.This talk is based on our joint work with Xinyi Yuan and Shouwu Zhang,and work of Smith.
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