椭圆曲线的有理数解  

Rational solutions of an elliptic curve

作　　者：陈亦飞[1] 

机构地区：[1]中国科学院数学与系统科学研究院,北京100190

出　　处：《科学通报》2016年第34期3638-3643,共6页Chinese Science Bulletin

摘　　要：椭圆曲线的研究历史悠久,其中一个基本问题就是对于一条椭圆曲线,找出其所有的有理数解.对椭圆曲线有理数解的研究也不断推动着数论中众多领域的发展.例如,椭圆曲线理论在证明费马大定理中起到了关键作用.1922年,莫德尔证明椭圆曲线的有理数解构成一个有限生成交换群.从而,椭圆曲线有无穷多解等价于这个群的秩大于0.与此相关的最著名的问题当属七大千禧年问题之一的贝赫(Birch)和斯维纳通-戴尔(SwinnertonDyer)猜想(BSD猜想):椭圆曲线的秩和哈斯-韦伊(Hasse-Weil)L函数在s=1处的阶相等.BSD猜想为判断椭圆曲线是否有无穷多有理数解提供了一个途径.然而,要证明这个猜想十分困难,数学家们仍在为此努力着.The study of elliptic curves is long standing.A fundamental problem for an elliptic curve is to find all rational solutions.The problem has played and continues to play a fundamental role in the development of many areas of number theory.The theory of elliptic curves was crucial in the proof of Fermat’s Last Theorem.In this paper,we introduce rational solutions of lines and conics first.A natural problem of rational solutions of plane cubic arises,which is the problem of rational solutions of an elliptic curve in most cases.There is a natural addition on the rational solutions of an elliptic curve.So the set of rational solutions of an elliptic curve forms an abelian group.In 1922,Mordell proved that the set of rational solutions of an elliptic curve was a finitely generated abelian group,which confirmed a conjecture of Poincaré in 1901.By the fundamental structure theorem of finitely generated abelian group,the group of rational solutions of an elliptic curve is completely determined by its rank and torsion subgroup.The rank of the rational solutions of an elliptic curve is called the rank of the elliptic curve,which is an import invariant of an elliptic curve.Therefore,an elliptic curve having infinitely many rational solutions is equivalent to the rank of elliptic curve being positive.After many efforts,people are clear about the torsion subgroup of the rational solutions of an elliptic curve.However,people know a little about the rank of an elliptic curve.There are many conjectures of the rank of elliptic curves.One of them says most elliptic curves has rank 0 or 1.Birch and Swinnerton-Dyer reduce an elliptic curve over rational numbers to the elliptic curve over a finite field.The advantage of an elliptic curve over a finite field is that its solutions are finite.Then Birch and Swinnerton-Dyer define Hasse-Weil L-function L（E,s） regarding of solutions of the elliptic curve over the finite field.The Hasse-Weil L-function L（E,s） has good analytic properties and is somehow computable.Birch and Swinne

关 键 词：椭圆曲线 BSD猜想 椭圆曲线的秩 

分 类 号：O182[理学—数学]