Rational points on elliptic curves over totally real fields and p-adic L-functions

全实域和 p 进 L 函数上椭圆曲线上的有理点

• 批准号: 0901289
• 负责人: Kenneth Ribet
• 金额: $ 2.11万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-10-01 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901289&HistoricalAwards=false
• 关键词: Rational; points; elliptic; curves; over

The PIs will generalize to the setting of totally real fields some recent work by Darmon and Bertolini for elliptic curves over the rational field. In this work, Darmon and Bertolini derive a p-adic analytic formula for Heegner points on elliptic curves that involves the central derivative of the two-variable p-adic L-function attached to the curve.The proposed work is related directly to the conjecture of Birch and Swinnerton-Dyer, an outstanding open conjecture that should shed light on the set of solutions to cubic polynomial equations in rational numbers (quotients of whole numbers). This conjecture relates the set of solutions to the behavior of the associated L-function, an auxiliary function that is defined in terms of the numbers of solutions to the polynomial equation when it is viewed as a congruence modulo varying prime numbers. The study of cubic equations via L- functions has been the center of fruitful mathematical research at least since the 1960s.

PI将把Darmon和Bertolini关于有理数域上椭圆曲线的一些最新工作推广到全实域的设置。在这项工作中，Darmon和Bertolini得到了椭圆曲线上Heegner点的p-进解析公式，它涉及附在曲线上的二元p-进L-函数的中心导数。所提出的工作直接与Birch和Swinnerton-Dyer的猜想有关，这是一个突出的公开猜想，它应该揭示有理数(整数的商)中三次多项式方程的解的集合。这个猜想将解的集合与相关的L函数的行为联系起来，该函数是一种辅助函数，当它被视为模变素数的同余时，它是根据多项式方程的解的个数来定义的。至少自20世纪60年代以来，利用L函数研究三次方程一直是取得丰硕成果的数学研究的中心。