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 最近针对有理域上的椭圆曲线所做的一些工作推广到完全实数域的设置。 在这项工作中，Darmon 和 Bertolini 推导了椭圆曲线上 Heegner 点的 p 进解析公式，其中涉及附加到曲线上的二变量 p 进 L 函数的中心导数。所提出的工作与猜想直接相关Birch 和 Swinnerton-Dyer 的论文，这是一个杰出的开放猜想，它应该阐明有理数（整数的商）的三次多项式方程的解集。 该猜想将解集与相关 L 函数的行为联系起来，L 函数是一个辅助函数，当多项式方程被视为模变质数的同余时，根据多项式方程的解数来定义。 至少自 20 世纪 60 年代以来，通过 L 函数研究三次方程一直是卓有成效的数学研究的中心。