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将概括为完全真实领域的设置，达尔蒙（Darmon）和贝托利尼（Bertolini）在理性领域的椭圆曲线进行了一些椭圆形曲线。 在这项工作中，达尔蒙（Darmon）和贝托利尼（Bertolini）为赫格纳（Heegner）指出了椭圆曲线上指向的p- ad式分析公式，涉及两种可变性的p- ad r函数的中心衍生物，该曲线附加到曲线上。拟议的工作直接与birch and Swinnerton-dyer dyer deprip to nopect of to nopect of nope the On nopect of On nopect of nopect的固定性相关，该设置是在良好的范围内，该构想的固定性是在弯曲的曲线上，该弯曲的态度是在曲线上，该曲线是在弯曲曲线上的良好态度。理性数字（整数的商）。 该猜想将解决方案集与关联的L功能的行为相关联，相关的L功能是一种辅助函数，该辅助函数是根据多项式方程的解决方案数量定义的，当它被视为一致性模量在变化的质量数字时。 至少自1960年代以来，通过L函数对立方方程的研究一直是富有成果的数学研究的中心。