Algebraic cycles, L-functions and rational points on elliptic curves

代数环、L 函数和椭圆曲线上的有理点

• 批准号: 1015173
• 负责人: Kartik Prasanna
• 金额: $ 8.26万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 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: 2009-11-16 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1015173&HistoricalAwards=false
• 关键词: Algebraic; cycles; functions; rational; points

This proposal aims to investigate relations between special values of L-functions, cycles and periods with important applications to some open conjectures about L-functions such as those of Birch-Swinnerton-Dyer and Bloch-Kato-Beilinson. Specifically, the investigator and his collaborators will (i) Study the algebraic cycles associated to Rankin-Selberg L-functions and their images under Abel-Jacobi maps, and apply these results to give new constructions of rational points on CM elliptic curves; (ii) Study p-adic L-functions and the Iwasawa main conjecture for CM Hida deformations; (iii) Explore methods to prove a conjecture of his relating periods of quaternionic modular forms to adjoint L-values, with applications to some cases of the Bloch-Kato conjecture (iv) Study the problem of constructing and counting invariant linear forms on a triple product of representations of the metaplectic group, thus generalizing results on triple product L-functions associated to modular forms of integral weight to the setting of modular forms of half-integral weight. The general focus of this proposal is the area of number theory. Number theory has to do with such objects as prime numbers and diophantine equations. Other than being perhaps the oldest branch of mathematics, it is of great significance in today's world, since many cryptographic protocols (needed for secure transmissions over the internet) and error correcting codes (needed for compact discs, hard discs and the like) are based on number theoretic methods. These practical applications in fact involve rather sophisticated geometrical objects such as elliptic curves. Over the last half-century, we have realized that one can gain a better understanding of these geometric objects by studying certain functions, called L-functions. Conjecturally one expects that very interesting information about the geometric object is encoded in the behavior of the associated L-function at certain special points. The investigator hopes to deepen our understanding of this connection through the work to be done in the current proposal. One of the concrete consequences of this project will be a new method to find solutions in rational numbers to certain cubic equations, a central problem in number theory.

该提案旨在调查L功能，周期和时期特殊值与重要应用的特殊价值之间的关系，以对某些关于L功能的开放猜想，例如Birch-Swinnerton-Dyer和Bloch-Kato-Beilinson。具体而言，研究者及其合作者将（i）研究与兰金·塞伯格（Rankin-Selberg L）及其在亚伯-Jacobi地图下的图像相关的代数周期，并将这些结果应用于CM椭圆形曲线上的理性点的新结构； （ii）研究P-ADIC L功能和用于CM HIDA变形的Iwasawa主要猜想； （iii）探索方法来证明其与伴随l值的Quaternion模块化形式相关时期的猜想，并适用于某些Bloch-kato猜想（IV）的某些情况，研究了建立和计数不变形式的构建和计数与转换群体的三重形式相关联的构造的不变形式的问题的问题，从而使整体构图的形式与转移的形式相关联，从而将属于转移的产品的形式相关联。一半综合体重。该提案的一般重点是数字理论领域。数字理论与素数和二磷剂方程等对象有关。除了成为数学最古老的分支外，它在当今世界上具有重要意义，因为许多加密协议（通过Internet上的安全传输所需）和错误纠正代码（紧凑型唱片，硬盘等所需）是基于数字理论方法。实际上，这些实际应用涉及相当复杂的几何对象，例如椭圆曲线。在过去的半个世纪中，我们已经意识到可以通过研究某些功能（称为L功能）来更好地理解这些几何对象。猜想的人期望有关几何对象的非常有趣的信息在某些特殊点上相关的L功能的行为中编码。研究人员希望通过当前的提案中的工作加深我们对这一联系的理解。该项目的具体后果之一将是一种新方法，可以找到某些立方方程的合理数字解决方案，这是数字理论的核心问题。