Arithmetic Aspects of Special Values of L-Functions

一个多世纪以来，理解椭圆曲线（本质上是一个三次方程）上有理点集合的结构一直是数论的一个目标。它与埋藏在古代的开放问题（如同余数问题）有联系。Birch和Swinnerton-Dyer （BSD）猜想预测了有理点和相关l函数之间的神秘关系。有趣的l函数出现在数学和物理学的各个领域。虽然表面上是解析的，但l -函数显示出与算术惊人的亲缘关系。PI旨在研究l函数的特殊值的某些算术方面。项目的第一部分旨在建立椭圆曲线具有解析秩（其中心相关l函数的消失阶）为1的p进判据。它将建立在BSD猜想（由于Skinner， Zhang和其他人）的最新进展的基础上，并旨在删除一些关键假设。项目的另一部分旨在研究l函数在垂直族和水平族中的特殊值的模p不消失。它将以过去二十年的进展为基础（由于Hida， Michel- Venkatesh和其他人）。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Understanding the structure of the set of rational points on an elliptic curve (essentially a cubic equation) has been an aim in number theory for over a century. It has connections to open problems (such as the congruent number problem) buried in antiquity. The Birch and Swinnerton-Dyer (BSD) conjecture predicts a mystifying relation among the rational points and the associated L-function. The intriguing L-functions appear in various areas of mathematics and also in physics. Even though analytic in appearance, the L-functions reveal striking affinity to arithmetic.The PI seeks to study certain arithmetic aspects of special values of L-functions. The first part of the project aims to establish a p-adic criterion for an elliptic curve to have analytic rank (the vanishing order of the associated L-function at its center) one. It will build on the recent progress towards the BSD conjecture (due to Skinner, Zhang and others) and aims to remove some of the key hypotheses. The other part of the project aims to study mod p non-vanishing of special values of L-functions in vertical and horizontal families. It will build on the progress during the last two decades (due to Hida, Michel--Venkatesh and others).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.