Arithmetic Aspects of Special Values of L-Functions

一个多世纪以来，了解椭圆曲线(本质上是三次方程)上有理点集的结构一直是数论的一个目标。它与埋藏在古代的公开问题(如同余数问题)有联系。Birch和Swinnerton-Dyer(BSD)猜想预言了有理点和相关的L函数之间一种神秘的关系。耐人寻味的L函数出现在数学的各个领域，也出现在物理领域。尽管表面上是解析的，但L函数表现出了与算术的惊人的亲和力。PI试图研究L函数特殊值的某些算术方面。该项目的第一部分旨在建立椭圆曲线具有解析阶(相关的L函数在其中心的消失阶)的p-进准则。它将建立在BSD猜想(由于斯金纳、张和其他人)的最新进展的基础上，并旨在消除一些关键的假设。本项目的另一部分目的是研究L函数在垂直和水平家庭中的特殊值的模非零性。它将在过去20年(由于Hida、Michel-Venkatesh和其他人)的进展基础上再接再厉。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.