Arithmetic Aspects of Special Values of L-Functions

理解椭圆曲线（本质上是一个三次方程）上有理点集的结构是数论的一个目标，已经有世纪了。它与埋藏在古代的开放问题（如全等数问题）有联系。Birch和Swinnerton-Dyer猜想（BSD）预言了有理点和相关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。尽管在外观上是解析的，但L-函数显示出与算术惊人的亲和力。PI旨在研究L-函数的特殊值的某些算术方面。该项目的第一部分旨在建立一个椭圆曲线的p-adic准则，使其具有解析秩（相关L-函数在其中心的消失阶）1。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，旨在消除一些关键假设。另一部分是研究垂直和水平族中L-函数特殊值的mod p非零性。它将建立在过去二十年的进展（由于希达，米歇尔-文卡特什和其他人）。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.