L-Values, Special Cycles, and Euler Systems

数论，特别是研究整数和有理数的性质以及方程的整数或有理数解，是数学最古老的分支之一。它也是数学的许多其他分支的聚会场所-分析，几何，表示论（仅举几例）-数论基本问题的进展继续揭示这些分支之间的惊人联系以及其他领域的新应用。 本项目的核心问题是确定一个代数簇或基元的L-函数的特殊值的算术性质，并将这些值与与该代数簇或基元相关的代数量的阶数联系起来（例如Chow群、类群或塞尔默群）。 一个重要的和激励性的例子是著名的伯奇和斯温纳顿-戴尔猜想的椭圆曲线，大大推广了布洛赫-加藤猜想。 这个项目将通过进一步发展以下之间的联系来扩展这个基本问题的进展：L-函数，伽罗瓦表示，自守形式，表示理论，L-值和模形式的p-adic变化，特殊循环（特别是Shimura变种），以及伽罗瓦上同调群中的特殊类。这一进展将在两个主要方向展开：（1）证明Bloch-Kato猜想的新实例或其结果（例如Birch和Swinnerton-Dyer公式的p部分的新情况），以及（B）开发新工具和新方法来证明这些结果（如表示论在存在性欧拉系统中的新应用）该奖项反映了NSF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。

Number theory, especially the study of properties of integers and rational numbers and the integer or rational solutions to equations, is one of the oldest branches of mathematics. It is also a meeting ground for many other branches of mathematics -- analysis, geometry, representation theory (to name a few) -- and progress on the fundamental problems in number theory continues to reveal surprising connections between these branches as well as new applications to other areas. This project is focused on one of these fundamental problems and aims to establish new instances of these connections.The fundamental problem at the heart of this project is to determine the arithmetic nature of the special values of L-functions of an algebraic variety or motive and to relate these values to the orders of algebraic quantities associated to the variety or motive (such as a Chow group, a class group, or a Selmer group). An important and motivating instance is the celebrated Birch and Swinnerton-Dyer Conjecture for elliptic curves, vastly generalized by the Bloch-Kato Conjectures. This project will extend progress toward this fundamental problem through further development of connections between: L-functions, Galois representations, automorphic forms, representation theory, p-adic variation of L-values and modular forms, special cycles (especially on Shimura varieties), and special classes in Galois cohomology groups. This progress will unfold in two primary directions: (A) the proof of new instances of the Bloch-Kato Conjectures or their consequences (such as new cases of the p-part of the Birch and Swinnerton-Dyer formula), and (B) the development of new tools and new methods for proving results towards these conjectures (such as new applications of representation theory to the existence Euler systems).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.