Higher Rank Selmer Groups

数论是数学中最古老的分支之一，研究整数的性质和隐藏结构。它的一个重要方面是研究方程或方程组，我们正在寻找整数的解决方案。这些方程被命名为丢番图方程后，丢番图，希腊数学家的罗马时代。丢番图方程至今仍是人们研究的热点。独立地，在19世纪，一个新的数学分支诞生了，它是由牛顿和莱布尼茨在世纪发明的微积分与16世纪世纪代数学家发现的复数结合而成的：复分析，研究所谓的全纯函数，其变量是复数。在1859年的一篇革命性论文中，黎曼发现这个新的分支可以成功地用于解决数论中以前难以解决的问题。在他的工作的基础上，数学家们理解了如何将任何丢番图方程与一个全纯函数联系起来，称为其L函数，并使其成为研究方程的基本工具。然而，经验上观察到的微妙关系的解决方案的数量，某些丢番图方程，和分析性质的全纯L-功能仍然未经证实和深深的神秘。该项目将采用新的方法来研究这些关系。 该项目将侧重于布洛赫-加藤猜想，该猜想预测丢番图方程的算术性质（或者更确切地说，它们的基本组成部分，称为动机）与它们的全纯L函数之间的关系。更准确地说，这个公式预言了一个动机的塞尔默群（它是一个附属于动机的算术对象，在某些基本情况下，与定义它的丢番图方程的解密切相关）的秩等于该动机的L函数在1处的阶。这个项目的目的是证明，在许多情况下，一个方向，在这个自然平等，即塞尔默集团的秩至少为零的顺序。所提出的方法包括研究特征簇，或自守形式的泛族，在临界点，Dasgupta和PI的临界p-adic L-函数的分解方法，以及PI和Chenevier，Bergdall开发的临界Galois表示的变形的局部研究，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, the study of properties and hidden structures of integers, is one of the oldest branch of mathematics. An important aspect of it is the study of equations, or systems of equations, for which we are looking for solutions that are integers. These equations are named diophantine equations after Diophantus, a Greek mathematician of the Roman era. Diophantine equations are still the object of intense study nowadays. Independently, in the 19th century, a new branch of mathematics, born of the marriage between the calculus invented by Newton and Leibniz in the 17th century, and complex numbers discovered by algebraists of the 16th century, was developed: complex analysis, the study of so-called holomorphic functions, whose variable is a complex number. In a revolutionnary paper of 1859, Riemann observed that this new branch could be successfully used to solve previously intractable questions in number theory. Building on his work, mathematicians understood how to associate to any diophantine equation an holomorphic function, called its L-function, and make it a fundamental tool in studying the equation. Yet empirically observed subtle relations between the number of solutions of certain diophantine equations, and the analytic property of their holomorphic L-functiona remain unproved and deeply mysterious. The project will employ new methods to study these relations. The project will focus on the conjecture of Bloch-Kato, which predicts a relation between the arithmetic properties of diophantine equations (or rather their elementary components, called motives) and their holomorphic L-function. More precisely, this conjectures predicts that the rank of the Selmer group of a motive (which is an arithmetic object attached to the motive, closely related, in some fundamental cases, to the solutions of the diophantine equation defining it) is equal to the order at 1 of the L-function of that motive. The aim of this project is to prove, in many cases, one direction in this conjectural equality, namely that the Selmer groups has rank at least the order of vanishing. The proposed method involves the study of eigenvarieties, or universal families of automorphic forms, at critical points, the method of decomposition of critical p-adic L-functions of Dasgupta and the PI, and the local study of deformation of critical Galois representations developed by the PI and Chenevier, Bergdall, Breuil 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.