Higher Rank Selmer Groups

数论研究整数的性质和隐藏结构，是数学中最古老的分支之一。它的一个重要方面是研究方程，或方程组，我们正在寻找整数解。这些方程以罗马时代的希腊数学家丢番图斯的名字命名为丢番图方程。丢番图方程至今仍是人们深入研究的对象。19世纪，牛顿和莱布尼茨在17世纪发明的微积分与16世纪的代数学家发现的复数相结合，形成了一个新的数学分支：复数分析，即研究变量为复数的所谓全纯函数。在1859年的一篇革命性论文中，Riemann观察到这一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。在他工作的基础上，数学家们了解了如何将一个全纯函数与任何丢番图方程联系起来，称为它的L函数，并使其成为研究方程的基本工具。然而，经验上观察到的某些丢番图方程的解的个数与其全纯L函数的解析性质之间的微妙关系仍未得到证明，而且是非常神秘的。该项目将采用新的方法来研究这些关系。这个项目将集中在Bloch-Kato的猜想上，该猜想预测了丢番图方程(或者更确切地说，它们的初等分量，称为动机)和它们的全纯L函数之间的算术性质之间的关系。更确切地说，这一猜想预测了动机(它是附在动机上的算术对象，在某些基本情况下与定义动机的丢番图方程的解密切相关)的塞尔默群的阶等于该动机的L函数的1的阶数。这个项目的目的是，在许多情况下，证明这个猜想等式中的一个方向，即Selmer群的排名至少是消失的次序。建议的方法包括研究临界点的特征簇或自同构形式的普适族，Dasgupta和PI的临界p-进L函数的分解方法，以及PI和Chenevier，Bergdall，Breuil等人开发的临界Galois表示的变形的局部研究。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。

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.