Higher Rank Selmer Groups

更高等级的塞尔默团体

基本信息

批准号：
1802440
负责人：
Joel Bellaiche
金额：
$ 15万
依托单位：
Brandeis University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802440&HistoricalAwards=false
关键词：
Higher Rank Selmer Groups

项目摘要

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.

数字理论是对整数的属性和隐藏结构的研究，是数学最古老的分支之一。它的一个重要方面是对方程式或方程式的研究，我们正在为其寻找整数的解决方案。这些方程式是在罗马时代的希腊数学家Diophantus之后命名为Diophantine方程式。当今，二磷剂方程仍然是深入研究的对象。独立地，在19世纪，数学的新分支是由牛顿和莱布尼兹在17世纪发明的微积分之间的婚姻，以及16世纪代数主义者发现的复杂数字，开发了复杂的分析，对所谓的霍洛型函数的研究，其可变数字是一个复杂的数字。在1859年的革命论文中，里曼观察到，这个新分支可以成功地用于解决数字理论中以前棘手的问题。在他的工作的基础上，数学家了解了如何与任何双态功能相关联，称为其L功能，并使其成为研究方程式的基本工具。然而，经验观察到的某些双磷剂方程的解决方案的数量与其全态L功能的分析特性之间的微妙关系仍然未经证实，并且非常神秘。该项目将采用新方法来研究这些关系。该项目将集中于Bloch-kato的猜想，该猜想预测了二聚体方程的算术特性（或者是其基本成分，称为动机）与其全态L功能之间的关系。更确切地说，该猜想预测，动机的Selmer组的等级（在某些基本情况下，与动机相关的算术对象（这是与定义其定义的二磷剂方程的解决方案）的算术对象相当于该动力的1个命令的命令。该项目的目的是在许多情况下证明在这种猜想平等的一个方向上，即Selmer群体至少排名消失的顺序。所提出的方法涉及研究特征变体或自动形式的普遍家庭，在临界点，在临界点分解Dasgupta和PI的关键padic l功能以及局部对关键Galois的变形研究的局部研究。通过使用基金会的知识分子和更广泛影响的评论标准来通过评估来支持。