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Constructions and properties of p-adic L-functions for GL(n)

GL(n) 的 p 进 L 函数的构造和性质

基本信息

批准号：
EP/T001615/1
负责人：
Christopher Williams
金额：
$ 39.5万
依托单位：
University of Warwick
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT001615%2F1
关键词：
Constructions properties adic functions GL

项目摘要

L-functions are fundamental mathematical objects that encode deep arithmetic information. Their study goes back centuries, and they are the subject of the two biggest unsolved problems in modern number theory, namely the Riemann hypothesis and the Birch and Swinnerton-Dyer (BSD) conjecture. The BSD conjecture predicts that the number of rational solutions of a cubic equation (defining an 'elliptic curve') is controlled by a value of an analytic L-function. This prediction, providing a mysterious bridge between the fields of arithmetic geometry and complex analysis, has since been hugely generalised in the Bloch-Kato conjectures. There has been much recent success in attacking such problems by changing the way we look at this bridge. In particular, by considering different notions of 'distance' between two numbers, we are able to build a whole array of different algebraic connections between arithmetic and analysis, and these have allowed us to build parts of the bridge required for BSD and Bloch-Kato. The distance in question is the 'p-adic' distance, where two numbers are very close if their difference is very divisible by a prime p (for example, the numbers 1 and 1,000,000,001 are very close 2-adically, since their difference is divisible by 2 nine times). For each prime p, there should be a p-adic version of the Bloch-Kato conjectures - known as 'Iwasawa main conjectures' - and each of these gives another crucial connection between arithmetic and analysis. Such connections depend absolutely on the existence of p-adic versions of L-functions. In addition to their utility in solving important conjectures, p-adic L-functions are beautiful objects in their own right. It is expected that for every L-function there is a p-adic version, but as they can be extremely difficult to construct, we are very far from reaching this goal. The aim of this proposal is to extensively push forward our understanding of this p-adic picture by constructing new p-adic L-functions, drawing together novel techniques from algebraic topology, geometry and representation theory to attack fundamental but historically intractable cases. In particular, I will use powerful new methods developed in my recent research to give some of the first constructions for higher-dimensional automorphic forms.

L函数是编码深层算术信息的基本数学对象。他们的研究可以追溯到几个世纪前，并且是现代数论中两个最大的未解决问题的主题，即Riemann假设和Birch and Swinnerton-Dyer（BSD）猜想。 BSD猜想预言了一个三次方程（定义了一条“椭圆曲线”）的有理解的个数由一个解析L函数的值控制。这一预测，提供了一个神秘的桥梁之间的领域算术几何和复杂的分析，已被极大地推广，在布洛赫-加藤aptures。最近，通过改变我们对这座桥的看法，在解决这些问题方面取得了很大的成功。特别是，通过考虑两个数字之间不同的“距离”概念，我们能够在算术和分析之间建立一系列不同的代数联系，这些使我们能够建立BSD和Bloch-Kato所需的桥梁。这里的距离是“p-adic”距离，如果两个数字的差可以被素数p整除，那么它们就非常接近（例如，数字1和1，000，000，001非常接近2-adic，因为它们的差可以被2整除9次）。对于每一个素数p，都应该有一个布洛赫-加藤定理的p-adic版本--被称为“岩泽主定理”--每一个都给出了算术和分析之间的另一个重要联系。这样的连接完全依赖于L-函数的p-adic版本的存在性。除了它们在解决重要问题中的效用之外，p-adic L-函数本身也是美丽的对象。我们期望每个L-函数都有一个p-adic版本，但是由于它们非常难以构造，我们离这个目标还很远。该提案的目的是通过构建新的p-adic L-函数，将代数拓扑，几何和表示论中的新技术结合起来，以攻击基本但历史上难以解决的情况，从而广泛推动我们对p-adic图像的理解。特别是，我将使用我最近的研究中开发的强大的新方法来给出高维自守形式的一些第一个构造。