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

• 批准号: EP/T001615/2
• 负责人: Christopher Williams
• 金额: $ 3.38万
• 依托单位: University of Nottingham
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FT001615%2F2
• 关键词: Constructions; properties; adic; functions; GL

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 函数是编码深度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，它们是现代数论中两个最大的未解决问题的主题，即黎曼假设和伯奇和斯温纳顿-戴尔（BSD）猜想。 BSD 猜想预测三次方程（定义“椭圆曲线”）的有理解数由解析 L 函数的值控制。这一预测在算术几何和复分析领域之间架起了一座神秘的桥梁，此后在布洛赫-加藤猜想中得到了广泛推广。最近，通过改变我们看待这座桥的方式，在解决此类问题方面取得了很多成功。特别是，通过考虑两个数字之间“距离”的不同概念，我们能够在算术和分析之间建立一系列不同的代数联系，这使我们能够构建 BSD 和 Bloch-Kato 所需的部分桥梁。 所讨论的距离是“p-adic”距离，如果两个数字的差值可以被素数 p 整除，则两个数字非常接近（例如，数字 1 和 1,000,000,001 的 2-adic 值非常接近，因为它们的差值可以整除2 九倍）。对于每个素数 p，应该有一个布洛赫-加藤猜想的​​ p 进版本 - 称为“岩泽主要猜想” - 并且每个猜想都给出了算术和分析之间的另一个重要联系。这种联系绝对取决于 L 函数的 p 进数版本的存在。 除了在解决重要猜想方面的实用性之外，p 进 L 函数本身就是美丽的对象。预计每个 L 函数都有一个 p 进数版本，但由于它们构建起来极其困难，我们距离实现这一目标还很远。 该提案的目的是通过构造新的 p 进 L 函数，结合代数拓扑、几何和表示论的新技术来解决基本但历史上棘手的情况，从而广泛推进我们对 p 进图的理解。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一些高维自同构形式的第一个构造。