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功能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以来，它们是现代数字理论中两个最大的未解决问题的主题，即Riemann假设以及桦木和Swinnerton-Dyer（BSD）猜想。 BSD猜想预测，立方方程（定义“椭圆曲线”）的有理解的数量由分析L功能的值控制。此预测在算术几何和复杂分析领域之间提供了一个神秘的桥梁，此后在Bloch-Kato猜想中已被广泛概括。通过改变我们看着这座桥的方式来攻击此类问题，最近取得了很大的成功。特别是，通过考虑两个数字之间的“距离”的不同概念，我们能够在算术和分析之间构建一系列不同代数连接的数组，这些数字使我们能够构建BSD和Bloch-Kato所需的桥梁的一部分。 所讨论的距离是“ P-ADIC”距离，如果它们的差分非常接近，则两个数字非常接近（例如，数字1和1,000,000,001的数字在2-Aped上非常接近，因为它们的差异可分配2九次）。对于每个Prime P，应该有一个P -ADIC版本的Bloch -Kato猜想（称为“ Iwasawa Main Supendures”），并且每一种都在算术和分析之间提供了另一个至关重要的联系。这种连接绝对取决于L功能的P-Adic版本的存在。 除了解决重要猜想的实用性外，P-Adic L功能本身就是美丽的物体。可以预期，对于每个L功能，都有一个P-ADIC版本，但是由于它们难以构造，因此我们远没有达到这个目标。 该提案的目的是通过构建新的P-Adic L功能来广泛推动我们对这一P-ADIC图片的理解，从代数拓扑，几何学和代表理论中汇总出新的技术来攻击基本但历史上棘手的案例。特别是，我将在最近的研究中使用强大的新方法来为高维自动形式提供一些最初的结构。