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

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以前，他们是现代数论中两个最大的悬而未决的问题的主题，即黎曼假设和Birch and Swinnerton-Dyer(BSD)猜想。BSD猜想预测一个三次方程(定义一条‘椭圆曲线’)的有理解数由一个解析的L函数的值控制。这一预测在算术几何和复杂分析领域之间架起了一座神秘的桥梁，此后在布洛赫-加藤猜想中得到了极大的推广。最近，通过改变我们看待这座桥的方式，在解决这类问题方面取得了很大成功。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，这使得我们能够建立BSD和Bloch-Kato所需的桥梁的一部分。所讨论的距离是‘p-adi’距离，其中，如果两个数字的差可以被素数p整除，则它们非常接近(例如，数字1和1,000,000,001非常接近2,000,000,001，因为它们的差可以被2 9次整除)。对于每个素数p，都应该有一个p进位版本的Bloch-Kato猜想--也就是众所周知的岩泽主猜想--而每一个猜想都给出了算术和分析之间的另一个重要联系。这种联系完全依赖于L函数的p-进形式的存在。除了在解决重要猜想方面的效用外，p-进L函数本身也是美丽的对象。人们期望每个L函数都有一个p元版本，但由于它们的构造极其困难，我们距离这个目标还很远。这个建议的目的是通过构造新的p-进L函数，结合代数拓扑学、几何学和表示论的新技术来解决基本但历史上难以解决的问题，从而广泛地推动我们对这一p-进图的理解。特别是，我将使用在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的第一个构造。

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.