Beyond L-functions: the Eisenstein Cocycle and Hilbert's 12th Problem

代数论涉及的是作为多项式方程的解的数字。这些解所在的数系称为代数数域。一个多世纪以来，数论的核心问题之一就是一个人是否能用分析技术构造某些特殊的代数数域。在1900年国际数学家大会上，希尔伯特将这一问题列为他著名的23个问题清单中的第12个问题。这个问题只在最简单的情况下得到了解决。这些解使用模形式的特定值，这些模形式是具有丰富对称性的某些解析函数。这个项目概述了一个在无限的新情况下解决希尔伯特的第12个问题的方案。其关键思想是结合数论中的其他高级技术，包括模形式、伽罗瓦表示和岩泽理论，使用一种现代的分析形式，称为p-adi分析。在这些新的情况下解决希尔伯特的第12个问题将是我们对代数数系理解的一大进步。PI与Spiess提出了一个猜想，即Gross-Stark单位的精确p-进解析公式。这些单位与其他容易编写的元素一起，生成了全实场的最大阿贝尔扩张。因此，解决这个问题可以看作是为全实领域提供了希尔伯特第12个问题的解决方案。通常的L函数猜想框架，如斯塔克猜想，并不能为希尔伯特第12问题提供这样的解。私家侦探将继续与卡德一起破解他的猜测。相对于以前的工作，两个新的想法是使用Taylor-Wiles的“水平岩泽理论”方法，以及引入模形式的群环族。其次，在与Spiess的合作中，PI提出了一个关于Gross调节器的主子式和特征多项式的猜想。PI计划与Spiess和Kakde合作，将上述技巧推广到更高的阶(特别是Taylor-Wiles方法的应用)，从而证明他关于主子函数的猜想，这再次超出了通常的p元L函数的框架。最后，PI将与Guido Kings合作，将艾森斯坦共循环应用于阿贝尔L--一般数域的函数的研究。要考虑的两个重要测试案例是几乎完全真实的地面领域和作为CM领域的阿贝尔延伸的地面领域。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic number theory concerns numbers that are solutions to polynomial equations. The number systems in which these solutions live are called algebraic number fields. One of the central motivating questions in number theory for over a century has been whether one can construct certain special algebraic number fields using analytic techniques. Hilbert stated this problem as the 12th in his famous list of 23 problems at the 1900 International Congress of Mathematicians. The problem has been solved in only the simplest cases. These solutions use special values of modular forms, which are certain analytic functions that have a rich supply of symmetries. This project outlines a program to give a solution to Hilbert's 12th problem in an infinite family of new situations. The key ideas are to use a modern form of analysis, called p-adic analysis, in conjunction with other advanced techniques in number theory including modular forms, Galois representations, and Iwasawa theory. Solving Hilbert's 12th problems in these new cases will be a major advance in our understanding of algebraic number systems. The PI has stated a conjecture with Spiess for an exact p-adic analytic formula for Gross-Stark units. These units, along with other easily written elements, generate the maximal abelian extension of totally real fields. Therefore, solving this problem can be viewed as providing a solution to Hilbert's 12th problem for totally real fields. Such a solution to Hilbert's 12th problem is not provided by the usual framework of conjectures for L-functions, such as Stark's conjectures. The PI will continue his work with Kakde on attacking his conjecture. Two new ideas relative to previous work on this topic are the use of the Taylor-Wiles "horizontal Iwasawa theory" method, as well as the introduction of group-ring families of modular forms. Next, in joint work with Spiess, the PI has stated a conjecture for the principal minors and characteristic polynomial of Gross' regulator. The PI plans to work together with Spiess and Kakde to generalize the techniques described above to higher rank (in particular the application of the Taylor-Wiles method) and thereby prove his conjecture on principal minors, which again goes beyond the usual framework of p-adic L-functions. Finally, the PI will work with Guido Kings to apply the Eisenstein cocycle to the study of abelian L-functions of general number fields. Two important test cases to consider are ground fields that are almost totally real and ground fields that are abelian extensions of CM fields.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.