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

代数数论涉及多项式方程解的数字。 这些解所在的数字系统称为代数数域。 一个多世纪以来，数论的核心问题之一是是否可以使用分析技术构造某些特殊的代数数域。 希尔伯特在 1900 年国际数学家大会上将这个问题列为他著名的 23 个问题清单中的第 12 个问题。 该问题仅在最简单的情况下得到解决。 这些解决方案使用模形式的特殊值，这些值是具有丰富对称性的某些解析函数。 该项目概述了一个程序，用于在无限的新情况族中解决希尔伯特第 12 个问题。 关键思想是使用一种称为 p 进分析的现代分析形式，结合数论中的其他先进技术，包括模形式、伽罗瓦表示和岩泽理论。 在这些新情况下解决希尔伯特第 12 个问题将是我们对代数数系统理解的重大进步。 PI 与 Spiess 一起提出了关于 Gross-Stark 单位的精确 p-adic 解析公式的猜想。 这些单元与其他易于编写的元素一起生成完全实数域的最大阿贝尔扩张。 因此，解决这个问题可以被看作是为全实数域提供希尔伯特第十二个问题的解决方案。 L 函数的常见猜想框架（例如 Stark 猜想）无法提供希尔伯特第 12 个问题的这种解决方案。 PI 将继续与 Kakde 合作，破解他的猜想。 与该主题的先前工作相关的两个新想法是使用泰勒-怀尔斯“水平岩泽理论”方法，以及引入模形式的群环族。 接下来，PI 与 Spiess 合作，提出了 Gross 调节器的主次式和特征多项式的猜想。 PI 计划与 Spiess 和 Kakde 合作，将上述技术推广到更高阶（特别是 Taylor-Wiles 方法的应用），从而证明他对主次函数的猜想，这再次超出了 p 进 L 函数的通常框架。最后，PI将与Guido Kings合作，将爱森斯坦余循环应用于一般数域的阿贝尔L函数的研究。 需要考虑的两个重要测试案例是几乎完全真实的地面场和作为 CM 场的阿贝尔扩展的地面场。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值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.