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

代数数论关注的是多项式方程的解。 这些解所在的数制称为代数数域。 一个中央激励问题在数论超过世纪一直是人们是否可以构造某些特殊的代数数域使用分析技术。 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个问题中的第12个。 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殊值，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程序，以解决希尔伯特的第12个问题，在一个无限的家庭的新情况。 主要思想是使用一种现代形式的分析，称为p-adic分析，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。 在这些新的情况下解决希尔伯特第12问题将是我们理解代数数系的一个重大进展。 PI与Spiess一起提出了一个关于Gross-Stark单位的精确p-adic解析公式的猜想。 这些单位，沿着与其他容易写的元素，生成全真实的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真实的域提供希尔伯特第12问题的解决方案。 希尔伯特第12问题的这种解决方案并不是由L-函数的一般结构（如斯塔克结构）提供的。 PI将继续与Kakde一起攻击他的猜想。 两个新的想法相对于以前的工作就这一主题是使用泰勒-怀尔斯“水平岩泽理论”的方法，以及引进群环族的模形式。 接下来，在与Spiess的联合工作中，PI提出了关于格罗斯调节器的主要子式和特征多项式的猜想。 PI计划与Spiess和Kakde一起工作，将上述技术推广到更高的秩（特别是泰勒-怀尔斯方法的应用），从而证明他关于主子式的猜想，这再次超出了通常的p-adic L-函数框架。最后，PI将与Guido Kings合作，将Eisenstein上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CM场的阿贝尔扩展地面场。该奖项反映了NSF的法定使命，并通过使用基金会的知识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.