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

超越 L 函数：爱森斯坦余循环和希尔伯特第 12 个问题

基本信息

批准号：
1901939
负责人：
Samit Dasgupta
金额：
$ 18.6万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901939&HistoricalAwards=false
关键词：
Beyond functions Eisenstein Cocycle Hilbert

项目摘要

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.

代数数论关注的是多项式方程的解。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是能否利用解析技术构造某些特殊的代数数域。希尔伯特在1900年的国际数学家大会上列出了他著名的23个问题中的第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殊值，模形式是某些具有丰富对称性的解析函数。这个项目概述了一个程序，在无限的新情况下给出希尔伯特第12问题的解。关键思想是使用一种现代形式的分析，称为p进分析，结合数论中的其他先进技术，包括模形式、伽罗瓦表示和岩川理论。在这些新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个重大进步。PI已经和spess一起提出了一个关于Gross-Stark单位的精确p进解析公式的猜想。这些单元，连同其他易于编写的元素，生成全实数域的最大阿贝尔扩展。因此，解决这个问题可以看作是为全实场提供了希尔伯特第12个问题的解。对于希尔伯特第12问题的解，通常的l -函数的猜想框架，如Stark的猜想，并没有提供这样的解。私家侦探将继续与卡克德合作，攻击他的猜想。与此主题的先前工作相关的两个新想法是使用Taylor-Wiles的“水平Iwasawa理论”方法，以及引入模形式的群环族。其次，在与spess的合作中，PI对Gross调节器的主次多项式和特征多项式提出了一个猜想。PI计划与spess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），从而证明他关于主次函数的猜想，这再次超出了通常的p进l函数框架。最后，PI将与Guido Kings合作，将爱森斯坦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用例是几乎完全真实的接地域和CM域的阿贝尔扩展接地域。该奖项反映了美国国家科学基金会的法定使命，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。