The Brumer-Stark Conjecture and its Refinements

这个项目涉及代数数论，它是数学的一个分支，旨在研究由多项式的根(称为数域)产生的基本数系的性质。数论家感兴趣的是对其对称群(称为伽罗瓦群)具有交换性质的数域进行分类，并给出生成这些特殊数域的公式。现代方法已经证明了数域和某些被称为L函数的相关函数之间的联系，这些函数的值编码了数论中许多最重要的猜想。首席研究员Samit Dasgupta博士在最近的工作中在这些专题上取得了重要进展，目前的提案旨在进一步推动这一方向。达斯古普塔博士还计划继续和扩大他的活动，向所有年龄段和职业阶段的学生和学者传播数学。达斯古普塔博士在不同的数学俱乐部为本科生讲授说明性课程，为研究生教授迷你课，参与研究生和博士后咨询，参与当地会议的组织，并是几家期刊的编辑委员会成员。所有这些活动都与达斯古普塔博士的目标有关，他的目标是在社会上全面推广数学，特别是支持传统上代表性不足的各种群体。更严格地说，达斯古普塔博士的工作是由现代代数数论中的两个核心问题推动的：经典的和p-进的L-函数作为代数对象的调节器的特定值的表达，以及通过基场固有的解析手段生成数域的阿贝尔扩张，如希尔伯特的第12个问题中所述。Dasgupta博士以前的工作在Brumer-Stark猜想、Gross-Stark猜想和全实域的类场的显式解析构造方面取得了重大进展。达斯古普塔博士将继续他在这个方向上的探索，提出五个关于斯塔克单位、L-函数、模形式和伽罗瓦表示之间的联系的具体问题。这些工作将极大地提高我们对L函数特殊值与相关代数对象之间关系的认识。首先，他将通过处理p=2的局部化来完成Brumer-Stark猜想的证明。接下来，他将扩展他与Kakde的工作，证明包括在素数2处的全实域的CM阿贝尔扩张的负部分的等变Tamagawa数猜想。在与Spiess的合作中，Dasgupta博士将证明他们关于Gross调节矩阵的特征多项式的联合猜想。另外，他将与Darmon和Charollois合作，将Darmon、Vonk和Pozzi的策略扩展到实二次域，以给出Dasgupta博士对任意全实域上Brumer-Stark单位的显式解析公式的纯p元解析证明。Dasgupta博士将与Victor Rotger一起研究Harris和Venkatesh的一个猜想，该猜想将由Venkatesh定义的派生Hecke算子与Galois扩展中的Stark单位相关联，该单位由附加在权重一形式上的Galois表示所切割。该奖项反映了NSF的法定使命，并已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。

This project concerns algebraic number theory, a branch of mathematics that aims to study properties of the basic number systems arising from roots of polynomials (called number fields). Number theorists are interested in classifying number fields whose symmetry groups (called Galois groups) have the commutative property, and in producing formulas to generate these special number fields. Modern methods have demonstrated the connection between number fields and certain associated functions called L-functions, whose values encode many of the most important conjectures in number theory. The principal investigator, Dr. Samit Dasgupta, has made important progress on these topics in recent work, and the current proposal aims to push further in this direction. Dr. Dasgupta also plans to continue and expand his activities to disseminate mathematics to students and academics of all age groups and career stages. Dr. Dasgupta gives expository lectures for undergraduates in various math clubs, teaches minicourses for graduate students, is involved in graduate and postdoctoral advising, is involved in local conference organizing, and is on the editorial board of several journals. All these activities connect to Dr. Dasgupta’s goal to promote mathematics holistically in society, with a particular view toward supporting various groups that have been traditionally underrepresented. More technically, Dr. Dasgupta’s work is motivated by two central problems in modern algebraic number theory: the expression of special values of classical and p-adic L-functions as regulators of algebraic objects, and the generation of abelian extensions of number fields through analytic means intrinsic to the ground field, as codified in Hilbert's 12th problem. Dr. Dasgupta’s prior work has made significant progress on the Brumer-Stark Conjecture, the Gross-Stark Conjecture, and the explicit analytic construction of class fields of totally real fields. Dr. Dasgupta will continue his explorations in this direction with five specific questions on the connections between Stark units, L-functions, modular forms, and Galois representations. All these projects will advance our knowledge in a significant way on the relationship between special values of L-functions and associated algebraic objects. Firstly, he will complete the proof of the Brumer-Stark conjecture by handling the localization at p=2. Next, he will extend his work with Kakde to prove the Equivariant Tamagawa Number Conjecture for the minus part of CM abelian extensions of totally real fields, including at the prime 2. In joint work with Spiess, Dr. Dasgupta will prove their joint conjecture on the characteristic polynomial of Gross's regulator matrix. Separately, he will work with Darmon and Charollois on expanding the strategy of Darmon, Vonk, and Pozzi for real quadratic fields to give a purely p-adic analytic proof of Dr. Dasgupta's explicit analytic formula for Brumer-Stark units over arbitrary totally real fields. Dr. Dasgupta will work with Victor Rotger to study a conjecture of Harris and Venkatesh relating the derived Hecke operators defined by Venkatesh to Stark units in the Galois extension cut out by the adjoint of the Galois representation attached to weight one forms.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.