The Brumer-Stark Conjecture and its Refinements

布鲁默-斯塔克猜想及其改进

• 批准号: 2200787
• 负责人: Samit Dasgupta
• 金额: $ 55万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200787&HistoricalAwards=false
• 关键词: Brumer; Stark; Conjecture; its; Refinements

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.

这个项目涉及代数数论，数学的一个分支，旨在研究由多项式根（称为数域）产生的基本数制的性质。 数论学家对对称群（称为伽罗瓦群）具有交换性质的数域的分类以及产生这些特殊数域的公式很感兴趣。 现代方法已经证明了数域和某些称为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结构。 首席研究员Samit Dasgupta博士在最近的工作中在这些主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。Dasgupta博士还计划继续并扩大他的活动，向所有年龄组和职业阶段的学生和学者传播数学。Dasgupta博士在各种数学俱乐部为本科生提供临时讲座，为研究生教授迷你课程，参与研究生和博士后咨询，参与当地会议组织，并担任多家期刊的编辑委员会成员。所有这些活动都与Dasgupta博士在社会中全面推广数学的目标有关，特别是支持传统上代表性不足的各种群体。从技术上讲，Dasgupta博士的工作是由现代代数数论中的两个中心问题所激发的：古典和p-adic L函数的特殊值作为代数对象的调节器的表达，以及通过基场固有的分析手段生成数域的阿贝尔扩展，如希尔伯特的第12个问题中所编码的。 Dasgupta博士先前的工作在Brumer-Stark猜想、Gross-Stark猜想和全真实的场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的探索在这个方向上的五个具体问题之间的联系斯塔克单位，L-功能，模形式和伽罗瓦表示。 所有这些项目将推进我们的知识在一个显着的方式L-函数的特殊值和相关的代数对象之间的关系。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，他将扩大他的工作与Kakde证明等变玉川数猜想的负部分CM阿贝尔扩展的完全真实的领域，包括在总理2。 在与Spiess的合作中，Dasgupta博士将证明他们对格罗斯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，扩展Darmon、Vonk和Pozzi针对真实的二次域的策略，以给出Dasgupta博士针对任意完全真实的域上的Brumer-Stark单位的显式分析公式的纯p-adic分析证明。 Dasgupta博士将与维克托罗特尔格研究一个猜想的哈里斯和Venkatesh有关的衍生Hecke运营商所定义的Venkatesh斯塔克单位在伽罗瓦扩展削减了伴随的伽罗瓦表示附加到重量一形式。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。