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.

该项目涉及代数数理论，这是数学的一个分支，旨在研究由多项式根（称为数字字段）引起的基本数字系统的属性。数字理论家对对对称组（称为Galois组）具有交换性属性的数字字段感兴趣，并且对生成这些特殊数字字段的公式产生了公式。现代方法已经证明了数字字段与某些相关功能之间的联系，称为L功能，其值编码了数字理论中许多最重要的猜想。首席研究员萨米特·达斯古普塔（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朝这个方向发展。 Dasgupta博士还计划继续并扩大他的活动，以将数学传播给所有年龄段和职业阶段的学生和学者。 Dasgupta博士为各种数学俱乐部的本科生提供外部讲座，教学生很小的学院，参与研究生和博士后咨询，参与本地会议组织，并且是多个期刊的编辑委员会。所有这些活动都与达斯古普（Dasgupta）博士在社会上整体促进数学的目标相关，并特别认为传统上人为不足的各个群体。从技术上讲，Dasgupta博士的工作是由现代代数数理论中的两个核心问题进行的：经典和Padic L功能作为代数对象的调节剂的特殊价值观的表达，以及通过分析数字的Abelian扩展的产生，分析含义对基地领域的本质，如希尔伯特（Hilbert）的第12个问题所致。 Dasgupta博士的先前工作在Brumer-Stark的建筑，总体建筑以及完全真实领域的班级领域的明确分析结构方面取得了重大进展。达斯古普塔（Dasgupta）博士将继续朝这个方向进行探索，其中有五个具体问题，涉及史塔克单元，l功能，模块化形式和galois表示之间的联系。所有这些项目将以重要的方式促进我们的知识，以了解L功能的特殊价值与相关代数对象之间的关系。首先，他将通过处理p = 2的本地化来完成布鲁姆stard猜想的证明。接下来，他将扩大与卡克德（Kakde）的合作，以证明塔玛川数字的猜想是CM Abelian Extensions的负分部分，包括在Prime 2上。另外，他将与Darmon和Charollois合作，扩大Darmon，Vonk和Pozzi的策略，以提供实际的二次领域，以提供Dasgupta博士对Brumer-Stark-Stark-Stark博士在任意完全真实领域的明确分析公式的纯粹P-ADIC分析证明。 Dasgupta博士将与Victor Rotger合作研究Harris和Venkatesh的猜想，该猜想将Venkatesh定义的派生的Hecke运营商与Galois扩展中的Stark单位定义为由Galois代表的伴随，而Galois表示的重量一形式。该奖项反映了NSF的法定使命，并通过使用基金会的知识分子优点和更广泛的影响评估标准来评估我们被认为是诚实的支持。