LEAPS-MPS: Elliptic Dedekind Sums, Eisenstein Cocycles, and p-adic L-Functions

LEAPS-MPS：椭圆戴德金和、爱森斯坦余循环和 p 进 L 函数

• 批准号: 2212924
• 负责人: Tian An Wong
• 金额: $ 12.18万
• 依托单位: Regents of the University of Michigan - Dearborn
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2212924&HistoricalAwards=false
• 关键词: LEAPS; MPS; Elliptic; Dedekind; Sums

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).Number theory studies the properties of prime numbers and has important applications in physics, computer science, cyber security, and other areas. There are important classes of numbers extending the usual integers and corresponding classes of functions known as modular forms. The goal of this project is to extend known results about classical modular forms over the usual integers to more general settings. It is expected that different phenomena will appear in this context, which will yield new and interesting applications to number theory and beyond.The project will develop explicit methods and results in relation to the arithmetic of Bianchi modular forms, elliptic Dedekind sums, and L-functions over imaginary quadratic fields. Building upon previous work, the PI will continue the study of this circle of ideas and explore their connections to open problems and conjectures, namely (1) a generalization of Sczech’s Eisenstein cocycles over imaginary quadratic fields with applications to p-adic L-functions, (2) elliptic Dedekind sums attached to discrete subgroups of SL(2,C) and arithmetic applications such as to conjectures of Sharifi, and (3) Bianchi period polynomials associated to binary Hermitian forms, with applications to special values of L-functions.The broader impacts of this project are through the research training and mentorship of undergraduate students in a Primarily Undergraduate Institution — many of whom being first-generation students and members of under-represented minorities.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.

该奖项全部或部分由2021年美国救援计划法案（公法117 - 2）资助。数论研究素数的性质，在物理学、计算机科学、网络安全等领域有着重要的应用。有一些重要的数类扩展了通常的整数和相应的函数类，称为模形式。这个项目的目标是将已知的关于通常整数上的经典模形式的结果扩展到更一般的设置。预计不同的现象将出现在这种情况下，这将产生新的和有趣的应用数论和beyond. The项目将开发明确的方法和结果有关的算术的比安奇模形式，椭圆Dedekind和，L-函数在虚二次域。在以前工作的基础上，PI将继续研究这个思想圈，并探索它们与开放问题和代数的联系，即（1）虚二次域上的Schzech Eisenstein上循环的推广及其对p-adic L-函数的应用，（2）SL（2，C）离散子群的椭圆Dedekind和以及算术应用，如Sharifi代数，（3）与二元厄米特形式相关的比安奇周期多项式，该项目的更广泛的影响是通过研究培训和指导的本科生在一个小学本科院校-其中许多是第一代学生和成员下，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。

项目成果

Kronecker's first limit formula for Kleinian groups

克莱尼群的克罗内克第一极限公式

• DOI: 10.7169/facm/1997
• 发表时间: 2023
• 期刊: Functiones et Approximatio Commentarii Mathematici
• 影响因子: 0.5
• 作者: Miao, Zihan;Nguyen, Anh;Wong, Tian An
• 通讯作者: Wong, Tian An

On partial information retrieval: the unconstrained 100 prisoner problem

关于部分信息检索：无约束的 100 名囚犯问题

• DOI: 10.1007/s00236-022-00436-y
• 发表时间: 2023
• 期刊: Acta Informatica
• 影响因子: 0.6
• 作者: Lodato, Ivano;Shekatkar, Snehal M.;Wong, Tian An
• 通讯作者: Wong, Tian An