Obstructed deformation rings and modularity of Galois representations

受阻变形环和伽罗瓦表示的模块化

• 批准号: 2200390
• 负责人: Chandrashekhar Khare
• 金额: $ 28.8万
• 依托单位: University of California-Los Angeles
• 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 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200390&HistoricalAwards=false
• 关键词: Obstructed; deformation; rings; modularity; Galois

This project investigates the relationship between algebra (Galois representations), analysis (automorphic forms), and geometry (motives). The study of this relationship in the past has had a major impact in solving classical problems in number theory, for example in the solution by Andrew Wiles in 1995 of Fermat's Last Theorem, a problem that had remained unresolved for more than 350 years. The work in this project will advance the paradigmatic Langlands program, which drives a lot of the current research in modern number theory. This has broad implications which might also be useful to applications of number theory where reciprocity laws can provide a powerful computational tool. The project provides training opportunities for graduate students.One of the broad themes of algebraic number theory is to prove reciprocity laws which give a way to understanding the splitting behavior of primes in number fields, or L-functions of varieties, in terms of more computable and apparently unrelated objects like automorphic forms. Such reciprocity laws go back to Gauss and his Law of Quadratic Reciprocity, and have a continuing life in current mathematics in the guise of the Langlands program. The methods of this proposal will prove novel cases of such reciprocity laws and establish new relations between the much studied triad of motives, Galois representations and automorphic forms. Reciprocity laws have implications for Diophantine geometry that studies solutions of polynomial equations in integers.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.

这个项目研究代数（伽罗瓦表示），分析（自守形式）和几何（动机）之间的关系。过去对这种关系的研究对解决数论中的经典问题产生了重大影响，例如安德鲁·怀尔斯在1995年解决了费马大定理，这是一个350多年来一直没有解决的问题。 该项目的工作将推进范式朗兰兹纲领，该纲领推动了现代数论中的许多当前研究。这具有广泛的影响，这也可能是有用的应用数论的互惠法律可以提供一个强大的计算工具。该项目为研究生提供了培训机会。代数数论的一个广泛主题是证明互惠定律，这为理解数域中素数的分裂行为提供了一种方法，或者品种的L-函数，在更可计算和明显无关的对象方面，如自守形式。这种互反定律可以追溯到高斯和他的二次互反定律，并以朗兰兹纲领的形式在现代数学中继续存在。这个建议的方法将证明新的情况下，这种互惠法律和建立新的关系，研究的三重动机，伽罗瓦表示和自守形式。互易定律对研究整数多项式方程解的丢番图几何有影响。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。