Obstructed deformation rings and modularity of Galois representations

这个项目调查了代数(伽罗瓦表示法)、分析(自同构形式)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Andrew Wiles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。这个项目中的工作将推进范例朗兰兹计划，该计划推动了现代数论中的许多当前研究。这具有广泛的含义，这对数论的应用也可能是有用的，在那里互易定律可以提供强大的计算工具。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易定律，该定律提供了一种方法，通过更可计算的和表面上无关的对象，如自同构形，来理解素数在数域中的分裂行为，或者是簇的L函数。这样的互易定律可以追溯到高斯和他的二次互易定律，并以朗兰兹计划的名义在当前的数学中继续存在。这一提议的方法将证明这种互惠定律的新案例，并在被广泛研究的动机、伽罗瓦表示和自同构形的三元组之间建立新的关系。互易定律对研究整数中多项式方程的解的丢番图几何学有影响。该奖项反映了NSF的法定使命，并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 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.