Automorphy lifting theorems and generalizations of Serre's conjecture

自同构提升定理和塞尔猜想的推广

• 批准号: 1101483
• 负责人: Francesco Calegari
• 金额: $ 4.54万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101483&HistoricalAwards=false
• 关键词: Automorphy; lifting; theorems; generalizations; Serre

This project investigates some of the mod p and p-adic aspects of the Langlands program, and their links with p-adic Hodge theory. Specifically, the PI proposes to prove new cases of Serre's conjecture, to formulate precise generalizations of Serre's conjecture to arbitrary reductive groups, to prove new automorphy lifting theorems, and to establish results towards the Gouvea-Mazur conjecture. In particular, the PI proposes to investigate the ubiquity of the property of ``potential diagonalizability'' for potentially crystalline Galois representations, which will give powerful new automorphy lifting theorems thanks to previous work of him and his collaborators. He also proposes to use functoriality and these new automorphy lifting theorems to establish cases of Serre's conjecture over real quadratic fields.The Langlands program is a branch of mathematics that includes ideas from number theory (the study of equations in whole numbers), representation theory and analysis. The number theoretic part of the program consists of a vast set of interlinked conjectures relating the solutions of equations in whole numbers to other apparently unrelated mathematical objects. In recent years it has become clear that there should be another general framework in which the number theoretic part of the program fits, a ``p-adic and mod p Langlands program''. The PI proposes to formulate some precise conjectures in the mod p program, and to use a technique originally developed by Andrew Wiles in his work on Fermat's Last Theorem to prove cases of these conjectures, and other more classical conjectures in number theory.

这个项目调查了朗兰兹计划的一些mod p和p进方面，以及它们与p进Hodge理论的联系。具体地说，PI建议证明Serre猜想的新情况，将Serre猜想精确地推广到任意约化群，证明新的自同构提升定理，并建立针对Gouvea-Mazur猜想的结果。特别是，PI建议研究潜在结晶Galois表示的“潜在对角化”性质的普遍性，这将给出强大的新的自同构提升定理，这要归功于他和他的合作者以前的工作。他还建议使用函数性和这些新的自同构提升定理来建立实二次域上的Serre猜想的例子。朗兰兹计划是数学的一个分支，包括数论(研究整数方程)、表示理论和分析。该程序的数论部分由大量相互关联的猜想组成，这些猜想将整数方程的解与其他明显无关的数学对象联系在一起。近年来，很明显，应该有另一种通用框架来适应该程序的数论部分，即“p进和mod p朗兰兹程序”。PI建议在mod p程序中形成一些精确的猜想，并使用Andrew Wiles在他的费马大定理工作中发展的一种技术来证明这些猜想的情况，以及数论中其他更经典的猜想。