权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Langlands Functoriality in Nonsolvable and Relative Settings

不可解和相对设置中的朗兰兹函数性

基本信息

批准号：
1405708
负责人：
Jayce Getz
金额：
$ 15.3万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2018-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405708&HistoricalAwards=false
关键词：
Langlands Functoriality Nonsolvable Relative Settings

项目摘要

In elementary school one learns how to solve quadratic equations using the quadratic formula; it involves taking a certain square root. One can also find solutions for degree three polynomials and degree four polynomials by taking radicals, but in general by the Abel-Ruffini theorem there is no formula for the roots of polynomial of degree five or higher in terms of radicals. This theorem, first proven in the 1800's, is an indication of a profound difference between so-called solvable polynomials (those that can be solved by radicals) and nonsolvable polynomials (those that cannot be solved by radicals). The latter are much harder to understand. Existing techniques in the so-called Langlands program are often limited to the solvable setting. Part of this proposal is dedicated to providing new techniques in the Langlands program that will work in nonsolvable settings.Though only proven in special (albeit important) cases the Langlands functoriality conjectures have become a cornerstone of modern mathematics. The conjectures had their genesis in the area of automorphic forms, which lies at the intersection of number theory, representation theory, and harmonic analysis on Lie groups. The theory developed to understand and prove cases of the conjectures has become crucial to understanding these subjects, and has reached beyond them, leading to important applications in other areas such as topology, algebraic geometry and mathematical physics. This proposal aims to develop tools to resolve important cases of the conjectures and their analogues in other contexts. The proposed research has two parts. First, it will investigate Langlands' so-called ``Beyond Endoscopy'' idea in settings designed to establish descent and base change of automorphic representations along nonsolvable Galois extensions. Second, it will develop relative analogues of the theory of twisted endoscopy. This will lead to a better understanding of special cycles on Shimura varieties with a view towards establishing cases of the Tate and Beilinson-Bloch conjectures.

在小学里，人们学习如何用二次公式解二次方程;它涉及到取一定的平方根。人们也可以通过取根式找到三次多项式和四次多项式的解，但一般来说，根据阿贝尔-鲁菲尼定理，五次或更高次多项式的根在根式方面没有公式。该定理于1800年代首次得到证明，它表明了所谓的可解多项式（可以通过根式求解的多项式）和不可解多项式（无法通过根式求解的多项式）之间的深刻差异。后者更难理解。在所谓的朗兰兹程序中的现有技术通常限于可解的设置。该计划的一部分致力于为朗兰兹纲领提供新的技术，这些技术将在不可解的情况下工作。尽管只有在特殊（尽管重要）的情况下才能证明，但朗兰兹函子性证明已经成为现代数学的基石。这些理论起源于自守形式领域，它位于数论、表示论和李群调和分析的交叉点。为理解和证明几何的情况而发展的理论对理解这些学科至关重要，并且已经超越了它们，导致了其他领域的重要应用，如拓扑学，代数几何和数学物理。该提案旨在开发工具，以解决在其他情况下出现类似情况的重要案例。拟议的研究有两个部分。首先，它将研究朗兰兹所谓的“超越内窥镜”的想法，旨在建立下降和自同构表示沿沿着不可解伽罗瓦扩展的基础变化的设置。其次，它将发展扭曲内窥镜理论的相对类似物。这将导致更好地了解志村品种的特殊周期，以期建立泰特和贝林森-布洛赫的案例。