Langlands Functoriality in Nonsolvable and Relative Settings

在小学，人们学习如何使用二次方程解二次方程；它涉及到取一定的平方根。三次多项式和四次多项式也可以通过求根来求解，但根据Abel-Ruffini定理，一般没有关于根的五次或五次以上多项式的根的公式。这个定理在1800年的S首次得到证明，它表明了所谓的可解多项式(可用根解的多项式)和不可解多项式(不能用根解的多项式)之间的深刻区别。后者则更难理解。在所谓的朗兰兹计划中，现有的技术通常局限于可解的环境。这项提议的一部分是致力于在朗兰兹计划中提供在不可解环境下工作的新技术。尽管朗兰兹函数猜想只在特殊(尽管重要)情况下得到证明，但朗兰兹函数猜想已经成为现代数学的基石。这些猜想起源于自同构型领域，它位于李群上的数论、表示论和调和分析的交叉点上。为了理解和证明猜想的情况而发展起来的理论已经成为理解这些主题的关键，并且已经超越了这些主题，导致了在其他领域的重要应用，如拓扑学、代数几何和数学物理。这项提案旨在开发工具，以解决猜想及其在其他情况下的类似情况的重要案例。拟议的研究包括两个部分。首先，它将在旨在沿着不可解的伽罗瓦扩张建立自同构表示的下降和基变化的设置中，研究朗兰兹的所谓“超越内窥镜”的思想。其次，它将发展扭曲内窥镜理论的相关类似物。这将有助于更好地理解下村品种的特殊周期，以期建立泰特和贝林森-布洛赫猜想的案例。

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.