Shimura Varieties with Parahoric and Deeper Level Structure

朗兰兹计划是一个影响深远的猜想网络，将数学中看似无关的学科联系在一起：数论、表示论、分析和代数几何。更具体地说，朗兰兹互易猜想试图揭示算术/数论性质的函数与分析/表示理论性质的函数之间的神秘身份。这样的恒等式揭示了产生函数的对象中隐藏的对称性，这样的恒等式的每个实例通常都有深刻的后果。例如，Wiles和他的学派证明了朗兰兹互易猜想的一个特例，最终得到了费马大定理的证明。费马大定理涉及一个特定多项式方程的整数解。多项式的更一般的解集被称为代数变分，在这些变分中，志村变分在建立朗兰兹设想的案例中发挥了核心作用。他们的研究汇集了来自不同领域的技术，挑战了来自许多不同方向的数学家。本项目以志村品种研究为中心，作为朗兰兹互易猜想的试验场。在此过程中，他的博士生和博士后将参与PI，并将继续他在传播基础科学，以及在不同数学部门和不同专业领域的数学家之间建立互动方面的工作。更具体地说，PI将从几何和表示理论的角度研究Shimura变量，利用各个领域的最新突破，以便用自同构l函数表示非常一般的Shimura变量的Hasse-Weil zeta函数。重点将放在志村类型的阿贝尔类型上，PI将解决由于这些类型上存在奇点而引起的各种困难。这些坏约简问题在旁水平结构的情况下最容易研究，这将首先进行研究，建立在PI的先前工作以及Kisin-Shin-Zhu在良好约简（无奇点）情况下的最新方法的基础上。PI还将解决更深层次的结构情况，引入新的p-adic几何技术，这是由Scholze和Fargues-Scholze提出的。特别重要的将是Fargues-Scholze对局部朗兰兹对应的几何化，以及它与PI对更深层次志村变量实施朗兰兹- kottwitz方法的想法的相互作用。这一不良约简现象的研究将为实现志村变异上同调中的伽罗瓦表示与自同构表示之间的联系提供最终的步骤。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The Langlands program is a far-reaching web of conjectures linking seemingly unrelated subjects in mathematics: number theory, representation theory, analysis, and algebraic geometry. More specifically, the Langlands reciprocity conjectures seek to uncover mysterious identities relating functions of arithmetic/number-theoretic nature with functions of analytic/representation-theoretic nature. Such identities reveal hidden symmetries in the objects that give rise to the functions, and each instance of such identities usually has deep consequences. For example, a special case of Langlands' reciprocity conjectures was proved by Wiles and his school, finally resulting in a proof of Fermat's Last Theorem. Fermat's Last Theorem concerned integer solutions of a particular polynomial equation. More general solution sets of polynomials are called algebraic varieties, and among those, Shimura varieties have been central in establishing cases of Langlands' vision. Their study brings together techniques from diverse fields and challenges mathematicians from many different directions. This project centers on the study Shimura varieties as a testing ground of Langlands' reciprocity conjectures. Along the way, the PI will involve his PhD students and Postdocs, and will continue his work on disseminating fundamental science, and on building interactions between different mathematics departments and between mathematicians with diverse fields of expertise.More specifically, the PI will study Shimura varieties from a geometric and representation-theoretic viewpoint, using recent breakthroughs in various domains, in order to express Hasse-Weil zeta functions of very general Shimura varieties in terms of automorphic L-functions. The focus will be on Shimura varieties of abelian type, and the PI will solve various difficulties arising from the presence of singularities on these varieties. These bad reduction issues are easiest to study in the case of parahoric level structure, which will be investigated first, building on prior work of the PI along with recent methods of Kisin-Shin-Zhu in the good reduction (no singularities) cases. The PI will also address deeper level structure situations, bringing in new p-adic geometry techniques due to Scholze and Fargues-Scholze. Of particular importance will be the Fargues-Scholze geometrization of the local Langlands correspondence, and the way this interacts with the PI's ideas on implementing the Langlands-Kottwitz method for deeper level Shimura varieties. This study of bad reduction phenomena will provide steps toward the ultimate goal of realizing links between Galois representations and automorphic representations in the cohomology of Shimura varieties.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.