Shimura Varieties with Parahoric and Deeper Level Structure

具有旁隐和更深层次结构的志村品种

• 批准号: 2200873
• 负责人: Thomas Haines
• 金额: $ 21万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200873&HistoricalAwards=false
• 关键词: Shimura; Varieties; Parahoric; Deeper; Level

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.

朗兰兹纲领是一个意义深远的知识网络，它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，朗兰兹互易定理试图揭示算术/数论性质的函数与解析/表示论性质的函数之间的神秘恒等式。这种恒等式揭示了产生函数的对象中隐藏的对称性，而这种恒等式的每一个实例通常都有深刻的后果。例如，怀尔斯和他的学派证明了朗兰兹互易定理的一个特例，最终证明了费马大定理。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簇，其中，志村簇在朗兰兹观点的建立中起了核心作用。他们的研究汇集了来自不同领域的技术，并从许多不同的方向挑战数学家。本研究以志村品种为中心，作为朗兰兹互惠理论的试验场。沿着这条路，PI将让他的博士生和博士后参与进来，并将继续他的工作，传播基础科学，建立不同数学部门之间的互动，以及不同专业领域的数学家之间的互动。更具体地说，PI将从几何和表示论的角度研究志村簇，利用各个领域的最新突破，为了用自守L-函数表示非常一般的Shimura簇的Hasse-Weil zeta函数。重点将是志村品种的阿贝尔型，和PI将解决各种困难所产生的奇点存在这些品种。这些坏的减少问题是最容易研究的情况下，parahoric水平结构，这将首先进行调查，建立在以前的工作的PI沿着与最近的方法Kisin-Shin-Zhu在良好的减少（无奇点）的情况下。PI还将解决更深层次的结构情况，引入新的p-adic几何技术，由于Scholze和Festeres-Scholze。特别重要的将是本地朗兰兹对应的Festhes-Scholze几何化，以及这与PI关于实施更深层次的志村品种的朗兰兹-Kottwitz方法的想法相互作用的方式。这项研究的不良减少现象将提供步骤实现之间的联系伽罗瓦表示和自守表示的上同调志村品种的最终目标。这一奖项反映了NSF的法定使命，并已被认为是值得支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。