Analytic Continuation of p-adic automorphic forms and applications to the Langlands program

p 进自守形式的解析延拓及其在朗兰兹纲领中的应用

• 批准号: RGPIN-2014-06640
• 负责人: LKassaei, Payman
• 金额: $ 2.03万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=589135
• 关键词: Analytic; Continuation; adic; automorphic; forms

The overarching aim of this research proposal is to use methods of p-adic analysis and p-adic geometry to make important advances in the p-adic Langlands Program. Much of our work will be devoted to making bridges between p-adic and classical Langlands programs, enabling us to make important progress in the classical Langlands program using p-adic analytic methods. The methodology is largely through analytic and algebraic geometry and a main underlying technique will be that of p-adic analytic continuation of automorphic forms. The technique of p-adic analytic continuation of automorphic forms first emerged in the groundbreaking work of Buzzard-Taylor. After Buzzard-Taylor's work, we invented a method for proving classicality criteria using p-adic analytic continuation which established analytic continuation as an indispensable tool in the theory of p-adic automorphic forms. Our method, though our work and others', has produced many classicality criteria, some of which had been missing from the literature for too long, despite the availability of other aspects of the theory. One of the main long term goals of our proposal is to prove classicality criteria in the many cases that remain open despite recent flurry of progress. In fact, one strand of research in our proposal is devoted to the study of various analytic continuation techniques and direct applications. Under this umbrella, another long term objective is to study domains of automatic analytic continuation for p-adic automorphic forms, a notion that has played a crucial role in our recent proof of the Artin conjecture over certain totally real fields. Under this strand of research, a major objective is the study of directional classicality and integrability introduced by Breuil which have applications to the ongoing effort towards a p-adic local Langlands correspondence beyond the case of GL_2(Q_p). Another strand of research in our proposal is inspired by our proof of the Artin conjecture. In 1999, Buzzard and Taylor proved modularity of Galois representations in weight one using p-adic modular forms. Their work led to a proof of many cases of the classical Artin conjecture. After more than 15 years of resistance, this work has been recently generalized by us to the case of totally real fields. Our work and its subsequent generalizations by Pilloni, Sasaki, Stroh, and Tian have almost settled the Artin conjecture over totally real fields. Our long term objective here is to use our knowledge of p-adic automorphic forms and their analytic continuation to prove automorphy of Galois representation in low weights. This encompasses a vast possibility of research generalizing our work on the Artin conjecture over totally real fields. We intend to focus our attention to the case of partial weight one Hilbert modular forms. We also plan to study a mod-p version of this phenomenon which would provide many remaining cases of the refined conjecture of Serre over totally real fields. An essential feature of our work will be to view p-adic automorphic forms as sections of vector bundles over Shimura varieties: this avails us of powerful techniques in algebraic and analytic geometry. For example, Buzzard-Taylor's approach and our generalization of it are feasible only through this geometric interpretation. A major strand of research in our proposal is, hence, devoted to the study of mod-p and p-adic Geometry of Shimura varieties. We plan to study stratifications on the mod-p Shimura varieties of interest from angles that are guided by our study of p-adic analytic continuation of p-adic automorphic forms. These results will be then used to study the p-adic geometry of Shimura varieties and the p-adic dynamics of the U_p operators on them.

本研究计划的总体目标是利用 p 进分析和 p 进几何方法在 p 进朗兰兹纲领中取得重要进展。 我们的大部分工作将致力于在 p-adic 和经典 Langlands 方案之间建立桥梁，使我们能够使用 p-adic 分析方法在经典 Langlands 方案中取得重要进展。该方法主要是通过解析几何和代数几何，主要的基础技术是自守形式的 p-adic 解析延拓。 自守形式的 p-adic 解析延拓技术首先出现在 Buzzard-Taylor 的开创性工作中。在 Buzzard-Taylor 的工作之后，我们发明了一种使用 p-adic 解析延拓来证明经典性准则的方法，该方法使解析延拓成为 p-adic 自守形式理论中不可或缺的工具。我们的方法，尽管我们的工作和其他人的工作，已经产生了许多经典标准，其中一些标准已经在文献中缺失太久了，尽管理论的其他方面是可用的。我们提案的主要长期目标之一是在许多案例中证明经典性标准，尽管最近取得了一系列进展，但这些标准仍然悬而未决。事实上，我们提案中的一项研究致力于研究各种分析连续技术和直接应用。在此框架下，另一个长期目标是研究 p 进自同构形式的自动解析延拓域，这一概念在我们最近证明某些完全真实域上的 Artin 猜想中发挥了至关重要的作用。在这方面的研究中，一个主要目标是研究 Breuil 引入的方向经典性和可积性，这可应用于超越 GL_2(Q_p) 情况的 p-adic 局部朗兰兹对应的持续努力。 我们提案中的另一项研究受到我们对 Artin 猜想的证明的启发。 1999 年，Buzzard 和 Taylor 使用 p 进模形式证明了权重一的伽罗瓦表示的模块化。他们的工作证明了经典阿廷猜想的许多案例。 经过 15 年多的抵制，这项工作最近被我们推广到完全真实的领域的情况。我们的工作以及随后 Pilloni、Sasaki、Stroh 和 Tian 的推广几乎解决了完全真实域上的 Artin 猜想。 我们的长期目标是利用我们对 p-adic 自同构形式及其解析延拓的知识来证明低权重下伽罗瓦表示的自同构。这包含了在完全真实的领域中推广我们关于阿廷猜想的工作的巨大研究可能性。我们打算将注意力集中在部分权重一希尔伯特模形式的情况上。我们还计划研究这种现象的 mod-p 版本，这将提供 Serre 在完全真实的域上的精致猜想的许多剩余案例。 我们工作的一个基本特征是将 p-adic 自同构形式视为 Shimura 簇上向量丛的部分：这利用了代数和解析几何中强大的技术。例如，Buzzard-Taylor 方法和我们对它的推广只有通过这种几何解释才可行。因此，我们提案中的一个主要研究方向是致力于研究 Shimura 品种的 mod-p 和 p-adic 几何形状。我们计划从我们对 p-adic 自同构形式的 p-adic 分析延拓研究的指导角度来研究感兴趣的 mod-p Shimura 变体的分层。这些结果将用于研究 Shimura 簇的 p 进几何及其 U_p 算子的 p 进动力学。