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

这项研究计划的主要目的是利用p-进分析和p-进几何的方法在p-进兰兰兹计划中取得重要进展。我们的大部分工作都将致力于在p-进和经典朗兰兹程序之间架起桥梁，使我们能够使用p-进分析方法在经典朗兰兹程序中取得重要进展。方法论主要是通过解析和代数几何，一种主要的基本技术将是自同构形式的p进位解析延拓。自同构形的p向解析延拓技术最早出现在Buzzard-Taylor的开创性工作中。在Buzzard-Taylor的工作之后，我们发明了一种用p-进解析延拓来证明经典判据的方法，从而使解析延拓成为p-进自同构形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，其中一些已经从文献中缺失了太长时间，尽管该理论的其他方面可用。我们建议的主要长期目标之一是在许多情况下证明经典标准，尽管最近取得了一系列进展，但这些标准仍然开放。事实上，在我们的建议中，有一项研究致力于各种解析延拓技术和直接应用的研究。在这个保护伞下，另一个长期目标是研究p-进自同构形的自动解析连续的域，这一概念在我们最近在某些全实域上的Artin猜想的证明中发挥了关键作用。在这一研究思路下，一个主要的目标是研究Breuil所引入的方向经典和可积性，它们对于正在进行的超越GL_2(Q_P)情形的p-进局部Langland对应的努力具有应用。我们提案中的另一项研究灵感来自我们对Artin猜想的证明。1999年，Buzzard和Taylor利用p-进模形式证明了权为1的Galois表示的模性。他们的工作导致了经典Artin猜想的许多情况的证明。经过15年多的抗拒，这项工作最近被我们推广到完全实场的情况。我们的工作以及随后由Pilloni、Sasaki、Stroh和Tian推广的结果几乎解决了Artin猜想在全实域上的问题。我们在这里的长期目标是利用我们关于p-进自同构形式及其解析延拓的知识来证明伽罗瓦表示在低权下的自同构。这包含了将我们关于Artin猜想的工作推广到完全真实的领域的研究的巨大可能性。我们打算把注意力集中在部分权一的Hilbert模形式的情形。我们还计划研究这一现象的mod-p版本，它将提供全实域上Serre精化猜想的许多剩余情况。我们工作的一个基本特征将是将p-进自同构形看作是Shimura簇上的向量丛的截面：这将使我们在代数和解析几何中获得强大的技巧。例如，Buzzard-Taylor的方法和我们对它的推广只有通过这种几何解释才是可行的。因此，在我们的建议中，一个主要的研究方向是致力于研究Shimura簇的mod-p和p-adi几何。我们计划在我们对p-进自同构形的p-进解析延拓的研究的指导下，从一些角度来研究mod-p下村系的分层。这些结果将被用来研究Shimura簇的p-进几何以及它们上的U_p算子的p-进动力学。

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.