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

这项研究计划的首要目标是使用p-adic分析和p-adic几何的方法在p-adic朗兰兹计划中取得重要进展。 我们的大部分工作将致力于使p-adic和经典朗兰兹程序之间的桥梁，使我们能够在经典朗兰兹程序使用p-adic分析方法取得重要进展。该方法主要是通过解析和代数几何和一个主要的基本技术将是P进解析继续自守形式。 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，我们发明了一种用p-adic解析延拓证明经典性准则的方法，从而确立了解析延拓作为p-adic自守形式理论中不可或缺的工具。我们的方法，虽然我们的工作和其他人的，产生了许多经典的标准，其中一些已经从文献中消失了太久，尽管理论的其他方面的可用性。我们提案的主要长期目标之一是在许多情况下证明经典性标准，尽管最近取得了一系列进展，但这些情况仍然开放。事实上，我们的建议中的一个研究链是致力于各种解析延拓技术和直接应用的研究。在这一保护伞下，另一个长期目标是研究自动解析连续域的p进自守形式，一个概念，发挥了至关重要的作用，在我们最近证明的阿丁猜想在某些完全真实的领域。在这一系列的研究中，一个主要的目标是研究Breuil引入的方向经典性和可积性，它们在GL_2（Q_p）之外的p-adic局部Langlands对应中有应用。 在我们的提案中，另一个研究方向受到了阿廷猜想证明的启发。1999年，Buzzard和Taylor使用p-adic模形式证明了权为1的伽罗瓦表示的模性。他们的工作导致了经典阿廷猜想的许多情况下的证明。 经过15年多的努力，我们最近把这项工作推广到了全真实的场的情形。我们的工作以及Pilloni、Sasaki、斯特罗和Tian等人的推广几乎解决了全真实的域上的Artin猜想。 我们在这里的长期目标是使用我们的知识的p-adic自守形式和他们的解析延续证明低权重的伽罗瓦表示的自同构。这包含了一个巨大的可能性，研究推广我们的工作对阿廷猜想完全真实的领域。我们打算把我们的注意力集中在部分重量的情况下希尔伯特模形式。我们还计划研究一个mod-p版本的这种现象，这将提供许多剩余的情况下，精制猜想塞尔在完全真实的领域。 我们的工作的一个重要特点将是查看p-adic自守形式的部分向量丛志村品种：这有利于我们强大的技术在代数和解析几何。例如，只有通过这种几何解释，Buzzard-Taylor的方法和我们对它的推广才是可行的。在我们的建议的一个主要的研究链是，因此，致力于研究的mod-p和p-adic几何志村品种。我们计划研究分层的mod-p志村品种的兴趣，从我们的研究的p-adic解析延续的p-adic自守形式的指导下的角度。这些结果将被用来研究Shimura簇的p-adic几何和它们上的U_p算子的p-adic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.