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Mod p and p-adic Geometry of Shimura Varieties, Canonical Subgroups of Abelian Varieties, and Applications to Automorphic Forms.

Shimura 簇的 Mod p 和 p-adic 几何、阿贝尔簇的规范子群以及自守形式的应用。

基本信息

批准号：
EP/H019537/1
负责人：
Payman Kassaei
金额：
$ 12.06万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2010
资助国家：
英国
起止时间：
2010 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FH019537%2F1
关键词：
Mod adic Geometry Shimura Varieties

项目摘要

Automorphic forms, typically defined analytically, are objects obtained from certain representations of algebraic groups. Langlands program is a web of conjectures predicting precise relationships between automorphic forms/representations and representations of Galois groups, thereby providing a profound link between analysis and algebra. It is the motivating force behind much of the research in Number theory today.Classical modular forms are examples of automorphic forms defined for the group GL(2,Q). In the 70's J.-P. Serre defined p-adic modular forms: objects that are obtained from p-adic analytic variation of modular forms. Ever since their introduction by Serre, p-adic methods have been pivotal in progress in the theory of automorphic forms.Ground-breaking results of Hida (80's) and Coleman (90's) for p-adic modular forms prompted a surge in research applying p-adic methods to the study of automorphic forms for groups other than GL(2,Q). Along with the recent progress in the p-adic representation theory of Galois groups, this research has led to the emergence of the beginnings of a p-adic Langlands philosophy. This is mostly a mystery at the moment but it informs a lot of research in the area. A link between the classical theory and the p-adic theory is given by criteria of classicality : they tell us which p-adic modular forms are classical modular forms and are in general hard to prove.A powerful approach to the study of p-adic automorphic forms is via geometry: more precisely, the study of the p-adic geometry of Shimura varieties. It was Katz who highlighted the power of a geometric approach: he recast Serre's theory of p-adic modular forms using the geometry of spaces called modular curves and provided many applications. Many of the recent approaches to the construction and study of spaces of p-adic automorphic forms are of a geometric nature. At the core of this proposal lies a plan to study aspects of the (p-adic and mod p) geometry of certain Shimura varieties (and of maps between them) that have emerged as essential in the study of p-adic automorphic forms, and especially in proving classicality criteria for them. In addition, we plan to use such results to construct and study canonical subgroups of abelian varieties. These are objects at the heart of our method for proving classicality. We intend to end the proposal by demonstrating applications to classicality of p-adic automorphic forms.

自守形式（英语：Automorphic forms）是从代数群的某些表示中得到的对象。朗兰兹程序是一个预测自守形式/表示和伽罗瓦群表示之间精确关系的网络，从而提供了分析和代数之间的深刻联系。它是当今数论研究背后的推动力。经典模形式是为群GL（2，Q）定义的自守形式的例子。在70年代，J P. Serre定义了p-adic模形式：从模形式的p-adic解析变分中获得的对象。自从Serre引入p-adic方法以来，p-adic方法在自守形式理论中一直是关键的进展。希达（80年代）和科尔曼（90年代）关于p-adic模形式的开创性结果促使了将p-adic方法应用于研究GL（2，Q）以外的群的自守形式的研究热潮。沿着伽罗瓦群的p-adic表示理论的最新进展，这一研究导致了p-adic朗兰兹哲学的开端。目前这主要是一个谜，但它为该领域的许多研究提供了信息。经典理论和p-adic理论之间的联系是由经典性准则给出的：它们告诉我们哪些p-adic模形式是经典模形式，并且通常很难证明。研究p-adic自守形式的一个强有力的方法是通过几何：更准确地说，是研究Shimura簇的p-adic几何。正是卡茨强调了几何方法的力量：他使用称为模曲线的空间几何来重铸塞尔的p-adic模形式理论，并提供了许多应用。许多最近的方法来建设和研究空间的p-adic自守形式是一个几何性质。在这个建议的核心在于一个计划，研究方面的（p-adic和mod p）几何的某些志村品种（和地图之间）已成为必不可少的研究p-adic自守形式，特别是在证明经典标准。此外，我们计划利用这些结果来构造和研究交换簇的典型子群。这些是我们证明古典性方法的核心。我们打算通过展示p-adic自守形式的经典性的应用来结束这个提议。