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

自同构形式，通常是解析定义的，是由代数群的某些表示得到的对象。朗兰兹程序是一个猜想网，预测自同构形式/表示与伽罗瓦群表示之间的精确关系，从而提供了分析与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）定义的自同构形式的例子。在70年代j.p。自定义的p进模形式：由模形式的p进解析变分得到的对象。自Serre引入p进方法以来，p进方法一直是自同构形式理论发展的关键。Hida（80年代）和Coleman（90年代）关于p进模形式的突破性成果促使了将p进方法应用于GL以外群的自同构形式研究的热潮（2，Q）。随着近年来伽罗瓦群的p进表象理论的进展，这一研究导致了p进朗兰兹哲学的出现。目前这基本上还是一个谜，但它为该领域的许多研究提供了信息。经典理论和p进理论之间的联系是由经典准则给出的：它们告诉我们哪些p进模形式是经典模形式，并且通常很难证明。研究p进自同构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变体的p进几何。正是卡茨强调了几何方法的力量：他用称为模曲线的空间几何重新定义了Serre的p进模形式理论，并提供了许多应用。最近关于p进自同构形式空间的构造和研究的许多方法都具有几何性质。这个建议的核心是研究某些志村变体（以及它们之间的映射）的（p进和模p）几何的各个方面，这些方面在p进自同构形式的研究中已经出现，特别是在证明它们的经典标准方面。此外，我们计划利用这些结果来构造和研究阿贝尔变体的正则子群。这些对象是我们证明古典性方法的核心。我们打算通过证明p进自同构形式的经典性的应用来结束这个提议。

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.