Shimura varieties - intersection theory, rigid geometry, stratifications and p-adic modular forms

数论致力于发掘数字的性质，从整数到多项式方程的解。在提出的主要问题中，有一些是关于通过某种确定的程序产生的数字集合中的模式的问题。例如，质数，其中一个主要问题是它们之间的距离有多大。或者，一个多项式方程a + bx + cx^2 +...+ fx^n = 0的解，这里的问题是关于解的对称性，即方程的所谓伽罗瓦群，以及它的各种“影子”或“表示”。 过去60年的一个伟大成就是数论和几何之间的综合。几何学现在可以用来研究一些关于数字模式的基本问题。这种综合的描述需要大量的背景知识，但它的要点是，几何通知数论，反之亦然。因此，我们可以使用几何演绎来了解数字的模式。这一领域被称为算术几何，一类特殊的几何空间志村变种在其中扮演着关键的角色，它们就像我们的罗塞塔石碑一样，破译了这些联系，这要归功于罗伯特·朗兰兹、让-皮埃尔·塞尔、志村吾郎、皮埃尔·德利涅、亚历山大·格罗滕迪克等许多有远见的人。 志村簇理论是数论、代数几何、调和分析和复刚性分析的综合。因此，它的研究需要使用多种技术，但作为回报，它提供了丰富的应用和与其他领域的联系。在Shimura簇的背景之外，我们的建议将对丢番图几何，伽罗瓦表示和类域理论的研究产生影响。我们考虑几个研究方向： (i)志村品种的交叉理论：在志村品种上，有一个杰出的子类，这些子类是可访问的，同时对应用程序最有用。它们要么来自其他志村变种，要么来自这些变种上的向量丛。每对这种特殊的互补维数的交集是一个整数，这样形成的整数集合可以组织成非常特殊的模式。它被证明，相同的模式产生于某些模块的形式-功能所产生的统计长度的向量在格。我们将证明这些图案的特殊情况;此外，除了表明两种起源（几何和晶格）非常不同的图案是相同的之外，我们还将使用变形理论和复数乘法的技术来研究图案本身。 (ii)有一类特殊的志村变种-它们与自旋群有关。正是针对这些人，我们计划研究上述问题。同时，在前人工作的基础上，通过交换簇上的Hodge圈的变形理论研究这些簇的无穷小结构，以及如何局部描述某些分层，是可取的。 (iii)此外，我们将有兴趣研究的p-adic动力学的某些运营商的Shimura品种。复数的动力学在一定程度上从不同的方向进行了探索-群论，测度论，遍历。相比之下，当所使用的度量是p-adic度量时，对该动态知之甚少。沿着这些路线的结果沿着将是有用的分数有趣的问题，从算术（类域理论，p-adic模形式和规范子群），图论和密码学（通过拉马努金图和iskanuvolcanoes，例如）。

Number theory is occupied with unearthing the properties of numbers, from numbers that are integers to numbers that are solutions to polynomial equations. Among the main questions asked are questions about patterns in collections of numbers that arise via some definite procedure. For example, prime numbers, where one of the main questions is how much apart they are. Or, the solutions to a polynomial equation a + bx + cx^2 + …+ fx^n = 0, where the question is about the symmetries of the solutions, the so-called Galois group of the equation, and its various "shadows" or "representations". A great achievement of the last 60 years is the synthesis between number theory and geometry. Geometry can now be used to study certain basic questions about patterns in numbers. The description of this synthesis requires a lot of background, but the gist of it is that geometry informs number theory and vice-versa. Thus, we can use geometric deductions to inform ourselves about patterns in numbers. This area is called Arithmetic Geometry and a special class of geometric spaces, the Shimura varieties, plays a pivotal role in it. They serve as our Rosetta stone to deciphering these connections, thanks to visionaries such as Robert Langlands, Jean-Pierre Serre, Goro Shimura, Pierre Deligne, Alexander Grothendieck and others too numerous to name. The theory of Shimura varieties, is a synthesis of number theory, algebraic geometry, harmonic analysis and complex and rigid analysis. Its study thus necessitates the use of multiple techniques but in return offers a rich array of applications and connections to other fields. Beyond the context of Shimura varieties, our proposal will have implications to the study of Diophantine geometry, Galois representations and class field theory. We consider several research directions: (i) Intersection theory on Shimura varieties: On a Shimura variety there is a distinguished class of subvarieties that are accessible and at the same time are the most useful for applications. These arise either from other Shimura varieties, or from vector bundles on those. The intersection of each pair of such special varieties of complementary dimension is an integer and the collection of the integers formed this way can be organized into very particular patterns. It is conjectured that the same patterns arise from certain modular forms - functions arising from the statistics of lengths of vectors in lattices. We shall prove particular cases of these conjectures; furthermore, besides showing that two patterns, of very different origins (geometric and lattices) are the same, we shall use techniques from deformation theory and complex multiplication to study the patterns themselves. (ii) There is a special class of Shimura varieties - they are associated to Spin groups. It is for those that we plan to study the problems mentioned above. At the same time, based on previous work, it is desirable to study the infinitesimal structure of these varieties via deformation theory of Hodge cycles on abelian varieties and how certain stratifications are described locally. (iii) In addition, we will be interested in studying the p-adic dynamics of certain operators acting on Shimura varieties. The dynamics over the complex numbers was explored to an extent from various directions - group theoretic, measure theoretic, ergodic. In contrast, little is known about that dynamics when the metric used is the p-adic metric. Results along these lines would be useful to a score of interesting problems ranging from arithmetic (class field theory, p-adic modular forms and canonical subgroups), to graph theory and cryptography (through Ramanujan graphs and isogeny volcanoes, for instance).