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

Shimura 品种 - 相交理论、刚性几何、分层和 p-adic 模形式

• 批准号: RGPIN-2014-05614
• 负责人: Goren, Eyal
• 金额: $ 2.04万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567280
• 关键词: Shimura; varieties; intersection; theory; rigid

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).

数论致力于挖掘数字的属性，从整数的数字到多项式方程解的数字。提出的主要问题包括有关通过某种确定的程序产生的数字集合的模式的问题。例如，素数，主要问题之一是它们之间的距离有多少。或者，多项式方程 a + bx + cx^2 + …+ fx^n = 0 的解，其中问题是关于解的对称性、方程的所谓伽罗瓦群及其各种“影子”或“表示”。过去60年的一项伟大成就是数论与几何的综合。几何现在可以用来研究有关数字模式的某些基本问题。这种综合的描述需要大量的背景知识，但其要点是几何为数论提供信息，反之亦然。因此，我们可以使用几何演绎来了解数字的模式。这个领域被称为算术几何，一类特殊的几何空间，志村变体，在其中发挥着关键作用。它们成为我们破译这些联系的罗塞塔石碑，这要归功于罗伯特·朗兰兹、让-皮埃尔·塞尔、志村五郎、皮埃尔·德利涅、亚历山大·格洛腾迪克等远见卓识者，以及其他不计其数的人。志村簇的理论，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，但反过来又提供了丰富的应用程序以及与其他领域的联系。除了志村簇的背景之外，我们的建议还将对丢番图几何、伽罗瓦表示和类场论的研究产生影响。我们考虑几个研究方向：（i）关于志村品种的交叉理论：在志村品种上，有一个独特的亚品种类别，这些亚品种是可访问的，同时也是对应用最有用的。它们要么来自其他志村品种，要么来自这些品种的载体束。每对这种互补维数的特殊变体的交集是一个整数，并且以这种方式形成的整数的集合可以被组织成非常特殊的模式。据推测，相同的模式源自某些模形式——由格中向量长度的统计产生的函数。我们将证明这些猜想的具体情况；此外，除了表明起源截然不同（几何和晶格）的两种模式是相同的之外，我们还将使用变形理论和复数乘法的技术来研究模式本身。 (ii)有一类特殊的志村品种——它们与旋转组相关。我们打算研究上述问题就是为了那些人。同时，基于先前的工作，需要通过阿贝尔簇的霍奇循环变形理论来研究这些簇的无穷小结构以及如何局部描述某些分层。 (iii) 此外，我们将有兴趣研究某些算子作用于 Shimura 品种的 p-adic 动力学。从不同的方向——群论、测度论、遍历——对复数的动力学进行了一定程度的探索。相比之下，当使用的度量是 p 进度量时，人们对这种动态知之甚少。这些结果对于解决一系列有趣的问题很有用，从算术（类域论、p-进模形式和规范子群）到图论和密码学（例如，通过拉马努金图和同源火山）。