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性质，从整数到多项式方程的解的数字。在提出的主要问题中，有一些是关于通过某种特定程序产生的数字集合中的模式的问题。例如，素数，其中一个主要问题是它们之间的距离有多大。或者，多项式方程a+bx+cx^2+…的解+Fx^n=0，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“阴影”或“表示”。过去60年的一大成就是数论和几何的综合。几何学现在可以用来研究有关数字模式的某些基本问题。对这种合成的描述需要大量的背景知识，但它的要点是几何告诉数论，反之亦然。因此，我们可以使用几何演绎来了解数字中的模式。这一领域被称为算术几何，而一类特殊的几何空间--下村簇在其中起着举足轻重的作用。他们是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·瑟尔、下村五郎、皮埃尔·德莱恩、亚历山大·格罗森迪克等不计其数的梦想家。下村变种理论是数论、代数几何、调和分析和复刚体分析的综合。因此，它的研究需要使用多种技术，但反过来也提供了丰富的应用和与其他领域的联系。除了Shimura簇的背景之外，我们的建议将对丢番图几何、伽罗瓦表示和类场理论的研究产生影响。我们考虑了几个研究方向：(I)关于Shimura变种的交集理论：在Shimura变种上，存在一类特殊的亚种，这些亚种是可访问的，同时也是最有应用价值的。它们要么来自其他下村品种，要么来自那些品种上的媒介束。每对这种互补维度的特殊变体的交集是一个整数，以这种方式形成的整数的集合可以被组织成非常特定的模式。据推测，相同的模式产生于某些模形式--由格中向量的长度统计产生的函数。我们将证明这些猜想的特殊情况；此外，除了证明两个起源非常不同的图案(几何和晶格)是相同的之外，我们还将使用形变理论和复数乘法的技巧来研究图案本身。(2)有一类特殊的下村品种--它们与自旋基团有关。对于那些我们计划研究上述问题的人来说。同时，在前人工作的基础上，利用交换簇上Hodge循环的形变理论以及某些层结的局域描述来研究这些簇的无穷小结构是可取的。(Iii)此外，我们将有兴趣研究作用于Shimura簇的某些算子的p-进动力学。从群论、测度论、遍历等不同角度对复数上的动力学进行了一定程度的探索。相比之下，当所用的度量是p-进度量时，人们对这种动力学知之甚少。沿着这些思路的结果将对一系列有趣的问题有用，从算术(类域理论、p-进模形式和典范子群)到图论和密码学(例如，通过Ramanujan图和同源火山)。