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，从整数到多项式方程的解。在被问到的主要问题中，有一些是关于通过某种特定程序产生的数字集合的模式的问题。例如，质数，其中一个主要问题是它们相差多少。或者，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，即方程的伽罗瓦群，以及它的各种“阴影”或“表示”。近60年来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，但它的要点是几何告诉数论，反之亦然。因此，我们可以使用几何推理来了解数字的模式。这个领域被称为算术几何，一类特殊的几何空间，志村变量，在其中起着关键作用。它们是我们破译这些联系的罗塞塔石碑，这要感谢像罗伯特·朗兰兹、让-皮埃尔·塞尔、志村吾郎、皮埃尔·德列涅、亚历山大·格罗滕狄克和其他数不胜数的梦想家。志村变分理论，是数论、代数几何、调和分析、复刚体分析的综合。因此，它的研究需要使用多种技术，但反过来又提供了丰富的应用程序和与其他领域的联系。除了志村变体的背景之外，我们的建议将对丢番图几何、伽罗瓦表示和阶级场理论的研究产生影响。我们考虑了以下几个研究方向：(1)志村变种的交集理论：在志村变种上有一类可接近的同时又最有应用价值的子变种。它们要么来自其他志村变种，要么来自那些变种上的向量束。这种特殊的互补维数的每一对的交点是一个整数，以这种方式形成的整数的集合可以被组织成非常特殊的图案。据推测，相同的模式产生于某些模形式——由格中向量长度统计产生的函数。我们将证明这些猜想的具体案例；此外，除了证明两种起源非常不同的图案（几何和晶格）是相同的，我们还将使用变形理论和复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Spin基团有关。正是为了这些人，我们计划研究上述问题。同时，在前人工作的基础上，利用阿贝尔变异上的Hodge旋回变形理论研究这些变异的无限小结构以及如何局部描述某些分层是很有必要的。（iii）此外，我们将有兴趣研究作用于Shimura变元的某些算子的p进动力学。从群论、测度论、遍历论等多个方向对复数上的动力学进行了一定程度的探讨。相比之下，当使用的度量是p进度量时，对这种动态知之甚少。沿着这些路线的结果将对许多有趣的问题有用，从算术（类场论、p进模形式和正则子群）到图论和密码学（例如，通过拉马努金图和等成因火山）。