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解的数字，其中提出的主要问题是有关通过某些确定过程产生的数字集合中的模式的问题，例如素数。 ，其中主要问题之一是它们相距多少，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。所谓的伽罗瓦方程组，及其各种“影子”或“表示”，过去 60 年的一项伟大成就是数论和几何之间的综合，现在可以用来研究有关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，这个领域称为算术几何。和一个特殊类型的几何空间，志村品种，在其中发挥着关键作用，它们作为我们的罗塞塔石碑来破译这些联系，这要感谢罗伯特·朗兰兹、让-皮埃尔·塞尔、志村五郎、皮埃尔·德利涅、亚历山大·格洛腾迪克等有远见的人。志村簇的理论是数论、代数几何、调和分析以及复杂和刚性分析的综合，因此它的研究需要使用多种技术，但是。作为回报，除了志村簇的背景之外，我们的建议还将对丢番图几何、伽罗瓦表示和类域论的研究产生影响。我们考虑了几个研究方向：（i）交集。关于 Shimura 品种的理论：在 Shimura 品种中，存在一类可访问且同时对应用最有用的亚品种，这些亚品种要么来自其他 Shimura 品种，要么来自这些品种的向量束。一对这样的互补维度的特殊变体是一个整数，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特殊情况；此外，除了证明起源截然不同（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杂的进一步乘法技术来研究模式本身。 (ii)有一类特殊的志村品种——它们与旋转群相关，我们计划研究上述问题。同时，基于以前的工作，我们希望研究无穷小。 (iii) 此外，我们将有兴趣研究作用于 Shimura 复数的某些算子的 p 进动力学。在某种程度上，从不同的方向——群论、测度论、遍历学。相反，当使用的度量是 p 进度量时，人们对这种动态知之甚少。这些结果对于一系列有趣的问题很有用。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）。