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).

数字理论被占据数字的属性，从整数的数字到多项式方程的解决方案的数字。在询问的主要问题中，有关于通过某些定义程序出现的数字集合中的模式的问题。例如，主要问题之一是它们有多少相距。或者，多项式方程式A + Bx + Cx^2 +… + fx^n = 0，其中问题是关于解决方案的对称性，等式的所谓Galois组及其各种“阴影”或“表示”。过去60年来的一个重大成就是数字理论与几何学之间的综合。现在，几何形状可以用于研究有关数字模式的某些基本问题。该综合的描述需要很多背景，但要点是几何信息编号理论，反之亦然。这是我们可以使用几何扣除来告知自己数字模式的信息。该区域称为算术几何形状和一类特殊的几何空间（Shimura品种）在其中起着关键作用。他们是我们的罗塞塔石头（Rosetta Stone）破译这些联系，感谢Robert Langlands，Jean-Pierre Serre，Goro Shimura，Pierre Deligne，Alexander Grothendieck等有远见的人，其他人太多了，无法透露姓名。 Shimura品种的理论是数字理论，代数几何，谐波分析以及复杂且刚性分析的综合。因此，它的研究需要使用多种技术，但作为回报，它提供了一系列与其他领域的应用程序和连接。除了Shimura品种的背景之外，我们的建议还将对研究养分几何，GALOIS表示和阶级田地理论有影响。我们考虑了几个研究方向：（i）关于Shimura品种的相交理论：在Shimura品种上，有一类杰出的亚变化类别可访问，同时对应用程序最有用。这些要么来自其他Shimura品种，要么来自这些品种的矢量束。每对此类特殊的互补维度的交点是整数，并且以这种方式形成的整数的集合可以组织成非常特殊的模式。据推测，相同的模式是由某些模块化形式产生的 - 函数是由晶格中向量长度的统计数据产生的。我们将证明这些猜想的特殊情况；此外，除了表明两种模式（几何和晶格）相同的两个模式外，我们还将使用变形理论和复杂乘法中的技术来研究模式本身。 （ii）有一类特殊的shimura品种 - 它们与自旋组相关。对于那些我们计划研究上述问题的人。同时，根据先前的工作，希望通过霍奇循环的变形理论在阿贝尔品种上研究这些品种的无限结构，以及如何在局部描述某些分层。 （iii）此外，我们将有兴趣研究作用于Shimura品种的某些操作员的P-ADIC动力学。从各个方向探索了复数上的动力学 - 群体理论，测量理论，ergodic。相比之下，当使用的度量是P-ADIC度量时，对这种动态知之甚少。沿这些线路的结果对于从算术（类场理论，P-Adic模块化形式和规范亚组）到图理论和密码学（例如，通过Ramanujan图和同等火山）等一系列有趣的问题很有用。