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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611738
• 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”。