Comparison isomorphisms and arithmetic applications

比较同构和算术应用

• 批准号: 261904-2011
• 负责人: Iovita, Adrian
• 金额: $ 1.31万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571168
• 关键词: Comparison; isomorphisms; arithmetic; applications

It is universally accepted that geometric objects are difficult to understand and classify. In order to describe them one usues linear-algebraic approximations of them, known as cohomology groups. They appear as families of groups (or vector spaces) with certain linear (or semi-linear) structure given by filtrations, gradings and operators. There are a number of cohomology theories each one attaching, by a specific and usually rather complicated recipe, to a geometric object a family of cohomology groups with their respective structures. A. Grothendieck predicted that there should be certain ''mysterious functors", or comparison isomorphisms relating these cohomology theories. The importance of comparing cohomology groups coming from different cohomology theories, beyond the fact that it is a natural question, is that certain questions about the geometric object are best formulated in terms of one cohomology theory and easier answered in terms of another. Our project is to use comparison isomorphisms in order to, on the one hand, construct and describe overconvergent modular forms and their Galois representations and on the other hand describe the geometry of curves over local fields.

人们普遍认为，几何对象是难以理解和分类。为了描述它们， 它们的线性代数近似，称为上同调群。它们表现为具有某些线性（或半线性）结构的群（或向量空间）族，这些结构由滤子、分次和算子给出。有许多上同调理论，每一个理论都通过一个特定的、通常相当复杂的配方，将一个几何对象与一个具有各自结构的上同调群族联系起来。A.格罗滕迪克预言应该存在某些“神秘的函子”， 或与这些上同调理论相关的比较同构。比较来自不同上同调理论的上同调群的重要性，除了它是一个自然问题的事实之外，还在于关于几何对象的某些问题最好用一种上同调理论来表述，而用另一种上同调理论来回答更容易。 我们的项目是使用比较同构，以便，一方面，构建和描述过收敛模形式及其伽罗瓦表示，另一方面描述局部域上的曲线几何。