Arithmetic Algebraic Geometry

算术代数几何

• 批准号: RGPIN-2018-06094
• 负责人: Murty, Vijayakumar
• 金额: $ 2.91万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758690
• 关键词: Arithmetic; Algebraic; Geometry

We will study several problems related to algebraic cycles and Abelian varieties, L-functions and mathematics arising from Information Technology.1) In the arithmetic and geometry of Abelian varieties, conjectures about algebraic cycles are fundamental. The classical conjectures of Hodge and Tate give us methods of recognizing subvarieties (and their linear combinations) in algebraic varieties. In the case of Abelian varieties, the additional group structure and the simple structure of the cohomology makes people more explicit formulations. Our recent work about understanding the relationship between cohomological spaces of cycles on an Abelian variety and on its reduction modulo a prime is aimed at increasing our understanding of algebraic cycles, though this goal may be far off. However, there are many problems which seem quite accessible (such as proving that a cohomology class can be declared to be a Tate class by checking the action of a finite number of Frobenius elements) and this project will focus on a few of those, based on recent work by the author.2) Another fundamental tool in number theory is that of L-function, an analytic object built from local data, which mysteriously learns about global data. In our project, we will actually look at spaces of L-functions that have additional algebraic structure (they form a ring). The purpose of studying collections of L-functions is to then ascertain what properties can actually be 'deformed' along a collection. Again, this kind of question is still too vague and too distant, and to get to the point where we might start thinking about such questions, we need to analyze the analysis and hopefully the geometry and topology of spaces of L-functions. Some first steps have been taken with the definitions of the Selberg and Lindelof classes. In this project, we will continue to work to clarify the algebraic structure of the Lindelof class. 3) There are many mathematical questions that are inspired by problems in information technology. One is to make explicit the arithmetic on Abelian varieties over finite fields. These are higher dimensional versions of elliptic curves and one might hope to form a public-key encryption scheme using such objects. There are of course some impediments to this such as new attacks on higher dimensional varieties and the possible construction of a quantum computer. Nevertheless, understanding explicitly, and hence computationally, the arithmetic on these higher dimensional varieties may for the foundation of a new kind of encryption scheme, not exactly modeled on the discrete logarithm, but perhaps on the more quantum-resistant model using isogenies. In addition to this kind of fundamental mathematical work, there are also problems involving algorithms to ensure security and privacy in transactions such as e-payments and e-health.

我们将研究与代数周期和阿伯利亚品种有关的几个问题，信息技术引起的l功能和数学。1）在亚伯利亚品种的算术和几何形状中，关于代数周期的猜想是基本的。 Hodge和Tate的经典猜想为我们提供了代数品种中识别次视角（及其线性组合）的方法。就阿贝里安品种而言，同时的额外组结构和简单的结构使人们更加明确。我们最近关于了解循环在亚洲品种中的共同点之间的关系以及在其还原模型上的关系旨在提高我们对代数周期的理解，尽管这一目标可能遥远。但是，似乎有很多问题似乎很容易获得（例如，可以通过检查有限数量的Frobenius元素的作用来宣布共同体课程为泰特阶级），并且该项目将重点放在作者的最新工作上。根据作者的最新工作。在我们的项目中，我们实际上将研究具有其他代数结构（它们形成环）的L功能的空间。研究L功能的集合的目的是确定在集合中实际上可以“变形”的属性。再说一次，这种问题仍然太模糊且太遥远了，要达到我们可能开始思考此类问题的地步，我们需要分析分析，并希望L-功能空间的几何形状和拓扑。 Selberg和Lindelof类的定义已经采取了一些第一步。在这个项目中，我们将继续努力阐明Lindelof类的代数结构。 3）有许多数学问题受到信息技术问题的启发。一种是在有限领域的阿贝尔品种上明确说明算术。这些是椭圆曲线的较高维度版本，人们可能希望使用此类对象形成一个公钥加密方案。当然，这有一些障碍，例如对更高维品种的新攻击以及量子计算机的可能构造。然而，在计算上明确理解这些较高维度的品种的算术可能是为一种新型的加密方案的基础而建立的，而不是基于离散对数，而是基于使用ISOGEN的更量子的模型。除了这种基本的数学工作外，还存在涉及算法的问题，以确保电子支付和电子卫生等交易中的安全性和隐私性。