Arithmetic Algebraic Geometry

我们将研究与代数圈和阿贝尔簇，L-函数和数学相关的几个问题，这些问题都是由信息技术引起的。1)在阿贝尔簇的算术和几何中，关于代数圈的图解是基本的。Hodge和Tate的经典代数给出了在代数簇中识别子簇（及其线性组合）的方法。在阿贝尔簇的情况下，额外的群结构和上同调的简单结构使人们有更明确的表述。我们最近的工作了解的关系上同调空间的周期的阿贝尔品种和其减少模一个总理的目的是增加我们的理解代数周期，虽然这个目标可能是遥远的。然而，有许多问题似乎很容易解决（例如通过检查有限数量的Frobenius元素的作用，证明上同调类可以被声明为Tate类），本项目将基于作者最近的工作重点关注其中的一些。2)数论中的另一个基本工具是L函数，一个从局部数据构建的分析对象，它神秘地学习全局数据。在我们的项目中，我们实际上将研究具有额外代数结构的L-函数空间（它们形成一个环）。研究L-函数集合的目的是确定什么性质可以沿着集合“变形”。同样，这类问题仍然太模糊和遥远，为了达到我们可能开始思考这些问题的地步，我们需要分析L-函数空间的分析，希望还有几何和拓扑。一些初步的步骤已经采取了Selberg和Lindelof类的定义。在这个项目中，我们将继续致力于阐明林德洛夫类的代数结构。**3）有许多数学问题是受信息技术问题的启发。一个是明确有限域上Abel簇的运算。这些是椭圆曲线的高维版本，人们可能希望使用这些对象形成公钥加密方案。当然，这也有一些障碍，比如对更高维度的新攻击以及量子计算机的可能构建。然而，明确地理解，因此计算，这些高维变量的算术可能是一种新的加密方案的基础，不完全是离散对数模型，但可能是更抗量子模型使用同源。除了这种基本的数学工作，还有一些问题涉及到确保电子支付和电子健康等交易的安全性和隐私性的算法。

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.