Arithmetic Algebraic Geometry

我们将研究与代数循环和阿贝尔变体、l函数以及信息技术产生的数学相关的几个问题。1)在阿贝尔变体的算术和几何中，关于代数循环的猜想是基本的。Hodge和Tate的经典猜想为我们提供了识别代数变量中的子变量（及其线性组合）的方法。在阿贝尔变体的情况下，附加的群结构和上同调的简单结构使人们的表述更加明确。我们最近的工作是理解阿贝尔变项上环的上同空间与其约模a素数之间的关系，旨在增加我们对代数环的理解，尽管这个目标可能还很遥远。然而，有许多问题似乎很容易理解（例如，证明上同类可以通过检查有限数量的Frobenius元素的作用来声明为Tate类），本项目将重点关注其中的一些问题，基于作者最近的工作。2)数论中的另一个基本工具是l -函数，它是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，我们将实际观察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沿着集合可以“变形”的性质。同样，这类问题仍然太模糊太遥远，为了达到我们开始思考这类问题的目的，我们需要分析l函数空间的分析，希望是几何和拓扑。对于Selberg和Lindelof类的定义已经迈出了一些第一步。在这个项目中，我们将继续努力澄清Lindelof类的代数结构。有许多数学问题受到信息技术问题的启发。一是明确有限域上阿贝尔变分的算法。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，这也存在一些障碍，比如对高维变种的新攻击，以及量子计算机的可能构建。然而，明确地理解，并因此计算，在这些高维变量上的算法可能是一种新的加密方案的基础，不是精确地以离散对数为模型，而是可能在使用等基因的更抗量子的模型上。除了这类基本的数学工作之外，还有一些涉及算法的问题，以确保电子支付和电子健康等交易中的安全性和隐私性。

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.