Arithmetic Algebraic Geometry

我们将研究与代数圈和阿贝尔簇、L函数和信息技术数学有关的几个问题。*1)在阿贝尔簇的算术和几何中，关于代数圈的猜想是基本的。Hodge和Tate的经典猜想给我们提供了识别代数簇中的亚簇(及其线性组合)的方法。在Abelian簇的情况下，上同调的附加群结构和简单结构使人们得到了更明确的表述。我们最近关于理解Abel簇上圈的上同调空间与模为素数的约化之间的关系的工作旨在增加我们对代数圈的理解，尽管这一目标可能还很遥远。然而，有许多问题似乎是很容易解决的(例如，通过检查有限个Frobenius元素的作用来证明一个上同调类可以被宣布为Tate类)，本项目将基于作者最近的工作集中于其中的几个。*2)数论中的另一个基本工具是L函数，这是一个从局部数据建立的分析对象，它神秘地学习全局数据。在我们的项目中，我们将实际查看L的空间--具有附加代数结构的函数(它们形成环)。研究L函数集合的目的是为了确定在集合上哪些性质实际上是可以“变形”的。同样，这类问题仍然太模糊和太遥远，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函数的空间的几何和拓扑。对于Selberg和Lindelof类的定义已经采取了一些初步步骤。在这个项目中，我们将继续致力于阐明Lindelof类的代数结构。有许多数学问题的灵感来自于信息技术中的问题。其一是给出有限域上阿贝尔簇的显式算法。这些是椭圆曲线的高维版本，人们可能希望使用这些对象来形成公钥加密方案。当然，这也有一些障碍，比如对更高维物种的新攻击，以及可能建造的量子计算机。然而，明确地理解，因此在计算上，这些高维变体上的算法可能会成为一种新的加密方案的基础，这种方案不是精确地建立在离散对数的基础上，而是可能建立在使用同源的更抗量子的模型上。除了这类基本的数学工作外，还存在涉及算法的问题，以确保电子支付和电子健康等交易的安全和隐私。

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.