Iwasawa theory for p-adic representations

椭圆曲线是可以用三次方程定义的曲线。对这些曲线的研究可以追溯到古希腊。椭圆曲线虽然定义简单，但却具有非常丰富的算术结构，使我们能够定义一种加密信息的密码系统。它们广泛用于在线通信和金融交易。因此，很好地理解这些曲线的算术性质是很重要的。在20世纪60年代，Birch和Swinnerton-Dyer提出了一个猜想，描述了任何椭圆曲线上可以有多少个点。它是数论中最重要的问题之一。2000年，它被克莱数学研究所选为7个千禧年奖问题之一，正确解决该问题的人将获得100万美元的奖金。今天，这仍然是一个悬而未决的问题，只有一些特殊的情况得到了解决。已经开发了许多工具来解决这个猜想。最富有成果的方法之一是Iwasawa理论，它研究椭圆曲线一次在一个素数处的行为。更具体地说，设E是一条椭圆曲线，p是一个固定的素数。我们研究当我们允许E上的点的坐标有不同的用p定义的代数结构时，这些点的数目是如何变化的。例如，设Q是有理数的集合。要研究的E上的自然点是坐标为Q的点，但我们也可以问，如果我们允许坐标为Q和平方根中的数字的表达式，那么有多少个点。如果我们进一步放宽这个条件允许四次方根呢？八根?如果我们一直这样做，它的渐近行为是什么？令人惊讶的是，我们能够通过非常明确的公式来描述这种行为，这要归功于数学家多年来在岩泽理论中开发的代数工具。在这个项目中，我们将研究其中的一些工具，并将它们应用于不同的数学对象。例如，我们将不仅仅研究椭圆曲线，而是研究椭圆曲线的高维化身——阿贝尔变体。这些抽象的几何对象具有与椭圆曲线相似的算术结构。但它们更复杂，更难以理解，因为它的维度可以任意大。因此，这些对象将来可能在密码学中有重要的应用。

Elliptic curves are curves that can be defined using cubic equations. The study of these curves can be traced back to the ancient Greeks. Despite its simple definition, elliptic curves possess very rich arithmetic structure, which enable us to define a cryptosystem for encrypting messages. They are used extensively in online communication and financial transactions. It is therefore important to have a good understanding of the arithmetic properties of these curves. In 1960’s, Birch and Swinnerton-Dyer formulated a conjecture that describes how many points there can be on any elliptic curves. It is one of the most important problems in Number Theory. In 2000, it has been chosen as one of the seven Millennium Prize Problems by the Clay Mathematics Institute, who will award one million US dollars for a correct solution to the problem. Today, it is still an open problem and only some special cases have been solved. Many tools have been developed to tackle this conjecture. One of the more fruitful approaches is Iwasawa Theory, which studies the behaviour of an elliptic curve at one prime number at a time. More specifically, let E be an elliptic curve and p a fixed prime number. We study how the number of points on E can vary when we allow the coordinates of these points to have different algebraic structures defined using p. For example, let Q be the set of rational numbers. The natural points on E to study are the ones with coordinates in Q. But we could also ask how many points there are if we allow the coordinates to be expressions of numbers in Q and a square root. What if we relax this condition further and allow fourth roots? Eight roots? What is the asymptotic behaviour if we keep on doing this forever? Surprisingly, we are able to describe this behaviour by very explicit formulae, thanks to the algebraic tools mathematicians in Iwasawa Theory have developed over the years. In this project, we will study some of these tools and apply them to different mathematical objects. For example, instead of just studying elliptic curves, we will study abelian varieties, which are higher-dimensional avatars of elliptic curves. These abstract geometric objects have similar arithmetic structures as elliptic curves. But they are more complex and more difficult to understand because its dimension can be arbitrarily large. As a result, these objects could potentially have important applications in cryptography in the future.