Iwasawa theory for p-adic representations

椭圆曲线是可以使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人。尽管椭圆曲线的定义很简单，但它具有非常丰富的算术结构，这使得我们能够定义一种加密消息的密码系统。它们被广泛用于在线交流和金融交易。因此，对这些曲线的算术性质有一个很好的了解是很重要的。1960年，S、伯奇和斯温纳顿-戴尔提出了一个猜想，它描述了任何椭圆曲线上可能有多少个点。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七个千禧年奖问题之一，该研究所将奖励100万美元奖励正确解决该问题。今天，它仍然是一个开放的问题，只解决了一些特殊情况。已经开发了许多工具来解决这一猜想。比较有成效的方法之一是岩泽理论，该理论一次只研究一个质数处椭圆曲线的行为。更具体地说，设E为椭圆曲线，p为固定素数。我们研究当我们允许E上的点的坐标具有使用p定义的不同的代数结构时，这些点的数目如何变化。例如，设q是有理数的集合。E上要研究的自然点是那些坐标在Q中的点。但我们也可以问，如果我们允许坐标是Q中的数字和平方根的表达式，那么E上的自然点有多少个。如果我们进一步放宽这一条件，允许第四根呢？八根？如果我们永远这样做，渐近行为是什么？令人惊讶的是，我们能够用非常明确的公式来描述这种行为，这要归功于岩泽理论中的数学家们多年来发展起来的代数工具。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，我们将不再仅仅研究椭圆曲线，而是研究阿贝尔簇，它是椭圆曲线的高维化身。这些抽象几何对象具有类似于椭圆曲线的算术结构。但它们更复杂，也更难理解，因为它的维度可以任意大。因此，这些对象在未来可能会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.**