Iwasawa theory for p-adic representations

p-adic 表示的 Iwasawa 理论

• 批准号: RGPIN-2015-05710
• 负责人: Lei, Antonio
• 金额: $ 1.82万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=680834
• 关键词: Iwasawa; theory; adic; representations

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.**

椭圆曲线是可以使用立方方程定义的曲线。这些曲线的研究可以追溯到古希腊人。尽管具有简单的定义，但椭圆曲线的潜在非常丰富的算术结构使我们能够定义一个加密消息的密码系统。它们广泛用于在线沟通和财务交易中。因此，重要的是要对这些曲线的算术特性有很好的了解。在1960年代，桦木和Swinnerton-Dyer提出了一个猜想，该猜想描述了任何椭圆曲线上可以有多少点。这是数字理论中最重要的问题之一。在2000年，克莱数学研究所（Clay Mathematics Institute）将其选为七个千年奖问题之一，他将授予一百万美元以解决该问题的正确解决方案。如今，这仍然是一个空旷的问题，只有一些特殊情况得到解决。已经开发了许多工具来解决该合同。伊瓦沙（Iwasawa）理论是最富有成果的方法之一，该理论一次研究椭圆曲线的行为。更具体地说，让E为椭圆曲线，P固定质数。当我们允许这些点的坐标具有使用p定义的不同代数结构时，我们研究E上的点的数量如何变化。例如，令Q为一组有理数。 E要研究的自然点是Q中具有坐标的自然点。但是，如果我们允许坐标为Q和平方根中的数字表达式，我们还可以问有多少点。如果我们进一步放松这种情况并允许第四根根源怎么办？八根根？如果我们继续这样做，什么不对称的行为是什么？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，感谢伊瓦沙（Iwasawa）理论中的代数工具多年来发展的代数工具。在这个项目中，我们将研究其中一些工具，并将其应用于不同的数学对象。例如，我们将不仅研究椭圆形曲线，还将研究Abelian品种，这些品种是椭圆曲线的较高尺寸的化身。这些抽象的几何对象具有与椭圆曲线相似的算术结构。但是它们更加复杂，更难理解，因为它的尺寸可能是任意的。结果，这些对象将来可能会在密码学中具有重要的应用。**