Number theory in function fields

函数域中的数论

• 批准号: 261908-2011
• 负责人: Liu, YuRu
• 金额: $ 1.89万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571169
• 关键词: Number; theory; function; fields

Classically, number theory is the study of arithmetic properties of integers. My research focuses on the interplay between this theory, and the theory of a related object, the set of polynomials over a finite field. The set of integers, and the aforementioned set of polynomials have ostensibly different structures. For example, given a non-zero integer, no multiple of it is zero. However, a fixed finite field always admits a prime number p for which any p-multiple of an element is zero, hence this property is carried over to polynomials on this field. Despite these apparent differences, it is a striking theme in arithmetic that there are remarkable similarities between these two sets. These similarities have many analogues that, in general, relate number fields to function fields. Since there are more structures in polynomials than in integers, the study of function fields has served as a stimulus to research in other branches of number theory. On the other hand, since finite fields admit multiples which are zero, many methods used to solve integer number theoretic problems fail to provide viable answers to their polynomial analogues. As new approaches are often required, number theory in function fields is of interest for its own sake. In addition, the theory of function fields is but another guise for the theory of algebraic curves. The latter plays an important role in the study of modern number theory, including the proof of Fermat's Last Theorem. The study of function fields and algebraic curves has many applications, a prominent one being the use of elliptic curves in cryptography, which facilitates secure transmissions of data.

经典上，数论是研究整数的算术性质。 我的研究重点是这个理论之间的相互作用，以及相关对象的理论，有限域上的多项式集。 整数的集合和上述多项式的集合具有表面上不同的结构。 例如，给定一个非零整数，它的倍数都不为零。 然而，一个固定的有限域总是允许一个素数p，对于这个素数p，元素的任何p-倍数都是零，因此这个性质被转移到这个域上的多项式。尽管有这些明显的差异，这是一个引人注目的主题，在算术之间有显着的相似之处，这两套。 这些相似性有许多类似物，一般来说，将数域与函数域联系起来。由于多项式中的结构比整数中的结构多，函数域的研究对数论的其他分支的研究起到了促进作用。另一方面，由于有限域允许倍数为零，许多用于解决整数数论问题的方法无法提供可行的答案，他们的多项式类似物。由于经常需要新的方法，函数域中的数论本身就很有趣。此外，函数域理论不过是代数曲线理论的另一个幌子。后者在现代数论的研究中起着重要的作用，包括费马大定理的证明。函数域和代数曲线的研究有许多应用，其中一个突出的应用是在密码学中使用椭圆曲线，这有助于数据的安全传输。