Mathematical Sciences: Arguments of Gauss Sums

数学科学：高斯和的论证

• 批准号: 9303220
• 负责人: David Grant
• 金额: $ 4.92万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-06-01 至 1996-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303220&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Arguments; Gauss; Sums

This award supports the research of Professor Grant to work in number theory, specifically, the theory of Gauss sums. Gauss sums are ubiquitous in number theory, playing a role in class field theory. The argument of the quadratic Gauss sum was found by Gauss himself in 1805 after a four-year search. No progress was made on the arguments of cubic Gauss sums for 170 years, when Cassels and Mathews related the argument to elliptic functions. Professor Grant proposes to study the argument of quintic Gauss sums using the arithmetic of curves of genus 2. Specifically, he plans to apply his previous work on the arithmetic of curves of genus 2 and their Jacobians. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it hlas become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.

该奖项支持格兰特教授在数论方面的研究，特别是高斯和理论。高斯和在数论中无处不在，在类场理论中起着重要的作用。二次高斯和的论证是由高斯本人在1805年经过四年的探索后发现的。当Cassel和Mathews将三次高斯和的论证与椭圆函数联系起来时，170年来一直没有取得任何进展。格兰特教授建议使用亏格2的曲线的算术来研究五次高斯和的论点。具体地说，他计划将他之前的工作应用于亏格2的曲线及其雅可比的算术上。这是数论领域的研究。数论始于整数和诸如一个整数被另一个整数整除的问题。它是最古老的数学领域之一，最初是出于纯粹的美学原因。然而，在过去的半个世纪里，它已经成为开发计算机科学的新算法和电子产品的新纠错码的必要工具。