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Grant教授在数字理论中的工作，特别是高斯总和理论。 高斯总和在数字理论上无处不在，在阶级字段理论中发挥了作用。 经过四年的搜索，高斯本人在1805年发现了二次高斯总和的论点。 当Cassels and Mathews将论点与椭圆函数相关联时，Coxic Gauss总和已有170年的参数没有取得进展。 格兰特教授建议使用第2属曲线的算术来研究五重奏高斯总和的论点。 这是数字理论领域的研究。 数字理论始于整个数字和问题，例如一个整数的划分。 它是数学最古老的领域之一，最初是出于纯粹的审美原因而追求的。 但是，在过去的半个世纪中，它成为开发用于计算机科学的新算法和纠正电子产品的新算法的重要工具。