Arithmetical Algebraic Geometry

算术代数几何

• 批准号: 9622801
• 负责人: Kenneth Ribet
• 金额: $ 18.89万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622801&HistoricalAwards=false
• 关键词: Arithmetical; Algebraic; Geometry

This award provides funds for studies of modular forms, abelian varieties, and Galois representations. Gerhard Frey's construction, which associates triples of integers to elliptic curves over the rational numbers, has led to a direct link between the investigator's subject and simple problems involving whole numbers. Frey's construction has led to a proof of Fermat's Last Theorem and a resolution of several allied problems in mathematics. Many questions remain to be resolved - the ABC conjecture is perhaps the best known. One expects that arithmetic algebraic geometry will continue to flourish; the investigator is working on some of the problems that are now "on the table" as well as other problems which emerge over time. This project falls into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.

该奖项为模形式、阿贝尔簇和伽罗瓦表示的研究提供资金。格哈德·弗雷 (Gerhard Frey) 的构造将整数三元组与有理数上的椭圆曲线联系起来，使研究者的主题与涉及整数的简单问题之间建立了直接联系。弗雷的构造导致了费马大定理的证明以及数学中几个相关问题的解决。许多问题仍有待解决——ABC 猜想也许是最著名的。人们期望算术代数几何将继续蓬勃发展；调查员正在研究目前"摆在桌面上"的一些问题以及随着时间的推移而出现的其他问题。 该项目属于算术几何的一般领域——这一学科融合了两个最古老的数学领域：数论和几何。 事实证明，这种结合非常富有成效——最近解决了几代人都面临的问题。 其众多后果之一就是新的纠错码。 此类代码对于现代计算机（硬盘）和光盘都是必不可少的。