Arithmetical Algebraic Geometry

算术代数几何

• 批准号: 9970593
• 负责人: Kenneth Ribet
• 金额: $ 13.5万
• 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: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970593&HistoricalAwards=false
• 关键词: Arithmetical; Algebraic; Geometry

9970593Kenneth Ribet intends to continue his work on the number theory associated with modular forms, modular curves, abelian varieties and Galois groups. Ribet is especially interested in number fields arising from torsion points on abelian varieties. While many questions in this subject are technical in nature, they are ultimately rooted in the classical problem of finding all whole number or fractional solutions to a family of equations.Kenneth Ribet studies the arithmetic of modular forms, Galois representations and abelian varieties. His research lies at the intersection of algebraic geometry and algebraic number theory, two flourishing fields of mathematics. Ribet is best known for his contribution to the proof of Fermat's Last Theorem: Ribet proved a technical result about Galois representations, sometimes known as Serre's epsilon conjecure, which relates Fermat's Last Theorem to the Shimura-Taniyama conjecture for elliptic curves. More recently, Ribet contributed to the proof of a Fermat's conjecture to the effect that three distinct positive perfect n'th powers (where n is bigger than 2) can never form an arithmetic progression.

9970593Kenneth Ribet打算继续他关于模形式、模曲线、阿贝尔变分和伽罗瓦群的数论研究。Ribet对由阿贝尔变体上的扭转点产生的数场特别感兴趣。虽然这门学科中的许多问题本质上是技术性的，但它们最终都植根于一个经典问题，即找到一组方程的所有整数或分数解。Kenneth Ribet研究模形式、伽罗瓦表示和阿贝尔变体的算法。他的研究是代数几何和代数数论这两个蓬勃发展的数学领域的交叉。里贝特最著名的贡献是他对费马大定理的证明：里贝特证明了一个关于伽罗瓦表示的技术结果，有时被称为Serre的epsilon猜想，它将费马大定理与椭圆曲线的Shimura-Taniyama猜想联系起来。最近，里贝对费马猜想的证明做出了贡献，证明了三个不同的正完全n次方（其中n大于2）永远不能形成等差数列。