Number Theory and Related Fields

数论及相关领域

• 批准号: 0403374
• 负责人: Barry Mazur
• 金额: --
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0403374&HistoricalAwards=false
• 关键词: Number; Theory; Related; Fields

ABSTRACT for award DMS-0403374Barry Mazur The investigator is working on four projects: To understand the solutions of cubic equations in the plane (i.e., points on elliptic curves) over number fields and how the collection of these solutions grows as one enlarges the number field. To study, by algebraic geometric methods as well as (p-adic and classical) analytic methods the nature of congruences between coefficients of Fourier coefficients of modular forms, a subject pioneered by Euler, Jacobi, Ramanujan and others. To understand the implications to arithmetic of the new work on rationally connected algebraic varieties. To make use of the connection between differential topology, homotopy theory and the arithmetic theory of modular forms (the work of Michael Hopkins) to explore issues in classical homotopy theory. ( Michael Hopkins and the PI are currently trying to write up a proof of the fact that the spectrum PL/O that classifies differential structures on compact combinatorial manifolds and the "kernel of the Eisenstein ideal" in the spectrum known as tmf ("topological modular forms") at level 2 are canonically isomorphic after appropriate completion at an odd prime number p; this would allow one to apply deep arithmetic results to analyze the structure of these spectra).The fields of arithmetic, algebraic geometry, complex and p-adic analysis, differential topology, and homotopy theory have-currently-exciting points of contact, where new ideas in one realm allow one to make inroads in others. All four projects in which the PI is engaged have the aim of strengthening these connections. To take one example, the most classical issue of finding rational solutions of cubic equations in two variables- which now is the mainstay of the theory and practice behind public-key encryption- finds itself as the core problem, center stage in all four of the PI's projects, even though these projects range through all the fields we have listed above. To take another example, the classical problem pioneered by Euler, Jacobi, Ramanujan and others, of finding and explaining congruences between Fourier coefficients of modular forms- one of the PI's projects- is now most powerfully addressed by viewing the modular forms in question as a dense set of points in a beautiful (p-analytic) space, whose geometric features are as intriguing as they are vital for an understanding of these congruences, as well as for other equally basic questions in analytic number theory.

DMS-0403374Barry Mazur奖的研究人员致力于四个项目：了解数域上平面上的三次方程(即椭圆曲线上的点)的解，以及这些解的集合如何随着数域的扩大而增长。用代数几何方法和(p-进的和经典的)分析方法研究模形式的傅里叶系数的系数之间同余的性质，这是由Euler，Jacobi，Ramanujan等人开创的一个课题。理解关于有理连通代数簇的新工作对算术的影响。利用微分拓扑、同伦理论和模形式算术理论(Michael Hopkins的工作)之间的联系来探索经典同伦理论中的问题。(Michael Hopkins和PI目前正在试图写出一个证明，即对紧致组合流形上的微分结构进行分类的谱Pl/O与水平2的被称为TMF(拓扑模形式)的谱中的“Eisenstein理想的核”在奇素数p处适当完成后典型同构；这将允许人们应用深度算术结果来分析这些谱的结构)。算术、代数几何、复数和p-进分析、微分拓扑学和同伦理论等领域有-当前-令人兴奋的接触点，其中一个领域的新思想允许一个领域的新思想在其他领域取得进展。国际和平协会参与的所有四个项目都旨在加强这些联系。举个例子，寻找两个变量的三次方程有理解的最经典问题--现在是公钥加密背后的理论和实践的支柱--发现自己是核心问题，在PI的所有四个项目中处于中心地位，尽管这些项目涵盖了我们上面列出的所有领域。再举一个例子，由Euler，Jacobi，Ramanujan和其他人开创的寻找和解释模形式的傅立叶系数之间的同余的经典问题-PI的项目之一-现在通过将所讨论的模形式视为美丽(p-解析)空间中的密集点集来最有力地解决，其几何特征对理解这些同余以及解析数论中的其他同样基本的问题同样有趣。