Number Theory and Related Fields

巴里·马祖尔（barry Mazur）研究员正在进行四个项目：了解平面（即椭圆曲线上的点）上的三次方程在数域上的解以及这些解的集合如何随着数域的扩大而增长。用代数几何方法以及（p进的和经典的）解析方法研究模形式的傅立叶系数系数之间的同余性质，这是由欧拉、雅可比、拉马努金等人开创的课题。了解合理连通代数变体的新工作对算术的意义。利用微分拓扑、同伦理论和模形式的算术理论（Michael Hopkins的作品）之间的联系，探讨经典同伦理论中的问题。(Michael Hopkins和PI目前正试图写出一个事实的证明，证明在紧组合流形上对微分结构进行分类的谱PL/O和在第2层被称为tmf（“拓扑模形式”）的谱中的“爱森斯坦理想核”在奇素数p处适当补全后是标准同构的；这将允许人们应用深度算术结果来分析这些光谱的结构)。算术、代数几何、复数和p进分析、微分拓扑和同伦理论等领域都有目前令人兴奋的接触点，其中一个领域的新思想允许一个人在其他领域取得进展。PI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的有理解这个最经典的问题——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，是PI所有四个项目的中心舞台，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，由欧拉、雅可比、拉马努金等人提出的经典问题，即寻找和解释模形式的傅里叶系数之间的同余式——PI的一个项目——现在通过将所讨论的模形式视为一个美丽（p解析）空间中的密集点集来有力地解决了这个问题，模形式的几何特征对理解这些同余式至关重要，同时也很有趣。以及解析数论中其他同样基本的问题。

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.