Number Theory and Related Fields

获奖摘要DMS-0403374 Barry Mazur研究者正在进行四个项目： 为了理解平面上三次方程的解（即，椭圆曲线上的点）以及这些解的集合如何随着数域的扩大而增长。 通过代数几何方法以及（p-adic和经典）分析方法研究模形式的傅立叶系数之间的系数同余的性质，这是由欧拉，雅可比，拉马努金和其他人开创的主题。理解有理连通代数簇的新工作对算术的影响。 利用微分拓扑学、同伦理论和模形式的算术理论（Michael霍普金斯的工作）之间的联系来探索经典同伦理论中的问题。（迈克尔霍普金斯和PI目前正试图写一个证明的事实，即频谱PL/O分类微分结构紧凑的组合流形和“核心的爱森斯坦理想”在频谱称为tmf（“拓扑模形式”）在第2级是规范同构后，适当完成一个奇素数p;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、复和p-adic分析、微分拓扑和同伦理论的领域目前有令人兴奋的接触点，其中一个领域的新思想允许人们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。举一个例子，最经典的问题是找到两个变量的三次方程的合理解-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，PI的所有四个项目的中心舞台，即使这些项目涉及我们上面列出的所有领域。再举一个例子，由欧拉、雅可比、拉马努金和其他人开创的寻找和解释模形式的傅里叶系数之间的同余关系的经典问题--PI的项目之一--现在通过将所讨论的模形式视为一个美丽的数学模型中的密集点集，（p-解析）空间，其几何特征是有趣的，因为他们是至关重要的理解这些同余，以及其他同样基本的问题，在解析数论。

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.