Number Theory and Related Fields

数论及相关领域

• 批准号: 0700580
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700580&HistoricalAwards=false
• 关键词: Number; Theory; Related; Fields

The Principal Investigator proposes to work on two distinct but related projects. The first projectis a study of Selmer groups and Mordell-Weil groups of elliptic curves over towers of number fields,and the second is to resolve certain deformational problems related to automorphic forms. Oneof the aims of the first project is to prove that Selmer rank grows at least as fast as would bepredicted by heuristics governing the signs of expected functional equations. One of the aims ofthe second is to understand the parameter spaces of automorphic forms that are usually referredto as "eigenvarieties."The first project pursued by the principal investigator is the fundamental arithmetic problem of understandingthe structure of triples of algebraic numbers that are solutions of given homogeneouspolynomials of degree three in three variables, this cubic case having an extraordinary amountof structure and playing a pivotal role in the larger project of understanding the arithmetic ofpolynomial equations in general. This type of number theory was critical, for example, in theestablishment of Fermat's Last Theorem (work of Wiles and Taylor-Wiles) over a dozen years ago,and - indeed - becomes ever more powerful and continues to be crucial for a wide range of applications,for example in cryptography. The second project proposed by the Principal Investigator hasits historical origin in classical work of Ramanujan, that dealt with the arithmetic properties of theFourier coefficients of modular forms. Ramanujan unearthed striking congruences that on the onehand contain important number theoretic information, and on the other suggest that a mysteriouscoherence underlies a large assortment of basic arithmetic phenomena; one of the goals of modernnumber theory is to expand this, and use its power for a range of applications.1

主要研究人员建议研究两个不同但相关的项目。第一个项目是对数字塔的椭圆曲线组和椭圆形曲线组的研究研究，第二个是解决与自动形式相关的某些变形问题。第一个项目的目标是证明Selmer等级的增长至少与控制预期功能方程式符号的启发式方法所预测的速度一样快。第二个目的之一是了解通常被称为“特征值”的自多态形式的参数空间。在更大的项目中，关键的作用是理解多种方程式的算术。这种数字理论至关重要，例如，在十几年前的Fermat的最后一个定理（Wiles和Taylor -Wiles的作品）中， - 的确， - 实际上 - 变得更加强大，并且对于广泛的应用中的广泛应用至关重要。首席研究者提出的第二个项目在拉玛努扬的古典工作中统治了历史起源，该项目涉及模块化形式的Fourier系数的算术特性。 Ramanujan发掘了一致性的一致性，其中包含重要的数字理论信息，另一方面，神秘的共同点是大量基本算术现象的大量基础。现代人理论的目标之一是扩展这一点，并将其力量用于一系列应用。1