Number Theory and Related Fields

首席调查员提议开展两个不同但相关的项目。第一个项目是研究数域塔上椭圆曲线的塞尔默群和Mordell-Weil群，第二个项目是解决与自守形式有关的某些变形问题。第一个项目的目的之一是证明塞尔默排名增长至少一样快，将beprochistics预测的迹象，预期的功能方程。第二部分的目的之一是理解自守形式的参数空间，这些自守形式通常被称为“特征簇”。“第一个项目所追求的主要研究人员是基本的算术问题的理解结构的三元组的代数数是解决方案的给定的三阶多项式的三个变量，这种立方的情况下有一个非凡的结构和发挥了关键作用，在更大的项目的理解算术的多项式方程一般。这种类型的数论是至关重要的，例如，在十几年前建立费马大定理（怀尔斯和泰勒-怀尔斯的工作），而且-确实-变得越来越强大，并继续在广泛的应用中至关重要，例如在密码学中。第二个项目提出的主要研究者hasits历史渊源在古典工作的拉马努金，处理算术性质的theFourier系数的模块化形式。拉马努金发现了一些引人注目的同余式，这些同余式一方面包含了重要的数论信息，另一方面也表明，大量的基本算术现象背后都有一种不寻常的一致性;现代数论的目标之一就是扩展这种一致性，并将其力量用于一系列应用。[1]

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