Number Theory and Related Fields

首席研究员建议在两个不同但相关的项目上工作。第一部分是对数域塔上椭圆曲线的Selmer群和Mordell-Weil群的研究，第二部分是解决与自同构形式有关的某些变形问题。第一个项目的目标之一是证明塞尔默秩的增长速度至少与控制预期函数方程符号的启发式预测的增长速度一样快。第二个目标之一是理解通常被称为“特征变”的自同构形式的参数空间。“首席研究员的第一个项目是理解三重代数数结构的基本算术问题，这些代数数是给定的三次三变量齐次多项式的解，这种三次情况具有非常多的结构，在理解多项式方程的算术这一更大的项目中起着关键作用。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（Wiles和Taylor-Wiles的工作），并且-确实-变得越来越强大，并继续对广泛的应用至关重要，例如在密码学中。由首席研究员提出的第二个项目有其历史渊源，在拉马努金的经典工作，处理的是模形式的傅里叶系数的算术性质。拉马努金发现了惊人的同余，这些同余一方面包含重要的数论信息，另一方面表明一种神秘的相干性隐藏在大量基本算术现象之下；现代数论的目标之一就是扩展这一点，并把它的力量用于一系列的应用

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