Number Theory and Related Fields

首席调查员建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Selmer群和Mordell-Weil群，第二个项目是解决与自同构形式有关的某些变形问题。第一个项目的目的之一是证明Selmer排名的增长速度至少与控制预期函数方程的符号的启发式预测的速度一样快。第二门课的目的之一是理解自同构形式的参数空间，通常被称为“本征簇”。主要研究者追求的第一个项目是理解代数数三元组的结构的基本算术问题，这些代数数是给定的三次三次多项式在三个变量中的解，这种立方情形具有非同寻常的结构，在理解一般多项式方程的算法的更大项目中发挥着关键作用。例如，这种类型的数论在十几年前费马最后定理(Wiles和Taylor-Wiles的工作)的建立中是至关重要的，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。第二个项目由首席研究员提出，它的历史起源于Ramanujan的经典工作，该工作涉及模形式的傅立叶系数的算术性质。Ramanujan发现了惊人的同余，一方面包含重要的数论信息，另一方面表明一种神秘的一致性是一大类基本算术现象的基础；现代数论的目标之一是扩展这一点，并将其力量用于一系列应用。

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