Iwasawa Theory

格林伯格教授打算研究一系列与塞尔默群、椭圆曲线和岩泽理论有关的问题。塞尔默群一直是研究数域上椭圆曲线的Mordell-Weil群的重要工具。 他们还在证明Birch和Swinnerton-Dyer猜想的特殊情况中发挥了重要作用，该猜想提供了椭圆曲线E的算术性质与E的Hasse-Weil L-函数的行为之间的关系。 近年来，人们已经清楚地认识到，塞尔默群的某些自然推广应该为更一般的L-函数的行为提供类似的拓扑关系。岩泽理论提供了一个框架，研究这些问题。在其本质上，这个想法是研究塞尔默群相关的一个家庭的代表性的绝对伽罗瓦群的数域。在一般情况下，这些公式的制定导致了一些基本问题。一个问题是找到一种简单的方法来测量塞尔默群有多大。 它们的大小应该由某个环中的元素来测量。但是很难找到一种方法来定义这个元素。该项目的目标之一是在两种截然不同的环境中解决这一问题。在一种情况下，环很容易描述，但它是非交换的。在另一种情况下，环是交换的，但我们对它的实际结构知之甚少。 这个项目的另一个目标是更好地理解某些间接反映塞尔默群结构的量的行为，特别是在非交换的情况下。这些量是某些“多重性”。通常有无数个这样的量。格林伯格教授希望展示如何从有限数量的量中确定所有这些量。 数论的基本问题之一是研究代数方程的解。这个问题的难度取决于方程的阶数和变量的个数。从古代起，人们就知道当次数为1或2，变量数也为1或2时，如何研究这个问题。然而，当我们求解3次方程时，即使变量数只有2，这个问题就变得微妙得多了。尽管从那时起已经取得了相当大的进展，但这个猜想仍然没有得到解决。这样的方程定义了一类称为“椭圆曲线”的曲线。“研究它们的性质已被证明是重要的密码学-设计代码的安全传输信息。格林伯格教授打算继续他的研究“塞尔默群”，这一直是一个传统的工具，在理解算术性质的椭圆曲线和研究的猜想，伯奇和Swinnerton-Dyer。最终的目标是要实现更深入的理解的解决方案的代数方程，定义一个椭圆曲线，并制定一个更普遍的观点有关类似于Birch和Swinnerton-Dyer猜想的猜想。

Professor Greenberg intends to study a diverse set of problems related toSelmer groups, elliptic curves, and Iwasawa theory. Selmer groups have traditionally been an important tool for studying the Mordell-Weil group of an elliptic curve over a number field. They have also played an important role in proving special cases of the Birch and Swinnerton-Dyer conjecture which provides a relationship between arithmetic properties of an elliptic curve E and the behavior of the Hasse-Weil L-function for E. In recent years, it has become clear that certain natural generalizations of the Selmer group should provide similar conjectural relationships to the behavior of much more general kinds of L-functions. Iwasawa theory provides a framework for studying these conjectures. In its essence, the idea is to study Selmer groups associated to a family of representations of the absolute Galois group of a number field. The formulation of these conjectures in a general setting leads to some fundamental problems. One problem is to find a simple way to measure how large the Selmer groups are. Their size should be measured by an element in a certain ring. But it is difficult to find a way to define that element. One of the objectives of this project is to tackle that question in two contrasting settings. In one setting, the ring is rather easy to describe, but is non-commutative. In the other setting, the ring is commutative, but we know very little about its actually structure. Another objective of this project is to better understand the behavior of certain quantities which indirectly reflect the structure of Selmer groups, especially in the non-commutative setting. These quantities are certain ``multiplicities.'' There are typically an infinite number of these quantities. Professor Greenberg hopes to show how to determine all of these quantities from just a finite number of them. One of the fundamental questions in the theory of numbers is the studyof solutions of an algebraic equation. The difficulty of this questiondepends on the degree of the equation and the number of variables. Ithas been understood since antiquity how to study this question when thedegree is one or two and the number of variables is also one or two.However, the question becomes much more subtle when one considersequations of degree three, even if the number of variables is just two.A fundamental conjecture concerning this question was formulated in the1960s by Birch and Swinnerton-Dyer. Although considerable progress hasbeen made since then, the conjecture remains unresolved. Such equationsdefine a class of curves known as "elliptic curves." The study of theirproperties has proved to be of importance in cryptography - designing codesfor the secure transmission of information. Professor Greenberg intends tocontinue his study of "Selmer groups" which have been a traditional tool inunderstanding the arithmetic properties of elliptic curves and in studyingthe conjecture of Birch and Swinnerton-Dyer. The ultimate goal is toachieve a deeper understanding of the solutions to the algebraic equationsthat define an elliptic curve, and to develop a more general point of viewconcerning conjectures analogous to the Birch and Swinnerton-Dyer conjecture.