Iwasawa Theory

格林伯格教授打算研究一系列与Selmer群、椭圆曲线和岩泽理论有关的问题。Selmer群传统上是研究数域上椭圆曲线的Mordell-Weil群的重要工具。它们还在证明Birch和Swinnerton-Dyer猜想的特例中发挥了重要作用，该猜想提供了椭圆曲线E的算术性质和E的Hasse-Weil L函数的行为之间的关系。近年来，很明显，Selmer群的某些自然推广应该提供类似于更一般类型的L函数的行为的猜想关系。岩泽理论为研究这些猜想提供了一个框架。在其本质上，其思想是研究与数域的绝对伽罗华群的表示族相关的塞尔默群。这些猜想在一般情况下的表述会导致一些基本问题。一个问题是找到一种简单的方法来衡量Selmer群体的规模。它们的大小应该由某一环上的一个元素来测量。但很难找到一种方法来定义这一元素。这个项目的目标之一是在两个截然不同的环境中解决这个问题。在一种情况下，环很容易描述，但却是不可交换的。在另一种情况下，环是可交换的，但我们对其实际结构知之甚少。这个项目的另一个目标是更好地理解某些量的行为，这些量间接反映了Selmer群的结构，特别是在非对易环境中。这些量具有一定的“多重性”。这些量通常有无穷多个。格林伯格教授希望展示如何从有限的数量中确定所有这些量。数论中的一个基本问题是研究代数方程的解。这道题的难度取决于方程的程度和变量的数量。自古以来，人们就知道如何研究次数为一次或二次、变量数为一次或二次的问题。然而，当考虑三次方程时，即使变量数只有两次，这个问题也变得微妙得多。关于这个问题的一个基本猜想是由Birch和Swinnerton-Dyer在20世纪60年代提出的。尽管自那时以来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被称为“椭圆曲线”的曲线。研究它们的性质对于密码学设计用于信息安全传输的码具有重要意义。格林伯格教授打算继续他对“塞尔默群”的研究，“塞尔默群”是理解椭圆曲线的算术性质和研究Birch和Swinnerton-Dyer猜想的传统工具。最终目的是更深入地理解定义椭圆曲线的代数方程的解，并对类似于Birch和Swinnerton-Dyer猜想的猜想发展更一般的观点。

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.