Iwasawa Theory

格林伯格教授打算研究一系列与塞尔默群、椭圆曲线和岩川理论相关的问题。Selmer群一直是研究椭圆曲线在数域上的莫德尔-韦尔群的重要工具。他们也在证明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，很明显，Selmer群的某些自然推广应该提供与更一般类型的l -函数的行为类似的猜想关系。岩川理论为研究这些猜想提供了一个框架。从本质上讲，其思想是研究与数字域的绝对伽罗瓦群的表示族相关的塞尔默群。在一般情况下提出这些猜想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塞尔默群体的规模。它们的大小应该由某个环中的一个元素来测量。但是很难找到一种方法来定义这个元素。这个项目的目标之一是在两个不同的环境中解决这个问题。在一种情况下，环很容易描述，但它是不可交换的。在另一种情况下，环是可交换的，但我们对它的实际结构知之甚少。该项目的另一个目标是更好地理解间接反映Selmer群结构的某些数量的行为，特别是在非交换设置中。这些数量是一定的“多重”。“这些数量通常是无限的。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。这个问题的难易程度取决于方程的程度和变量的数量。当度是一或二，变量的数量也是一或二时，如何研究这个问题，自古以来就已为人所知。然而，当考虑到三级时，即使变量的数量只有两个，这个问题就变得更加微妙了。关于这个问题的一个基本猜想是在20世纪60年代由Birch和Swinnerton-Dyer提出的。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曲线，称为“椭圆曲线”。研究它们的性质在密码学中具有重要的意义——为信息的安全传输设计密码。Greenberg教授打算继续他的“Selmer群”的研究，这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。最终目标是对定义椭圆曲线的代数方程的解有更深的理解，并对类似于Birch和Swinnerton-Dyer猜想的猜想有更普遍的看法。

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.