Selmer Groups

研究者打算从事一系列与塞尔默群、椭圆曲线、模形式、p-adic L-函数和岩泽理论相关的项目。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。它们也在证明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，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-函数类的行为提供类似的拓扑关系。Iwasawa理论通过允许L-函数在由素数p的模幂同余定义的族中变化，为研究这些猜想提供了一个框架。这导致了ap-adic L-函数的概念，以及将这种p-adic L-函数的行为与相应的塞尔默群联系起来的类似的代数关系的公式化。本研究者与几位合作者打算在一些新的背景下研究这种拓扑关系：（1）经典Artin L-函数的p-adic类似物，（2）与某些不同维数的Galois表示族相关联的L-函数族。此外，本文还研究了与模形式有关的塞尔默群与某些数域的理想类群之间的关系，并研究了关于p-adic L-函数的导数的一些尚未解决的问题。这个问题的难度取决于方程的阶数和变量的个数。当阶数为1或2，变量数也为1或2时，如何研究这个问题，自古以来就已为人们所理解。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。虽然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Swinnnerton-Dyer。最终目标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猜想的猜想提出更普遍的观点。

The investigator intends to pursue a diverse set of projects related toSelmer groups, elliptic curves, modular forms, p-adic L-functions andIwasawa theory. Selmer groups have traditionally been an important toolfor studying the Mordell-Weil group of an elliptic curve over a numberfield. They have also played an important role in proving special casesof the Birch and Swinnerton-Dyer conjecture which provides a relationshipbetween arithmetic properties of an elliptic curve E and the behavior ofthe Hasse-Weil L-function for E. In recent years it has become clear thatcertain natural generalizations of the Selmer group should provide similarconjectural relationships to the behavior of much more general kinds ofL-functions. Iwasawa theory provides a framework for studying theseconjectures by allowing the L-functions to vary in families defined bycongruences modulo powers of a prime p. This leads to the notion of ap-adic L-function and to the formulation of analogous conjecturalrelationships relating the behavior of such p-adic L-functions to thecorresponding Selmer groups. The investigator, together with severalcollaborators, intends to study this kind of conjectural relationship insome new settings: (1) p-adic analogues of classical Artin L-functions,(2) families of L-functions associated to certain families of Galoisrepresentations of varying dimension. In addition, the investigatorintends to study relationships between Selmer groups associated to modularforms and ideal class groups of certain number fields and to study someunresolved questions about the derivatives of p-adic L-functions.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 kind of equation was formulatedin the 1960s by Birch and Swinnerton-Dyer. Although considerable progresshas been 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 crytography - 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 Swinnnerton-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.