Selmer Groups

研究人员打算进行一系列与Selmer群、椭圆曲线、模形式、p-进L函数和岩泽理论有关的项目。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与更一般类型的L-函数的行为类似的猜想关系。岩泽理论允许L函数在素数p的模幂同余所定义的族中变化，从而为研究这些猜想提供了一个框架.这就引出了p-进的L-函数的概念，并建立了将这类p-进的L-函数的行为与相应的Selmer群联系起来的类似的猜想关系.作者和几位合作者打算在一些新的背景下研究这种猜想关系：(1)经典Artin L函数的p-进类似；(2)与某些变维Galois表示族相关的L函数族。此外，作者还研究了某些数域的模形式Selmer群与理想类群之间的关系，并研究了p元L函数导数的一些悬而未决的问题。这道题的难度取决于方程的程度和变量的数量。自古以来，当次数为1次或2次，变量数为1或2时，如何研究这一问题一直是人们所理解的。然而，当考虑三次方程时，即使变量数只有2次，这个问题就变得微妙得多。关于这类方程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Swinnnerton-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。

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.