Selmer Groups

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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.