The conjecture of Birch and Swinnerton-Dyer, which relates an analytic invariant of an elliptic curve to the arithmetic of the curve, is unquestionably among the most important open problems in number theory today. Inspired by Kummer's attempt to solve Fermat's Last Theorem and the mysterious connection between ideal class groups and zeta values it displayed, Iwasawa developed in 1959 an idea which later evolved into a fundamental branch of modern number theory. Iwasawa theory has been applied to a wide range of problems in which values of L-functions (or zeta functions) play a key role, and proved to be one of the most fruitful ways of understanding the Birch-Swinnerton-Dyer conjecture. The aim of this talk is to give an overview of the origin and the basic ideas of Iwasawa theory, and to discuss some important applications.
org: BARROERO Fabrizio