Iwasawa theory, and especially its main conjectures, is the main tool for
studying the mysterious exact formulae in number theory linking the very
different mathematical worlds of purely arithmetic questions on the one
hand, with special values of complex L-functions on the other (typified by
the conjecture of Birch and Swinnerton-Dyer). My lecture will attempt to
explain how, in the special case of elliptic curves, non-commutative
phenomena which arise in each of these worlds lead to very unexpected
consequences in the other world.