Number/Representation Theory

Iwasawa conjectured in the 1960s that the mu-invariant vanishes for abelian number fields. This theorem was originally proven by Ferrero-Washington using explicit formulae for abelian p-adic L-functions in terms of Stickelberger elements. Shortly after, Sinnott gave a different proof, also using an explicit formula in terms of rational functions. Greenberg subsequently conjectured that the mu-invariant vanishes for the non-abelian extensions of Q associated to ordinary representations of GL(2) coming from elliptic curves, but this conjecture has remained open, since there are no explicit formulae in this situation.

I will present a proof of Iwasawa’s conjecture using phi-gamma modules. This proof does not rely on explicit formulae, but rather on a certain global invariance property enjoyed by the phi-gamma modules in question. Our proof not only unifies the apparently dissimilar proofs of Ferrero-Washington and Sinnott, but also is amenable to generalization to GL(2). We will show how to extend the method to GL(2) and deduce Greenberg’s conjecture from certain formal properties enjoyed by a modular symbol constructed from phi-gamma modules.