On the Μ-invariant in Iwasawa Theory

The aim of this expository article is to discuss the μ-invariant associated to finitely generated modules over Iwasawa algebras. This is an important invariant which was first discovered by Iwasawa in the 1960’s and occurs in his seminal work on cyclotomic fields [9], [10]. In fact, Iwasawa conjectured that his μ-invariant was always zero for the pprimary subgroup of the ideal class group of the field obtained by adjoining all p-power roots of unity to a given finite extension F of Q, p being a fixed prime number. This was subsequently proven by Ferrero and Washington when F is an abelian extension of Q, but remains open in general. Mazur found the first simple examples of a positive μ-invariant in the case of the Iwasawa theory of elliptic curves. For example, let A denote the 5-primary subgroup of the Tate-Shafarevich group of the elliptic curve