Finite Submodules and Iwasawa μ-Invariants Original Research Article

Given a Zp-extension of number fields K∞/K and a GK-module A which is cofree as a Zp-module, we can define a Selmer group SA(K∞). If A satisfies certain ordinariness conditions, then SA(K∞) is a cofinitely generated, cotorsion Λ-module. In this paper, we study the effect that finite submodules of A can have on the μ-invariant of SA(K∞). For each finite submodule αsubset ofA, we use the Galois cohomology of α to define an invariant δ(α) which is a lower bound for μ(SA(K∞). Then we define an invariant m(A) that organizes the information that we get from all of the finite αsubset ofA. These invariants will lead to an elegant way of explaining some previously known examples where the μ-invariant is positive and they will provide us with new kinds of examples where μ is positive.