Variation of Iwasawa invariants in Hida families
  Let r : G_Q -> GL_2(Fpbar) be a p-ordinary and p-distinguished irreducible residual modular Galois representation. We show that the vanishing of the algebraic or analytic Iwasawa mu-invariant of a single modular form lifting r implies the vanishing of the corresponding mu-invariant for all such forms. Assuming that the mu-invariant vanishes, we also give explicit formulas for the difference in the algebraic or analytic lambda-invariants of modular forms lifting r. In particular, our formula shows that the lambda-invariant is constant on branches of the Hida family of r. We further show that our formulas are identical for the algebraic and analytic invariants, so that the truth of the main conjecture of Iwasawa theory for one form in the Hida family of r implies it for the entire Hida family. 
Matthew Emerton, Robert Pollack and Tom Weston