  We study the variation of mu-invariants in Hida families with residually
reducible Galois representations. We prove a lower bound for these invariants
which is often expressible in terms of the p-adic zeta function. This lower
bound forces these mu-invariants to be unbounded along the family, and
moreover, we conjecture that this lower bound is an equality. When U_p-1
generates the cuspidal Eisenstein ideal, we establish this conjecture and
further prove that the p-adic L-function is simply a power of p up to a unit
(i.e. lambda=0). On the algebraic side, we prove analogous statements for the
associated Selmer groups which, in particular, establishes the main conjecture
for such forms.
