Image of [Lambda]-adic Galois representations modulo p

Let p>=5 be a prime. If an irreducible component of the spectrum of the 'big' ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its mod p Galois representation contains an open subgroup of [InlineEquation not available: see fulltext.] for the canonical "weight" variable T. This fact appears to be deep, as it is almost equivalent to the vanishing of the [mu]-invariant of the Kubota-Leopoldt p-adic L-function and the anticyclotomic Katz p-adic L-function. Another key ingredient of the proof is the anticyclotomic main conjecture proven by Rubin/Mazur-Tilouine.[PUBLICATION ABSTRACT]