A note on Greenberg’s conjecture and the abc conjecture

For any totally real number field k k and any prime number p p , Greenberg’s conjecture for ( k , p ) (k,p) asserts that the Iwasawa invariants λ p ( k ) λ _p(k) and μ p ( k ) μ _p(k) are both zero. For a fixed real abelian field k k , we prove that the conjecture is “affirmative” for infinitely many p p (which split in k ) k) if we assume the abc conjecture for k k .