A note on Greenberg’s conjecture and the abc conjecture

Abstract

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