On a bound of λ and the vanishing of μ of ℤ_p-extensions of an imaginary quadratic field

Abstract

Let p be an odd prime number. To ask the behavior of λ- and μ-invariants is a basic problem in Iwasawa theory of ℤ_p-extensions. Sands showed that if p does not divide the class number of an imaginary quadratic field k and if the λ-invariant of the cyclotomic ℤ_p-extension of k is 2, then μ-invariants vanish for all ℤ_p-extensions of k, and λ-invariants are less than or equal to 2 for ℤ_p-extensions of k in which all primes above p are totally ramified. In this article, we show results similar to Sands' results without the assumption that p does not divide the class number of k. When μ-invariants vanish, we also give an explicit upper bound of λ-invariants of all ℤ_p-extensions.