A weak form of Greenberg's generalized conjecture for imaginary quadratic fields

Abstract

Let p be an odd prime number and k an imaginary quadratic field in which p splits. In this paper, we consider a weak form of Greenberg's generalized conjecture for p and k, which states that the non-trivial Iwasawa module of the maximal multiple Zp-extension field over k has a non-trivial pseudo-null submodule. We prove this conjecture for p and k under the assumption that the Iwasawa λ-invariants vanish for the Zp-extensions over k in which one of the prime ideals of k lying above p do not ramify and that the characteristic ideal of the Iwasawa module associated to the cyclotomic Zp-extension over k has a square-free generator.