Some remarks on pseudo-null submodules of tamely ramified Iwasawa modules

Abstract

We will give several observations about the structure of tamely ramified Iwasawa modules for a ℤ p -extension (or a multiple ℤ p -extension) of an algebraic number field. In the present paper, we consider the question whether a given tamely ramified Iwasawa module has a non-trivial finite (or pseudo-null) submodule or not. For the cyclotomic ℤ p -extension of ℚ (with odd p), we can obtain a complete answer to this question. We also give sufficient conditions for having a non-trivial pseudo-null submodule for the case of the ℤ p ⊕2 -extension of an imaginary quadratic field. We also give an application of our results to the “non-abelian Iwasawa theory” in the sense of Ozaki.