Finite \Lambda -submodules of Iwasawa modules for a CM-field for p=2

Abstract

Let p be a prime, X F ∞ - the minus quotient of the Iwasawa module, which we define to be the Galois group of the maximal unramified abelian pro-p-extension over the cyclotomic ℤ p -extension over a CM field F. If p is an odd prime, it is well known that X F ∞ - has no non-trivial finite ℤ p 〚Gal(F ∞ /F)〛-submodule. But X F ∞ - has non-trivial finite ℤ p 〚Gal(F ∞ /F)〛-submodule in some cases for p=2. In this paper, we study the maximal finite ℤ p 〚Gal(F ∞ /F)〛-submodule of X F ∞ - for p=2. We determine the size of the maximal finite ℤ 2 〚Gal(F ∞ /F)〛-submodule of X F ∞ - under some mild assumptions.