Pseudo-Null Submodules of the Unramified Iwasawa Module for {\\mathbb Z}p2-Extensions

Abstract

In this article, we study generalized Greenberg's conjecture for imaginary quadratic fields. In particular, for the prime number p ¼ 3, we show some examples.

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

“…From this fact, if D S (F ) is non-trivial, then D S (k) is also. Combining with the result in (1), we obtain (2).…”

“…Since L ∅ (K n ) is an intermediate field of L S (K n )/K n , we see that D S (K n ) is also non-trivial. Then the assertion follows from Lemma 4.1 (2). Remark 4.9.…”

“…Minardi [17] gave examples such that the unramified Iwasawa module X ∅ (k) has a non-trivial pseudo-null submodule when p is inert in k by showing that the decomposition subgroup for a prime lying above p is not trivial. (See also [2].) The same idea can be applicable to our tamely ramified case.…”

“…Hence (1) follows. We shall show (2). Let k be the inertia field of k/k for the unique prime p lying above p. We remark that p splits completely in k , because k /k is an unramified abelian p-extension and the order of p in the ideal class group of k is not divisible by p. Of course, every prime lying above p is totally ramified in k/k .…”

“…When S = ∅, this question is weaker than GGC, and there are several relating works (see, e.g., [2,4,29]).…”