On Tamely Ramified Iwasawa Modules for $\Zp$-extensions of Imaginary Quadratic Fields

Abstract

We study the Iwasawa modules related to certain tamely ramified extensions (tamely ramified Iwasawa modules). Let p be an odd prime number, and k an imaginary quadratic field. In the present paper, we shall give some results concerning the μ-invariant of tamely ramified Iwasawa modules for Z p -extensions of k.

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

“…Satoshi Fujii, Tsuyoshi Itoh Note added in proof. Concerning Remark 4.9, the values of the tables in [13] had been confirmed to be correct.…”

“…After that paper was published, a problem was found in the computer programs. However, the values stated in the tables of [13,Section 5] Some remarks on pseudo-null submodules 551 seem correct (it has not been completely confirmed yet). The details will be reported elsewhere.…”

“…(The proof is more complicated because we consider the initial layer of k a /k. See also [13,Remark 4.7].) Theorem 4.7.…”

“…Remark 4.9. We mention about the computational results given in [13,Section 5]. (Some of those results are related to Question 1.2 via Lemma 4.1.)…”

“…In this case, k/k is a Z ⊕2 p -extension. Similar to the case of Q c /Q, any prime q ∈ S satisfying N q ≡ 1 (mod p) does not have an effect on the structure of X S ( k), where N q is the absolute norm of q (see also, e.g., [12,13]). Hence, it is sufficient to consider S satisfying the following condition.…”