# Journal of the Ramanujan Mathematical Society

Volume 36, Issue 3, September 2021 pp. 251–260.

On q-ramified abelian 3-extensions over the initial layer of the anti-cyclotomic Z{3}-extension of an
imaginary quadratic field

**Authors**:
Tsuyoshi Itoh

**Author institution:**Division of Mathematics, Education Center, Faculty of Social Systems Science, Chiba Institute of Technology, 2--1--1 Shibazono, Narashino, Chiba, 275--0023, Japan

**Summary: **
Let k be an imaginary quadratic field, k{a}{∞}/k the
anti-cyclotomic Z{3}-extension, and k{a}{1}/k the unique
cubic cyclic subextension of k{a}{∞}/k (k{a}{1} is often
called the initial layer of k{a}{∞}/k). For a prime number q
(≠ 3), we denote by X{q} (k{a}{1}) (resp. X'{q} (k{a}{1}))
the Galois group of the maximal q-ramified (resp. q-ramified
3-split) abelian 3-extension over k{a}{1}. We give a result
concerning the behavior of the orders of X{q} (k{a}{1}) and X'{q}
(k{a}{1}). This supplements the previous work by Takakura and the
author, which also considers X{q} (k{a}{1}) and X'{q} (k{a}{1}) in
the context of studying ``tamely ramified Iwasawa modules''.

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