On q-ramified abelian 3-extensions over the initial layer of the anti-cyclotomic Z₃-extension of an imaginary quadratic field

Let k be an imaginary quadratic field, k(infinity)(a)/ k the anti-cyclotomic Z(3)-extension, and k(1)(a)/k the unique cubic cyclic subextension of k(infinity)(a)/k (k(1)(a) is often called the initial layer of k(infinity)(a)/k). For a prime number q not equal 3), we denote by X-q(k(1)(a)) (resp. X-q'(k(1)(a))) the Galois group of the maximal q-ramified (resp. q-ramified 3-split) abelian 3-extension over k(1)(a). We give a result concerning the behavior of the orders of X-q (k(1)(a)) and X-q' (k(1)(a)). This supplements the previous work by Takakura and the author, which also considers X-q(k(1)(a)) and X-q'(k(1)(a)) in the context of studying "tamely ramified Iwasawa modules".