Galois coinvariants of the unramified Iwasawa modules of multiple $$\mathbb {Z}_p$$ -extensions
Tóm tắt
For a CM-field K and an odd prime number p, let
$$\widetilde{K}'$$
be a certain multiple
$$\mathbb {Z}_p$$
-extension of K. In this paper, we study several basic properties of the unramified Iwasawa module
$$X_{\widetilde{K}'}$$
of
$$\widetilde{K}'$$
as a
$$\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}'/K)\rrbracket $$
-module. Our first main result is a description of the order of a Galois coinvariant of
$$X_{\widetilde{K}'}$$
in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic
$$\mathbb {Z}_p$$
-extension of K under a certain assumption on the splitting of primes above p. The second result is that if K is an imaginary quadratic field and if p does not split in K, then, under several assumptions on the Iwasawa
$$\lambda $$
-invariant and the ideal class group of K, we determine a necessary and sufficient condition such that
$$X_{\widetilde{K}}$$
is
$$\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}/K)\rrbracket $$
-cyclic. Here,
$$\widetilde{K}$$
is the
$$\mathbb {Z}_p^2$$
-extension of K.
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