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Let $p$ be a prime number. We show that, there exists an infinite family ofimaginary abelian fields such that, the Iwasawa module of the maximal multiple${\Bbb Z}_p$-extension is non trivial and pseudo-null for each field in thefamily. We also discuss on an application to non-abelian Iwasawa theory in thesense of Ozaki.

Reports on families of imaginary abelian fields with pseudo-null unramified Iwasawa modules