Finite submodules and Iwasawa \(\mu\)-invariants
Let \(K\) be a number field and let \(K_{\infty}/K\) be a \(\mathbb Z_p\)-extension. Let \(G_K=\)Gal\((\overline{K}/K)\) and \(A\) be a \(G_K\)-module that is cofinitely generated and cofree as a \(\mathbb Z_p\)-module. Under certain ordinariness conditions, \textit{R. Greenberg} [in Algebraic number theory -- in honor of K. Iwasawa, Adv. Stud. Pure Math. 17, 97-137 (1989; Zbl 0739.11045)] showed how to define a Selmer group \(S_A(K_{\infty})\), which is cofinitely generated as a \(\mathbb Z_pT\)-module. The \(\mu\)-invariant \(\mu(A)\) of \(S_A(K_{\infty})\) is positive if and only if the \(\mathbb Z_p\)-cotorsion group is infinite. The present paper studies the effect that finite submodules of \(A\) have on this \(\mu\)-invariant for the cyclotomic \(\mathbb Z_p\)-extension of \(K\). If \(\alpha\) is a finite submodule of \(A\), the author defines an integer \(\delta(\alpha)\) that is a lower bound for \(\mu(A)\). Let \(m(A)\) be the maximum value of \(\delta(\alpha)\) for all such \(\alpha\subset A\). It was shown by \textit{B. Perrin-Riou} [in Algebraic number theory -- in honor of K. Iwasawa, Adv. Stud. Pure Math. 17, 347-358 (1989; Zbl 0747.11056)] how \(\mu(A)\) behaves under isogenies. The present author shows that \(m(A)\) behaves similarly; namely, under certain hypotheses, if there is an isogeny from \(A\) to \(B\) with kernel \(\alpha\), then \(\mu(A)-\mu(B)=m(A)-m(B)=\delta(\alpha)\). Moreover, \(A\) is isogenous to some \(B\) with \(m(A)=0\). The hope is that \(\mu(A)=m(A)\) when \(A\) is defined over \(\mathbb Q\), which would mean that positive \(\mu\)-invariants could be explained completely by finite submodules of \(A\). The author discusses the case when \(A\) is not defined over \(\mathbb Q\), which is more complicated than when \(A\) is defined over \(\mathbb Q\) since there can be finite submodules of the representation induced to \(\mathbb Q\) that are not induced from finite submodules defined over the number field. He uses this fact to give examples where the \(\mu\)-invariant is positive.
scientific article; zbMATH DE number 1753715