On \(\mu\)-invariants of Selmer groups of some CM elliptic curves (Q2840297)

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) and with CM by an imaginary quadratic field \(\mathbb{Q}(\sqrt{-D})\). Fix an odd prime number \(p\) unramified in \(\mathbb{Q}(\sqrt{-D})\) and of ordinary reduction for \(E\), and consider the fields \(K=\mathbb{Q}(E[p])\), \(K_\infty=K\mathbb{Q}_\infty\) the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\) and \(K(E[p^\infty])\). One of the basic conjectures of noncommutative Iwasawa theory (the \textit{\(\mathfrak{M}_H(G)\)-conjecture} in [\textit{J. Coates} et al., Publ. Math., Inst. Hautes Étud. Sci. 101, 163--208 (2005; Zbl 1108.11081)]) predicts that the Pontryagin dual of the \(p\)-part of the Selmer group of \(E\) over \(K(E[p^\infty])\) (denoted \(\text{Sel}(K(E[p^\infty]))^\vee\,\)) modulo its \(p\)-power torsion is a finitely generated module over the ring \(\mathbb{Z}_p[[\text{Gal}(K(E[p^\infty]/K_\infty)]]\). In the above, setting the last statement is equivalent to proving that the \(\mathbb{Z}_p[[\text{Gal}(K_\infty/K)]]\)-module \(\text{Sel}(K_\infty)^\vee\) has no \(p\)-torsion, i.e., its \(\mu\)-invariant vanishes. The author considers the case \(p=3\) and uses some results of \textit{R. Greenberg} [Iwasawa theory, projective modules, and modular representations. Mem. Am. Math. Soc. 992 (2011; Zbl 1247.11085)] on Selmer atoms (i.e., Selmer groups for \(E[p]\otimes \alpha\) as \(\alpha\) varies among the irreducible \(\mathbb{F}_p\)-representations of \(\text{Gal}(K/\mathbb{Q})\,\)) to show that the vanishing of \(\mu\) is equivalent to the finiteness of the Selmer groups over \(\mathbb{Q}_\infty\) for \(E[p]\) and \(E[p]\otimes E[p](-1)\) (the only relevant Selmer atoms for \(p=3\) and a CM elliptic curve with \(|\text{Gal}(K/\mathbb{Q})|=8\)). This can be checked (using the theory of adjoints) by looking at some coefficients of a \(p\)-adic \(L\)-function whose explicit formula is provided in \textit{A. Dabrowski} and \textit{D. Delbourgo} [Proc. Lond. Math. Soc., III. Ser. 74, No. 3, 559--611 (1997; Zbl 0871.11041)]. With computations done with the SAGE and/or MAGMA programs, the author verifies the vanishing of the \(\mu\)-invariant (hence the \(\mathfrak{M}_H(G)\)-conjecture) for 6 elliptic curve with CM by \(\mathbb{Q}(\sqrt{-8})\) or \(\mathbb{Q}(\sqrt{-11})\).