Prime-primary Selmer group of a CM elliptic curve (${\mathrm{Sel}}_{{\mathfrak{p}}^{\infty }}(E/F)$)

For a split CM prime $\mathfrak{p}$, use $E[{\mathfrak{p}}^{\infty }]$ in the Galois cohomology definition of the Selmer group of an elliptic curve, with local conditions given by its Kummer maps. The decomposition of the CM $p$-power torsion splits the full $p$-primary Selmer group into the two prime-primary pieces. Class-field descriptions must retain the CM-character twist and distinguish relaxed local conditions from the actual Selmer conditions.