= Prime-primary Selmer group of a CM elliptic curve
{title2=$\operatorname{Sel}_{\mathfrak p^\infty}(E/F)$}
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.
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