Let $E$ be an elliptic curve over $\mathbb{Q}$, let $K$ be an imaginary quadratic field, and let $K_{\infty }$ be a ${\mathbb{Z}}_p$-extension of $K$. Given a set $\Sigma$ of primes of $K$, containing the primes above $p$, and the primes of bad reduction for $E$, write $K_{\Sigma }$ for the maximal algebraic extension of $K$ which is unramified outside $\Sigma$. This paper is devoted to the study of the structure of the cohomology groups $H^i(K_{\Sigma }/K_{\infty },E_{p^{\infty }})$ for $i=1,2,$ and of the $p$-primary Selmer group Sel${}_{p^{\infty }}(E/K_{\infty })$, viewed as discrete modules over the Iwasawa algebra of $K_{\infty }/K.$

Let $f$ be a cuspidal newform with complex multiplication (CM) and let $p$ be an odd prime at which $f$ is non-ordinary. We construct admissible $p$-adic $L$-functions for the symmetric powers of $f$, thus verifying conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus $p$-adic $L$-functions and prove an analogue of Pollack’s decomposition of the admissible $p$-adic $L$-functions....