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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....

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\ge 3$ and $a_p=0$. We generalise the definition of Kobayashi’s plus/minus Selmer groups over $\mathbb{Q}\left({\mu }_{p^{\infty }}\right)$ to $p$-adic Lie extensions $K_{\infty }$ of $\mathbb{Q}$ containing $\mathbb{Q}\left({\mu }_{p^{\infty }}\right)$, using the theory of $(\varphi ,\Gamma )$-modules and Berger’s comparison isomorphisms. We show that these Selmer groups can be equally described using Kobayashi’s conditions via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer groups in the ordinary case....