Let
$E/\mathbf {Q}$
be an elliptic curve and
$p>3$
be a good ordinary prime for E and assume that
$L(E,1)=0$
with root number
$+1$
(so
$\text {ord}_{s=1}L(E,s)\geqslant 2$
). A construction of Darmon–Rotger attaches to E and an auxiliary weight 1 cuspidal eigenform g such that
$L(E,\text {ad}^{0}(g),1)\neq 0$
, a Selmer class
$\kappa _{p}\in \text {Sel}(\mathbf {Q},V_{p}E)$
, and they conjectured the equivalence
$$ \begin{align*} \kappa_{p}\neq 0\quad\Longleftrightarrow\quad{\textrm{dim}}_{{\mathbf{Q}}_{p}}\textrm{Sel}(\mathbf{Q},V_{p}E)=2. \end{align*} $$
In this article, we prove the first cases on Darmon–Rotger’s conjecture when the auxiliary eigenform g has complex multiplication. In particular, this provides a new construction of nontrivial Selmer classes for elliptic curves of rank 2.