On the nonvanishing of generalised Kato classes for elliptic curves of rank 2

Let E/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 ord(s=1)}L(E,s) >= 2 ). A construction of Darmon-Rotger attaches to E and an auxiliary weight 1 cuspidal eigenform g such that L(E, ad(0) (g),1) not equal 0 , a Selmer class kappa(p) is an element of Sel (Q,VpE), and they conjectured the equivalence kappa(p) not equal 0 double left right arrow dim(Qp) Sel (Q,VpE) = 2. 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.