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Creators/Authors contains: "Hsieh, Ming-Lun"

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  1. ; (, Forum of Mathematics, Sigma)
    Abstract 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. 
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