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

Francesc Castella ; Ming-Lun Hsieh

Forum of Mathematics, Sigma, Tome 10 (2022)

Résumé

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.

Détail

DOI : 10.1017/fms.2021.85