Nonvanishing of Castella’s CM Selmer classes

Establish the nonvanishing of Castella’s CM Selmer classes $\kappa_p\in\Sel(\mathbb{Q},V_pE)$ for at least one prime, or determine whether such nonvanishing occurs for CM elliptic curves over $\mathbb{Q}$ in the rank-two setting.

Background

The paper discusses several rank-two Selmer constructions that can imply finiteness of the p-primary Shafarevich–Tate group when a suitable class is nonzero. For CM curves over the rationals, Castella’s CM analogue produces classes κp\kappa_p with the relevant implication.

However, the paper states that nonvanishing has not been demonstrated at even a single prime, and that a related basis-producing result assumes the finiteness it is intended to help establish.

References

Castella's CM analogue constructs $\kappa_p \in \Sel(Q, V_pE)$ with the same implication Thm.~B, but its nonvanishing is not known at a single prime, and the statement there that produces a basis of the Selmer group assumes $(E/Q)[p\infty]$ finite Thm.~C.

— Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves  (2609.08431 - Banwait, 8 Sep 2026) in Section “Questions,” paragraph discussing rank-two Selmer constructions