For simplicity, take $p\ge7$ a prime and $E/\mathbb{Q}$ an elliptic curve with good anomalous reduction at $p$, i.e., $|E(\mathbb{F}_p)|=p$. There is a standard exact sequence for the group of points over $\mathbb{Q}_p$, $$ 0 \to \hat{E}(p\mathbb{Z}_p) \to E(\mathbb{Q}_p) \to E(\mathbb{F}_p) \to 0. $$ The assumption that $p$ is anomalous implies that $E(\mathbb{F}_p)=\mathbb{Z}/p\mathbb{Z}$, while the formal group $\hat{E}(p\mathbb{Z}_p)$ is isomorphic to the formal group of the additive group, so we obtain an exact sequence $$ 0 \to p\mathbb{Z}_p^+ \to E(\mathbb{Q}_p) \to \mathbb{Z}/p\mathbb{Z} \to 0. \qquad(*) $$ In work I'm doing on elliptic pseudoprimes, the question of whether the sequence $(*)$ splits has become relevant. Questions:

- Are there places in the literature where the split-versus-nonsplit dichotomy for anomalous primes comes up, e.g., in the theory of $p$-adic modular forms?
- Is there a name for this dichotomy in the literature?
- Aside from the obvious observation that the sequence $(*)$ splits if and only if $E(\mathbb{Q}_p)$ has a $p$-torsion point, are there other natural necessary or sufficient conditions for splitting?