Structure of $E(Q_p)$ for elliptic curves with anomalous reduction modulo $p$

Question

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:

1. 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?
2. Is there a name for this dichotomy in the literature?
3. 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?

Answer 1

Hi, this is only a partial answer to (3), that was too long to be a comment.

In a recent post of mine, Felipe Voloch pointed out a very useful tameness criterion proved by Gross ("A tameness criterion for galois representations..." Duke J. 61 (1990) on page 514).

In your case, $E$ is good ordinary at $p$ and the criterion for tameness applies. If your sequence $(\ast)$ splits, then $E(\mathbb{Q}_p)$ has a point of order $p$, and therefore $\operatorname{Gal}(\mathbb{Q}_p(E[p])/\mathbb{Q}_p)$ is diagonalizable (and, in fact, of the form $[\ast,0;0,1]$). It follows from Gross' criterion that $j(E)\equiv j_0 \bmod p^2$, where $j_0$ is the $j$-invariant of the canonical lift of the reduction of $E$.

Unfortunately, this is only a necessary condition, as $\operatorname{Gal}(\mathbb{Q}_p(E[p])/\mathbb{Q}_p)$ may be diagonalizable but without the lower right corner being trivial.

Answer 2

If you work modulo $p^2$, the sequence $0 \to p\mathbb{Z}/p^2 \to E(\mathbb{Z}/p^2) \to E(\mathbb{Z}/p) \to 0$, splits iff $E$ is a canonical lift (as in Álvaro's answer). This comes up in cryptography (!) in the Smart-Satoh-Araki attack on the ECDLP for anomalous curves. This point is discussed in the Satoh-Araki paper, I believe.