TITLE: Lifting the p-torsion of a supersingular elliptic curve.
QUESTION [14 upvotes]: Let $K$ be a finite extension of $\mathbf{Q}_p$, with integer ring $R$ and residue field $k$. Say $G/R$ is a finite flat (commutative) group scheme of order $p^2$, killed by $p$. Say the special fibre of $G$ is isomorphic to the $p$-torsion in a supersingular elliptic curve over $k$. Is there some finite extension $L/K$ and an elliptic curve $E$ over $R_L$, the integers of $L$, such that $G$ becomes isomorphic to $E[p]$ over $R_L$?
The reason I ask is that in a lemma in Vincent Pilloni's thesis (which I unfortunately cannot find online) he proves that if the special fibre of $G$ is as above then the "degree" of such a $G$ is 1 (see Fargues' work on Harder-Narasimhan filtrations, for example, for the definition of "degree") via a brute force calculation. Pilloni's assertion would however also be a consequence of an affirmative answer to my question (and provided the motivation for my question). People are so good at deformation theory of finite flat group schemes nowadays that I thought this might be well-known to some people nowadays.

REPLY [5 votes]: This is a rough translation of Laurent F.'s answer (see this meta.MO thread). It's community wiki so you can improve the translation if you have 100 rep.

The answer is yes. The reason is as follows. If $S$ is a scheme where $p$ is locally nilpotent, then by definition (cf. Messing's thesis, Chapter I), a truncated Barsotti-Tate group of level of $1$ over $S$ is a finite locally free group scheme over $S$ annihilated by $p$ such that if $G_0$ denotes the reduction of $G$ modulo $p$ we have
$$
Im (F_{G_0}) = \ker (V_{G_0})
$$
as fppf sheaves on $S_0$, $S_0$ denoting the reduction modulo $p$ of $S$. However, it follows from the fiberwise criterion for flatness of EGA IV [Ed: he means EGA IV cor 11.3.11] that the morphism of $S_0$-group schemes of finite presentation
$$
F:G_0 \rightarrow \ker (V_{G_0})
$$
is faithfully flat if and only if it is fiberwise over $S_0$. From this we deduce that in the definition of a $BT_1$ one can replace $S_0$ by $S_{red}$! 
In the above considerations I took a scheme $S$ where $p$ is locally nilpotent, it follows that we have the same type of definition-result over a base which is a $p$-adic formal scheme.
We now return to the subject, namely the question of Kevin. We have that $G$ is a finite flat group scheme over the ring of integers of $K$ whose special fiber over the residue field of $K$ is a $BT_1$. The group scheme $G$ is hence a $BT_1$. [Ed: in fact this statement already implies Pilloni's result, by other results in the above cited article of Illusie.]
We will now use the following theorem (cf. the article of Illusie in Journees Arithmetiques de Rennes: "Déformations de groupes de Barsotti-Tate (d'après A. Grothendieck)", Astérisque 127). If $\mathcal{BT}_1$, resp. $\mathcal{BT}$ denotes the stack of truncated Barsotti-Tate groups of level $1$, resp. of Barsotti-Tate groups, over bases that are schemes in which $p$ is locally nilpotent, then the morphism "points of $p$-torsion"
$$
\mathcal{BT} \longrightarrow \mathcal{BT}_1
$$
is formally smooth. From this we deduce the following. Let $k$ be a perfect field of characteristic $p$ and $H$ a $p$-divisible group over $k$. Let $\mathfrak{X}$ be the space of deformations of $H$, a $spf (W(k))$-formal scheme non-canonically isomorphic to $spf \big(W(k)[[x_1,\dots,x_{d(h-d)}]]\big)$ where $h$ denotes the height of $H$ and $d$ dimension. Let $\mathcal{H}$ be the universal deformation of $\mathfrak{X}$. Then $\mathcal{H}[p]$ is a versal deformation of $H[p]$.
To conclude and answer the question of Kevin it suffices to invoke the theorem of Serre-Tate, which shows that if $H=E[p^\infty]$ where $E$ is a supersingular elliptic curve over $k$ then $\mathfrak{X}$ is the space of deformations of $E$. Applying the algebraization theorem of Grothendieck (GAGF) we deduce that if $\mathfrak{X}=spf(R)$, the universal deformation of the elliptic curve $E$ over $\mathfrak{X}$ actually becomes an elliptic curve over $spec(R)$. The result follows by specialization to $\mathcal{O}_K$.