# Tate module of an elliptic curve is Hodge-Tate

Let $E$ be an elliptic curve over $\mathbb{Q}_p$, and let $T_p(E)$ be its $p$-adic Tate module. Is there a simple way of seeing that $V_p(E):= T_p(E)\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$ is a Hodge-Tate representation,i.e., $B_{HT}$ admissible? Or maybe equivalently, is there a direct way of computing the Sen's operator for $V_p(E)\otimes \mathbb{C}_p$ when considered as a $\mathbb{C}_p$ representation of $G_{\mathbb{Q}_p}$. To clarify what I mean by simple, I wanted to avoid using Tate's general result on p-divisible groups or the complete machinary of hodge decomposition for etale cohomology of abelian varieties.

• You should ask this on MathOverflow. – Mathmo123 Nov 22 '16 at 17:09

• That discussion only treats the special case of a semistable elliptic curve, where the question reduces to Kummer theory. In fact, a similar approach, using Serre--Tate theory, will work for ordinary good reduction elliptic curves. For elliptic curves with super singular reduction, though, whose $p$-divisible group is a height $2$ formal group, it is less obvious that one can find a direct treatment that is more efficient than following Tate's general approach. – tracing Feb 13 '17 at 17:19