Let $E/\mathbf{Q}\_p$ be an elliptic curve having split multiplicative reduction. Then Tate uniformization gives a surjective homomorphism of $p$-adic analytic groups $G\_m \to E$, with infinite cyclic kernel. Is there an analogue of this fact for $E$ having nonsplit multiplicative reduction, perhaps replacing Gm with a nonsplit torus? E.g., can one uniformize $E$ over the quadratic extension where the reduction splits, and then somehow descend?


(My intuition was as follows. Take $E/\mathbf{Q}\_p$ with nonsplit multiplicative reduction, and let $K/\mathbf{Q}\_p$ be quadratic so that $E$ becomes split semistable over $K$, and let $E'$ be the $K$-twist of $E$ (which has split multiplicative reduction). Then one has a short exact sequence


$$0 \to Z \to \mathbf{G}\_m \to E' \to 0$$


(where $Z$ is the constant analytic group of integers). Extending scalars to $K$ then applying Weil restriction of scalars, we get


$$0 \to X \to T \to A \to 0$$


where $X$ is an etale-locally-constant analytic group, $T$ is a torus, and $A$ is an abelian variety, each of rank $2$ in the appropriate sense. The latter short exact sequence contains the former short exact sequence as a sub (direct factor?); the quotient sequence should be something like


$0 \to Z' \to \mathbf{G}\_m' \to E \to 0$,


where ' still denotes twisting by $K/\mathbf{Q}\_p$. Since $Z'$ has trivial $\mathbf{Q}\_p$-points, then, one should have something like $\mathbf{G}\_m'(\mathbf{Q}\_p) = E(\mathbf{Q}\_p)$, modulo any descent used in forming the quotient.


Does this sound sensical?


If anyone has access to Google Wave and wants to discuss, I've set up a wave here:
<https://wave.google.com/wave/#restored:wave:googlewave.com!w%252BQCn6fZTuZ>



Tate uniformization of nonsplit semistable elliptic curves
A form of this is contained in Silverman, second book, Chapter V, Corollary 5.4. I guess
that the image of Gm' in E (at the level of Q\_p-points) may have index 2. 


