Let `E/Q_p`

be an elliptic curve having split multiplicative reduction. Then the 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/Q_p`

with nonsplit multiplicative reduction, and let `K/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 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 Gm' \to E \to 0`

,

where ' still denotes twisting by `K/Q_p`

. Since `Z'`

has trivial `Q_p`

-points, then, one should have something like `G_m'(Q_p) = E(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