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?
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