  We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for
p-ordinary elliptic curves A over number fields K. It generalises the
one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K=Q and
the phenomenon of exceptional zeros. We discuss old and new theoretical
evidence towards our conjecture and in particular we fully prove it, under mild
conditions, in the following situation: K is imaginary quadratic, A=E_K is the
base-change to K of an elliptic curve over the rationals, and the rank of A is
either 0 or 1.
  The proof is naturally divided into a few cases. Some of them are deduced
from the purely cyclotomic case of elliptic curves over Q, which we obtain from
a refinement of recent work of Venerucci alongside the results of
Greenberg-Stevens, Perrin-Riou, and the author. The only genuinely
multi-variable case (rank 1, two exceptional zeros, three partial derivatives)
is newly established here. Its proof generalises to show that the
`almost-anticyclotomic' case of our conjecture is a consequence of conjectures
of Bertolini-Darmon on families of Heegner points, and of (partly conjectural)
p-adic Gross--Zagier and Waldspurger formulas in families.
