  The main goal of this article is to give an explicit rigid analytic
uniformization of the maximal toric quotient of the Jacobian of a Shimura curve
over the field of rational numbers at a prime dividing exactly the level. This
result can be viewed as complementary to the classical theorem of Cerednik and
Drinfeld which provides rigid analytic uniformizations at primes dividing the
discriminant. As a corollary, we offer a proof of a conjecture formulated by M.
Greenberg in his paper on Stark-Heegner points and quaternionic Shimura curves,
thus making Greenberg's construction of local points on elliptic curves over
the rationals unconditional.
