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 $\Bbb{Q}$ at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of \v{C}erednik 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 $\Bbb{Q}$ unconditional.