On Rigid Analytic Uniformizations of Jacobians of Shimura Curves

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 Q at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of Čerednik 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 Q unconditional.