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On rigid analytic uniformizations of Jacobians of Shimura curves
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 134, Number 5, October 2012
- pp. 1197-1246
- 10.1353/ajm.2012.0033
- Article
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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.