On rigid analytic uniformizations of Jacobians of Shimura curves

Details On rigid analytic uniformizations of Jacobians of Shimura curves On rigid analytic uniformizations of Jacobians of Shimura curves

2009-10-18

arxiv.org/abs/0910.3391v4

Mathematics

Number Theory

Corrected a few typos. Final version, to appear in American Journal of Mathematics

Scientific paper

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.

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