On rigid analytic uniformizations of Jacobians of Shimura curves

• Autores: Matteo Longo, Victor Rotger Cerdà, Stefano Vigni
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 134, Nº 5, 2012, págs. 1197-1246
• Idioma: inglés
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• Resumen

  • 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.