On maps between modular Jacobians and Jacobians of Shimura curves

On maps between modular Jacobians and Jacobians of Shimura curves

Fix a squarefree integer N , divisible by an even number of primes, and let Γ be a congruence subgroup of level M , where M is prime to N . For each D dividing N and divisible by an even number of primes, the Shimura curve X(Γ0(N/D)∩Γ ) associated to the indefinite quaternion algebra of discriminant D and Γ0(N/D) ∩ Γ -level structure is well defined, and we can consider its Jacobian J(Γ0(N/D) ∩...

On Relations between Jacobians of Certain Modular Curves

We confirm a conjecture of Merel describing a certain relation between the jacobians of various quotients of X(p) in terms of specific correspondences.

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 ...

Endomorphisms of Jacobians of Modular Curves

Endomorphisms of Jacobians of Modular Curves

Let XΓ = Γ\H∗ be the modular curve associated to a congruence subgroup Γ of level N with Γ1(N) ≤ Γ ≤ Γ0(N), and let X = XΓ,Q be its canonical model over Q. The main aim of this paper is to show that the endomorphism algebra End0Q(JX) of its Jacobian JX/Q is generated by the Hecke operators Tp, with p N , together with the “degeneracy operators” DM,d, D t M,d, for dM |N . This uses the fundament...