Shimura subgroups of 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) ∩...

Multiplicities of Galois Representations in Jacobians of Shimura Curves

Let p and q be distinct primes. The new part of Jo(pq) (defined, according to one’s taste, as a quotient or subvariety of Jo(pq)) is known to be isogenous to the Jacobian J of the Shimura curve derived from the rational quaternion algebra of discriminant pq. We show that J and the new subvariety of Jo(pq) differ in one significant respect. Namely, certain kernels which are 2-dimensional for the...

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 ...

Notes on Shimura curves

Singular Moduli of Shimura Curves

The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over Q. This paper shows that ...