Parametrizations of elliptic curves by Shimura curves and by classical modular curves.

Abstract

Fix an isogeny class of semistable elliptic curves over Q. The elements A of have a common conductor N, which is a square-free positive integer. Let D be a divisor of N which is the product of an even number of primes--i.e., the discriminant of an indefinite quaternion algebra over Q. To D we associate a certain Shimura curve X(0)D(N/D), whose Jacobian is isogenous to an abelian subvariety of J0(N). There is a unique A [symbol; see text] A in for which one has a nonconstant map piD : X(0)D(N/D) --> A whose pullback A --> Pic0(X(0)D(N/D)) is injective. The degree of piD is an integer deltaD which depends only on D (and the fixed isogeny class A). We investigate the behavior of deltaD as D varies.