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

Fix an isogeny class 𝒜 of semistable elliptic curves over Q. The elements of 𝒜 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 J(0)(N). There is a unique A ∈ 𝒜 for which one has a nonconstant map π(D) : X(0)(D)(N/D) → A whose pullback A → Pic(0)(X(0)(D)(N/D)) is injective. The degree of π(D) is an integer δ(D) which depends only on D (and the fixed isogeny class 𝒜). We investigate the behavior of δ(D) as D varies.