Colloquium Paper: Parametrizations of elliptic curves by Shimura curves and by classical modular curves

摘要

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 X0^D(N/D), whose Jacobian is isogenous to an abelian subvariety of J0(N). There is a unique A ∈ 𝒜 for which one has a nonconstant map πD : X0^D(N/D) → A whose pullback A → Pic⁰(X0^D(N/D)) is injective. The degree of πD is an integer δD which depends only on D (and the fixed isogeny class 𝒜). We investigate the behavior of δD as D varies.