  This paper gives a conjectural characterization of those elliptic curves over
the field of complex numbers which "should" be covered by standard modular
curves. The elliptic curves in question all have algebraic j-invariant, so they
can be viewed as curves over Q-bar, the field of algebraic numbers. The
condition that they satisfy is that they must be isogenous to all their Galois
conjugates. Borrowing a term from B.H. Gross, "Arithmetic on elliptic curves
with complex multiplication," we say that the elliptic curves in question are
"Q-curves." Since all complex multiplication elliptic curves are Q-curves (with
this definition), and since they are all uniformized by modular forms
(Shimura), we consider only non-CM curves for the remainder of this abstract.
We prove:
  1. Let C be an elliptic curve over Q-bar. Then C is a Q-curve if and only if
C is a Q-bar simple factor of an abelian variety A over Q whose algebra of
Q-endomorphisms is a number field of degree dim(A). (We say that abelian
varieties A/Q with this property are of "GL(2) type.")
  2. Suppose that Serre's conjecture on mod p modular forms are correct (Ref:
Duke Journal, 1987). Then an abelian variety A over Q is of GL(2)-type if and
only if it is a simple factor (over Q) of the Jacobian J_1(N) for some integer
N\ge1. (The abelian variety J_1(N) is the Jacobian of the standard modular
