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- W4299847495 abstract "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 Nge1. (The abelian variety J_1(N) is the Jacobian of the standard modular" @default.
- W4299847495 created "2022-10-03" @default.
- W4299847495 creator A5029473852 @default.
- W4299847495 date "1992-08-06" @default.
- W4299847495 modified "2023-09-26" @default.
- W4299847495 title "Abelian varieties over Q and modular forms" @default.
- W4299847495 doi "https://doi.org/10.48550/arxiv.alg-geom/9208002" @default.
- W4299847495 hasPublicationYear "1992" @default.
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