Bielliptic curves and symmetric products

We show that the twofold symmetric product of a nonhyperelliptic, nonbielliptic curve does not contain any elliptic curves. Applying a theorem of Faltings, we conclude that such a curve defined over a number field K K has only finitely many points over all quadratic extensions of K K . We illustrate our theory with the modular curves X 0 ( N ) , X 1 ( N ) , X ( N ) {X_0}(N),{X_1}(N),X(N) .