Curves with infinitely many points of fixed degree

Thed-th symmetric productC ^(d) of a curveC defined over a fieldK is closely related to the set of points ofC of degree ≤d. IfK is a number field, then a conjecture of Lang Hi] proved by Faltings Fa2] implies ifC ^(d) (K) is an infinite set, then there is aK-rational covering ofC → ℙ _|K ¹ of degree ≤2d. As an application one gets that for fixed fieldK and fixedd there are only finitely many primes ι such that the set of all elliptic curves defined over some extensionsL ofK with L∶K]≤d and withL-rational isogeny of degree ι is infinite.