CURVES WITH INFINITELY MANY POINTS OF FIXED DEGREE

The d-th symmetric product C-(d) of a curve C defined over a field K is closely related to the set of points of C of degree less than or equal to d. If K is a number field, then a conjecture of Lang [Hi] proved by Faltings [Fa2] implies if C-(d)(K) is an infinite set, then there is a K-rational covering of C --> (P1)(\K) of degree less than or equal to 2d. As an application one gets that for fixed field K and fixed d there are only finitely many primes l such that the set of all elliptic curves defined over some extensions L of K with [L : K] less than or equal to d and with L-rational isogeny of degree l is infinite.