Curves with infinitely many points of fixed degree
- Cite this article as:
- Frey, G. Israel J. Math. (1994) 85: 79. doi:10.1007/BF02758637
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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|K1 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.