Israel Journal of Mathematics

, Volume 85, Issue 1–3, pp 79–83 | Cite as

Curves with infinitely many points of fixed degree

  • Gerhard Frey


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 [LK]≤d and withL-rational isogeny of degree ι is infinite.


Elliptic Curf Abelian Variety Number Field Minimal Covering Symmetric Product 
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Copyright information

© Hebrew University 1994

Authors and Affiliations

  1. 1.Institut für Experimentelle MathematikUniversität GH EssenEssen 12Germany

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