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.
KeywordsElliptic Curf Abelian Variety Number Field Minimal Covering Symmetric Product
Unable to display preview. Download preview PDF.
- [A] D. Abramovich, Letter, 1992.Google Scholar
- [Fa2] G. Faltings,The general case of S. Lang's conjecture, Preprint, Princeton University (1992).Google Scholar
- [F-P-S] G. Frey, M. Perret and H. Stichtenoth,On the different of Abelian extensions of global fields, inCoding Theory and Algebraic Geometry (H. Stichtenoth and M. Tsfasman, eds.), Proceedings AGCT3, Luminy June 1991, Lecture Notes in Mathematics1518, Springer, Heidelberg, 1992, pp. 26–32.CrossRefGoogle Scholar