A note on elliptic curves over finite fields

@article{Rck1987ANO,
  title={A note on elliptic curves over finite fields},
  author={Hans-Georg R{\"u}ck},
  journal={Mathematics of Computation},
  year={1987},
  volume={49},
  pages={301-304}
}
Let E be an elliptic curve over a finite field k and let E(k) be the group of k-rational points on E. We evaluate all the possible groups E(k) where E runs through all the elliptic curves over a given fixed finite field k. 
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References

SHOWING 1-6 OF 6 REFERENCES
ABELIAN VARIETIES OVER FINITE FIELDS.
  • S. Lang
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1955
We shall generalize to Abelian varieties the well-known fact that an elliptic curve over a finite field always has a rational point (see Theorem 3). Our first theorem is purely algebraic and
Endomorphisms of abelian varieties over finite fields
Almost all of the general facts about abelian varieties which we use without comment or refer to as "well known" are due to WEIL, and the references for them are [12] and [3]. Let k be a field, k its
Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
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  • Variétés Abèliennes et Courbes Algébriques
  • 1948
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use
  • 1948