# On Group Structures Realized by Elliptic Curves over a Fixed Finite Field

@article{Farashahi2012OnGS, title={On Group Structures Realized by Elliptic Curves over a Fixed Finite Field}, author={Reza Rezaeian Farashahi and Igor E. Shparlinski}, journal={Experimental Mathematics}, year={2012}, volume={21}, pages={1 - 10} }

We obtain explicit formulas for the number of nonisomorphic elliptic curves with a given group structure (considered as an abstract abelian group) and the number of distinct group structures of all elliptic curves over a finite field. We use these formulas to derive some asymptotic estimates and tight upper and lower bounds for various counting functions related to classification of elliptic curves according to their group structure. Finally, we present results of some numerical tests that…

## 3 Citations

On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields

- MathematicsExp. Math.
- 2012

Some of the results are rigorous and are based on recent advances in analytic number theory; some are conditional under certain widely believed conjectures; and others are purely heuristic in nature.

Elliptic curves over finite fields : number theoretic and cryptographic aspects

- Mathematics, Computer Science
- 2013

Several natural questions about elliptic curves are presented, mostly over finite fields, that have led to some interesting number theoretic questions and whose solutions require rather involved techniques from various area of number theory.

On Totally Split Primes in High-Degree Torsion Fields of Elliptic Curves

- MathematicsInternational Mathematics Research Notices
- 2021

Analogously to primes in arithmetic progressions to large moduli, we can study primes that are totally split in extensions of ${\mathbb {Q}}$ of high degree. Motivated by a question of Kowalski we…

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© Bulletin de la S. M. F., 1988, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord…