# On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields

@article{Banks2012OnGS, title={On Group Structures Realized by Elliptic Curves over Arbitrary Finite Fields}, author={William D. Banks and Francesco Pappalardi and Igor E. Shparlinski}, journal={Experimental Mathematics}, year={2012}, volume={21}, pages={11 - 25} }

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection that correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are rigorous and are based on recent advances in analytic number theory; some are conditional under certain widely believed conjectures; and others are…

## 26 Citations

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## References

SHOWING 1-10 OF 37 REFERENCES

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

- MathematicsExp. Math.
- 2012

These formulas are used 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.

### On the group orders of elliptic curves over finite fields

- Mathematics
- 1993

Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over F_q that parametrizes elliptic curves over F_q along with…

### A note on elliptic curves over finite fields

- Mathematics
- 1987

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…

### Abelian Varieties over ℂ

- Mathematics
- 1986

These lecture notes present, in outline, the theory of abelian varieties over the complex numbers. They focus mainly on the analytic side of the subject. In the first section we prove some basic…

### Elliptic Curves: Number Theory and Cryptography, Second Edition

- Mathematics, Computer Science
- 2008

This accessible book prepares readers to tackle more advanced problems in the field of elliptic curves, before moving on to interesting applications, such as cryptography, factoring, and primality testing.

### A heuristic asymptotic formula concerning the distribution of prime numbers

- Mathematics
- 1962

Suppose fi, f2, -*, fk are polynomials in one variable with all coefficients integral and leading coefficients positive, their degrees being hi, h2, **. , hk respectively. Suppose each of these…

### The Distribution of Prime Numbers

- MathematicsNature
- 1933

THIS interesting “Cambridge Tract” is concerned mainly with the behaviour, for large values of x, of the function n(x), which denotes the number of primes not exceeding x. The first chapter gives…

### Elliptic Curves: Number Theory and Cryptography

- Mathematics, Computer Science
- 2003

This book discusses Elliptic Curve Cryptography, a Cryptosystem Based on Factoring and its Applications, and some of the techniques used to develop such systems.

### Bombieri-Vinogradov Type Theorem for Sparse Sets of Moduli

- Mathematics
- 2006

In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of…