# 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

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.

A Cohen–Lenstra phenomenon for elliptic curves

- MathematicsJ. Lond. Math. Soc.
- 2014

An asymptotic formula is obtained for counting the number of primes p for which the group of points modulo p is isomorphic to G under a certain conjecture concerning the distribution ofPrimes in short intervals.

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.

Abelian surfaces over finite fields with prescribed groups

- Mathematics
- 2013

Let A be an abelian surface over Fq , the field of q elements. The rational points on A/Fq form an abelian group A(Fq)≃Z/n1Z×Z/n1n2Z×Z/n1n2n3Z×Z/n1n2n3n4Z . We are interested in knowing which groups…

On the cyclicity of the rational points group of abelian varieties over finite fields

- Mathematics, Computer ScienceFinite Fields Their Appl.
- 2019

Poisson distribution of a prime counting function corresponding to elliptic curves

- Mathematics
- 2015

Let $E$ be an elliptic curve defined over rational field $\mathbb{Q}$ and $N$ be a positive integer. Now, $M_E(N)$ denotes the number of primes $p$, such that the group $E_p(\mathbb{F}_p)$ is of…

POINTS ON ELLIPTIC CURVE OVER FINITE FIELDS

- Mathematics
- 2015

We divide our study into two parts. In the first part we study the main topic of our interest, that is same as the title of this thesis. While in the second part we study a problem related to…

Mean-Value of Product of Shifted Multiplicative Functions and Average Number of Points on Elliptic Curves

- Mathematics
- 2014

Large sieve estimate for multivariate polynomial moduli and applications

- MathematicsMonatshefte für Mathematik
- 2021

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri–Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables…

On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions

- Mathematics
- 2022

. Let E be an elliptic curve deﬁned over Q and, for a prime p of good reduction for E let ˜ E p denote the reduction of E modulo p . Inspired by an elliptic curve analogue of Artin’s primitive root…

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