On group structures realized by elliptic curves over a fixed finite field

TL;DR

This paper derives explicit formulas for counting elliptic curves over finite fields with specific group structures, providing asymptotic estimates, bounds, and numerical insights into their distribution.

Contribution

It introduces explicit formulas for the enumeration of elliptic curves by their group structures over finite fields, advancing classification methods.

Findings

Explicit formulas for counting non-isomorphic elliptic curves with given group structures

Asymptotic estimates and bounds for classification functions

Numerical results revealing distribution phenomena of group structures

Abstract

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for 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 accordingly to their group structure. Finally, we present results of some numerical tests which exhibit several interesting phenomena in the distribution of group structures. We pose getting an explanation to these as an open problem.