
The Frequency of Elliptic Curve Groups Over Prime Finite Fields
Letting $p$ vary over all primes and $E$ vary over all elliptic
curves over the finite field $\mathbb{F}_p$, we study the frequency to
which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$.
It is wellknown that the only permissible groups are of the
form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$.
Given such a candidate group, we let $M(G_{m,k})$ be the frequency
to which the group $G_{m,k}$ arises in this way.
Previously, the second and fourth named authors determined an
asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes
in short arithmetic progressions. In this paper, we prove several
unconditional bounds for $M(G_{m,k})$, pointwise and on average. In
particular, we show that $M(G_{m,k})$ is bounded above by a constant
multiple of the expected quantity when $m\le k^A$ and that the
conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups
$G_{m,k}$ when $m\le k^{1/4\epsilon}$.
We also apply our methods to study the frequency to which a given
integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.
Keywords:average order, elliptic curves, primes in short intervals Categories:11G07, 11N45, 11N13, 11N36 