Published online by Cambridge University Press: 20 November 2018
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
$E\left( {{\mathbb{F}}_{p}} \right)$. It is well known 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
$M\left( {{G}_{m,\,k}} \right)$ be the frequency to which the group
${{G}_{m,\,k}}$ arises in this way. Previously, C.David and E. Smith determined an asymptotic formula for
$M\left( {{G}_{m,\,k}} \right)$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for
$M\left( {{G}_{m,\,k}} \right)$, pointwise and on average. In particular, we show that
$M\left( {{G}_{m,\,k}} \right)$ is bounded above by a constant multiple of the expected quantity when
$m\,\le \,{{k}^{A}}$ and that the conjectured asymptotic for
$M\left( {{G}_{m,\,k}} \right)$ holds for almost all groups
${{G}_{m,\,k}}$ when
$m\,\le \,{{k}^{1/4-\in }}$. We also apply our methods to study the frequency to which a given integer
$N$ arises as a group order
$\#E\left( {{\mathbb{F}}_{p}} \right)$.
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