5 research outputs found
The frequency of elliptic curve groups over prime finite fields
Letting vary over all primes and vary over all elliptic curves over
the finite field , we study the frequency to which a given group
arises as a group of points . It is well-known that the
only permissible groups are of the form . Given such a candidate group, we let be
the frequency to which the group arises in this way. Previously, the
second and fourth named authors determined an asymptotic formula for
assuming a conjecture about primes in short arithmetic
progressions. In this paper, we prove several unconditional bounds for
, pointwise and on average. In particular, we show that
is bounded above by a constant multiple of the expected quantity
when and that the conjectured asymptotic for holds for
almost all groups when . We also apply our
methods to study the frequency to which a given integer arises as the group
order .Comment: 40 pages, with an appendix by Chantal David, Greg Martin and Ethan
Smith. Final version, to appear in the Canad. J. Math. Major reorganization
of the paper, with the addition of a new section, where the main results are
summarized and explaine
Elliptic curves with a given number of points over finite fields
Given an elliptic curve and a positive integer , we consider the
problem of counting the number of primes for which the reduction of
modulo possesses exactly points over . On average (over a
family of elliptic curves), we show bounds that are significantly better than
what is trivially obtained by the Hasse bound. Under some additional
hypotheses, including a conjecture concerning the short interval distribution
of primes in arithmetic progressions, we obtain an asymptotic formula for the
average.Comment: A mistake was discovered in the derivation of the product formula for
K(N). The included corrigendum corrects this mistake. All page numbers in the
corrigendum refer to the journal version of the manuscrip