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 F-p, we study the frequency to which a given group G arises as a group of points E (F-p). It is well known that the only permissible groups are of the form G(m,k) := Z/mZ x Z/mkZ. 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, C. David and E. Smith 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 <= k(A) and that the conjectured asymptotic for M(G(m,k)) holds for almost all groups G(m,k) when m <= k(1/4-epsilon). We also apply our methods to study the frequency to which a given integer N arises as a group order #E(F-p).