On the group orders of elliptic curves over finite fields
Everett W. Howe
Abstract
Given a prime power q, for every pair of positive integers m and n with m dividing the GCD of n and q-1, we construct a modular curve over Fq that parametrizes elliptic curves over Fq along with Fq-defined points P and Q of order m and n, respectively, with P and (n/m)Q having a given Weil pairing. Using these curves, we estimate the number of elliptic curves over Fq that have a given integer N dividing the number of their Fq-defined points.
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