ON THE GROUP ORDERS OF ELLIPTIC-CURVES OVER FINITE-FIELDS

Given a prime power q, for every pair of positive integers m and n with m\gcd(n, q-1) we construct a modular curve over F(q) that parametrizes elliptic curves over F(q) along with F(q)-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 F(q) that have a given integer N dividing the number of their F(q)-defined points.