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On the group orders of elliptic curves over finite fields

Everett W. Howe
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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 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.

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Everett W. Howe. 2001-10-24. On the group orders of elliptic curves over finite fields. https://arxiv.org/abs/math/0110262

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