On the group orders of elliptic curves over finite fields

2001/10/24 by Everett W. Howe
Mathematics · #math.NT #math.AG #msc:11G20 #msc:14G15 #msc:14H52

paper · pdf

published as Compositio Math. 85 (1993) 229--247 · 18 pages, LaTeX. This is a preprint version from 1992

arxiv created 2001/10/24 · arxiv updated 2009/11/30

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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