The distribution of group structures on elliptic curves over finite prime fields
نویسندگان
چکیده
We determine the probability that a randomly chosen elliptic curve E/Fp over a randomly chosen prime field Fp has an `-primary part E(Fp)[`∞] isomorphic with a fixed abelian `-group H (`) α,β = Z/` × Z/`. We show that the probability agrees with the one predicted by a natural though unproven equidistribution hypothesis for Frobenius elements in GL(2,Z`). Probabilities for “|E(Fp)| divisible by n”, “E(Fp) cyclic” and expectations for the number of elements of precise order n in E(Fp) are derived, both for unbiased E/Fp and for E/Fp with p ≡ 1 (`). MSC 2000: 11 N 45, 11 G 20, 11 S 80
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