The distribution of the number of points modulo an integer on elliptic curves over finite fields
نویسندگان
چکیده
Let Fq be a finite field and let b and N be integers. We study the probability that the number of points on a randomly chosen elliptic curve E over Fq equals b modulo N . We prove explicit formulas for the cases gcd(N, q) = 1 and N = char(Fq). In the former case, these formulas follow from a random matrix theorem for Frobenius acting on the Ntorsion part of E, obtained by applying density results due to Chebotarev to the modular covering X(N) → X(1). As an additional application to this theorem, we estimate the probability that a randomly chosen elliptic curve has a point of order precisely N .
منابع مشابه
Elliptic curves with a given number of points over finite fields
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