Let p ≥ 5 be a prime and for a, b ∈ Fp, let Ea,b denote the elliptic curve over Fp with equation y 2 = x 3 + a x + b. As usual define the trace of Frobenius ap, a, b by #Ea,b(Fp) = p + 1 − ap, a, b. We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums P t∈Fp ap, t, b, P t∈Fp ap, a, t, Pp−1 t=0 a 2 p, t, b, Pp−1 t=0 a 2 p, a, t and Pp−1 t=0 a 3 p, t, b for primes p in various congruence classes. As an example of our results, we prove the following: Let p ≡ 5 (mod 6) be prime and let b ∈ F ∗ p. Then Xp−1 t=0 a 3 p, t, b = −p ( (p − 2) (−2 /p ) + 2p ) ( b /p ) .
Bulletin of the Australian Mathematics Society
Australian Mathematics Society
He, S., & McLaughlin, J. (2007). Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field. Bulletin of the Australian Mathematics Society, 75(1), 135-149. Retrieved from https://digitalcommons.wcupa.edu/math_facpub/58