On the elliptic curves modulo p

Abstract

We first prove Sun's three conjectures [Z.H. Sun, On the number of incongruent residues of x 4 +ax 2 +bx modulo p, J. Number Theory 119 (2006) 210-241; Z.H. Sun, http://sfb.hytc.edu.cn/xsjl/szh/, 2000, June] on the number of rational points of some elliptic curves over finite fields F p , which are related to the congruence cubic and quartic residue. And we provide some examples and comments concerning these conjectures.

On some congruence properties of elliptic curves

“…As indicated in Theorems 1 and 3 of [6] about this open question in the case (d, α) = (3, 0), the precise meaning of the KKP question may be explained as follows: For fixed integers d and α, can one find polynomials f (t), g(t) ∈ Z[t] in one variable t satisfying the following condition: For every integer k ∈ Z such that E : y 2 = x 3 + f (k)x + g(k) is an elliptic curve over the rational number field Q, one has E p (F p ) ≡ α (mod d) for a fixed integer d and for almost all primes p, where E p is the reduced curve of E at p.…”

“…In a recent paper [6], Kim, Koo, and Park presented a question about elliptic curves over finite fields as follows (in their original form): Open questions of Kim-Koo-Park: Let E p : y 2 = x 3 + f (k)x + g(k) be an elliptic curve over a finite field F p and α be a nonnegative integer. Can one find f (k), g(k) satisfying E p (F p ) ≡ α (mod d) for a fixed integer d and for almost all primes p?…”