On a congruence property of elliptic curves

Let d, α ∈ Z with d > 1. In this paper, we proved the following results about congruence properties of elliptic curves: (1) For elliptic curve E over the rational number field Q, if ♯Ẽp(Fp) ≡ α (mod d) hold for almost all primes p, then almost all supersingular primes p of E satisfy p ≡ α − 1 (mod d). In particular, α−1 is prime to d. Moreover, if φ(d) > 2, then E does not have complex multiplication. (2) There exists an elliptic curve E over Q such that ♯Ẽp(Fp) ≡ 0 (mod d) for almost all primes p if and only if d = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16. Depending on these results, we obtain solutions in many cases for an open question (see the text) presented recently by Kim, Koo and Park, e.g., this question in all cases (d, α) satisfying either d | α or gcd (α− 1, d) > 1 are solved completely. Some further questions and conjectures on congruence properties of elliptic curves are also presented and discussed.