Elliptic curves over finite fields with fibonacci numbers of points

For a prime power √q and√an elliptic curve E over Fq having q + 1 − a points, where a ∈ [−2 q, 2 q] let {#Em}m≥1 be the sequence of numbers whose mth term is the number of points of E over Fqm. In this paper, we determine all instances when #({#Em}m≥1 ∩ {Fn}n≥1 ) ≥ 2, where {Fn}n≥1 is the sequence of Fibonacci numbers. That is, we determine all six–tuples (a, q, m1, m2, n1, n2) such that #E = q + 1 − a, #Em1 = Fn1 and #Em2 = Fn2.