Arithmetic exceptionality of Lattès maps

Let F q denote a finite field of order q . A rational function r ( x ) ∈ Q ( x ) is said to be arithmetically exceptional if it induces a permutation on P 1 ( F p ) for infinitely many primes p . Based on some computational results, Odabaş conjectured that for each k ∈ N , the k -th Lattès map attached to an elliptic curve E / Q is arithmetically exceptional if and only if E has no k -torsion point whose x -coordinate is rational. In this paper, we prove that this conjecture is true for any elliptic curve E / Q having complex multiplication (CM) by an imaginary quadratic field other than Q ( − 11 ) . On the other hand, we show that the conjecture becomes invalid if E has CM by Q ( − 11 ) and 6 | k . Partial results for non-CM elliptic curves are also given.