Arithmetic exceptionality of Lattès maps

Let ${\mathbb{F}}_q$ denote a finite field of order $q$. A rational function $r(x)\in \mathbb{Q}(x)$ is said to be arithmetically exceptional if it induces a permutation on ${\mathbb{P}}^1({\mathbb{F}}_p)$ for infinitely many primes $p$. Based on some computational results, Odabaş conjectured that for each $k\in \mathbb{N}$, the $k$-th Lattès map attached to an elliptic curve $E\mathrm{/}\mathbb{Q}$ 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 $E\mathrm{/}\mathbb{Q}$ having complex multiplication by an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-11})\mathrm{.}$ On the other hand, we show that the conjecture becomes invalid if $E$ has CM by $\mathbb{Q}(\sqrt{-11})$ and $6\mid k$. Partial results for non-CM elliptic curves are also given.