Arithmetic exceptionality of Latt\`{e}s maps

TL;DR

This paper investigates when Lattès maps derived from elliptic curves over rationals permute finite fields, proving a conjecture for CM curves with certain exceptions and providing partial results for non-CM cases.

Contribution

It proves Odabaş's conjecture for elliptic curves with complex multiplication by imaginary quadratic fields other than (-11), and shows the conjecture fails for CM by (-11) when 6 divides k.

Findings

Conjecture holds for most CM elliptic curves.

Counterexamples exist for CM by (-11) with 6 dividing k.

Partial results obtained for non-CM elliptic curves.

Abstract

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, Odaba\c{s} conjectured that for each $k\in \mathbb{N}$, the $k$-th Latt\`{e}s map attached to an elliptic curve $E/\mathbb{Q}$ 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 $E/\mathbb{Q}$ having complex multiplication by an imaginary quadratic field other than $\mathbb{Q}(\sqrt{-11}).$ 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.