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arXiv · 2603.25014

Arithmetic exceptionality of Latt\`{e}s maps

Chatchawan Panraksa·Detchat Samart·Songpon Sriwongsa
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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.

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Chatchawan Panraksa, Detchat Samart, Songpon Sriwongsa. 2026-03-26. Arithmetic exceptionality of Latt\`{e}s maps. https://arxiv.org/abs/2603.25014

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