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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.25014 |
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| _version_ | 1866911545237176320 |
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| author | Panraksa, Chatchawan Samart, Detchat Sriwongsa, Songpon |
| author_facet | Panraksa, Chatchawan Samart, Detchat Sriwongsa, Songpon |
| contents | 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 $E/\mathbb{Q}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25014 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Arithmetic exceptionality of Lattès maps Panraksa, Chatchawan Samart, Detchat Sriwongsa, Songpon Number Theory 11T06, 11G05, 11G15 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 $E/\mathbb{Q}$ 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. |
| title | Arithmetic exceptionality of Lattès maps |
| topic | Number Theory 11T06, 11G05, 11G15 |
| url | https://arxiv.org/abs/2603.25014 |