Nondefinability results for elliptic and modular functions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909395161448448 |
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| author | McCulloch, Raymond |
| author_facet | McCulloch, Raymond |
| contents | Let $Ω$ be a complex lattice which does not have complex multiplication and $\wp=\wp_Ω$ the Weierstrass $\wp$-function associated to it. Let $D\subseteq\mathbb{C}$ be a disc and $I\subseteq\mathbb{R}$ be a bounded closed interval such that $I\capΩ=\emptyset$. Let $f:D\rightarrow\mathbb{C}$ be a function definable in $(\overline{\mathbb{R}},\wp|_I)$. We show that if $f$ is holomorphic on $D$ then $f$ is definable in $\overline{\mathbb{R}}$. The proof of this result is an adaptation of the proof of Bianconi for the $\mathbb{R}_{\exp}$ case. We also give a characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular $j$-function using similar methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_00613 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nondefinability results for elliptic and modular functions McCulloch, Raymond Logic 33E05, 03C64, 11F03 Let $Ω$ be a complex lattice which does not have complex multiplication and $\wp=\wp_Ω$ the Weierstrass $\wp$-function associated to it. Let $D\subseteq\mathbb{C}$ be a disc and $I\subseteq\mathbb{R}$ be a bounded closed interval such that $I\capΩ=\emptyset$. Let $f:D\rightarrow\mathbb{C}$ be a function definable in $(\overline{\mathbb{R}},\wp|_I)$. We show that if $f$ is holomorphic on $D$ then $f$ is definable in $\overline{\mathbb{R}}$. The proof of this result is an adaptation of the proof of Bianconi for the $\mathbb{R}_{\exp}$ case. We also give a characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular $j$-function using similar methods. |
| title | Nondefinability results for elliptic and modular functions |
| topic | Logic 33E05, 03C64, 11F03 |
| url | https://arxiv.org/abs/2307.00613 |