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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.06456 |
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| _version_ | 1866918459767521280 |
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| author | Fowler, Guy |
| author_facet | Fowler, Guy |
| contents | We prove an André--Oort-type result for a family of hypersurfaces in $\mathbb{C}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for the class number. We prove that, for $m, n \in \mathbb{Z}_{>0}$, there exists an effective constant $c(m, n)>0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $Δ_1, \ldots, Δ_n$ are such that $a_1 x_1^m + \ldots + a_n x_n^m \in \mathbb{Q}$ for some $a_1, \ldots, a_n \in \mathbb{Q} \setminus \{0\}$ and $\# \{ Δ_i : \mathbb{Q}(\sqrt{Δ_i}) = K_*\} \leq 1$, then $\max_i \lvert Δ_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in \mathbb{Q}$ for some $a_1, a_2, a_3 \in \mathbb{Q} \setminus \{0\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06456 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some uniform effective results on André--Oort for sums of powers in $\mathbb{C}^n$ Fowler, Guy Number Theory 11G18, 14G35 We prove an André--Oort-type result for a family of hypersurfaces in $\mathbb{C}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for the class number. We prove that, for $m, n \in \mathbb{Z}_{>0}$, there exists an effective constant $c(m, n)>0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $Δ_1, \ldots, Δ_n$ are such that $a_1 x_1^m + \ldots + a_n x_n^m \in \mathbb{Q}$ for some $a_1, \ldots, a_n \in \mathbb{Q} \setminus \{0\}$ and $\# \{ Δ_i : \mathbb{Q}(\sqrt{Δ_i}) = K_*\} \leq 1$, then $\max_i \lvert Δ_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in \mathbb{Q}$ for some $a_1, a_2, a_3 \in \mathbb{Q} \setminus \{0\}$. |
| title | Some uniform effective results on André--Oort for sums of powers in $\mathbb{C}^n$ |
| topic | Number Theory 11G18, 14G35 |
| url | https://arxiv.org/abs/2405.06456 |