Effective multiplicative independence of 3 singular moduli
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866918524077735936 |
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| author | Bilu, Yuri Gun, Sanoli Tron, Emanuele |
| author_facet | Bilu, Yuri Gun, Sanoli Tron, Emanuele |
| contents | Pila and Tsimerman proved in 2017 that for every $k$ there exists at most finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular moduli with the property "$x_1, \ldots,x_k$ are multiplicatively dependent, but any proper subset of them is multiplicatively independent". The proof was non-effective, using Siegel's lower bound for the Class Number. In 2019 Riffaut obtained an effective version of this result for $k=2$. Moreover, he determined all the instances of $x^my^n\in \mathbb Q^\times$, where $x,y$ are distinct singular moduli and $m,n$ non-zero integers. In this article we obtain a similar result for $k=3$. We show that $x^my^nz^r\in \mathbb Q^\times$ (where $x,y,z$ are distinct singular moduli and $m,n,r$ non-zero integers) implies that the discriminants of $x,y,z$ do not exceed $10^{10}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_05183 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Effective multiplicative independence of 3 singular moduli Bilu, Yuri Gun, Sanoli Tron, Emanuele Number Theory Pila and Tsimerman proved in 2017 that for every $k$ there exists at most finitely many $k$-tuples $(x_1,\ldots, x_k)$ of distinct non-zero singular moduli with the property "$x_1, \ldots,x_k$ are multiplicatively dependent, but any proper subset of them is multiplicatively independent". The proof was non-effective, using Siegel's lower bound for the Class Number. In 2019 Riffaut obtained an effective version of this result for $k=2$. Moreover, he determined all the instances of $x^my^n\in \mathbb Q^\times$, where $x,y$ are distinct singular moduli and $m,n$ non-zero integers. In this article we obtain a similar result for $k=3$. We show that $x^my^nz^r\in \mathbb Q^\times$ (where $x,y,z$ are distinct singular moduli and $m,n,r$ non-zero integers) implies that the discriminants of $x,y,z$ do not exceed $10^{10}$. |
| title | Effective multiplicative independence of 3 singular moduli |
| topic | Number Theory |
| url | https://arxiv.org/abs/2207.05183 |