p-Adically convergent loci in varieties arising from periodic continued fractions
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909999270199296 |
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| author | Capuano, Laura Mula, Marzio Terracini, Lea Veneziano, Francesco |
| author_facet | Capuano, Laura Mula, Marzio Terracini, Lea Veneziano, Francesco |
| contents | Inspired by several alternative definitions of continued fraction expansions for elements in $\mathbb Q_p$, we study $p$-adically convergent periodic continued fractions with partial quotients in $\mathbb Z[1/p]$. To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the $p$-adically convergent loci of these varieties, characterizing the zero and one-dimensional cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00739 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | p-Adically convergent loci in varieties arising from periodic continued fractions Capuano, Laura Mula, Marzio Terracini, Lea Veneziano, Francesco Number Theory 11J70, 11D88, !!Y65, 11D09 Inspired by several alternative definitions of continued fraction expansions for elements in $\mathbb Q_p$, we study $p$-adically convergent periodic continued fractions with partial quotients in $\mathbb Z[1/p]$. To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the $p$-adically convergent loci of these varieties, characterizing the zero and one-dimensional cases. |
| title | p-Adically convergent loci in varieties arising from periodic continued fractions |
| topic | Number Theory 11J70, 11D88, !!Y65, 11D09 |
| url | https://arxiv.org/abs/2402.00739 |