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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2311.06591 |
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| _version_ | 1866910542493384704 |
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| author | Akkarapakam, Sushmanth J. Morton, Patrick |
| author_facet | Akkarapakam, Sushmanth J. Morton, Patrick |
| contents | The periodic points of the algebraic function defined by the equation $g(x,y) = x^3(4y^2+2y+1)-y(y^2-y+1)$ are shown to be expressible in terms of Ramanujan's cubic continued fraction $c(τ)$ with arguments in an imaginary quadratic field in which the prime $3$ splits. If $w = (a+\sqrt{-d})/2$ lies in an order of conductor $f$ in $K$ and $9 \mid N_{K/\mathbb{Q}}(w)$, then one of these periodic points is $c(w/3)$, which is shown to generate the ring class field of conductor $2f$ over $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06591 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Ramanujan's cubic continued fraction Akkarapakam, Sushmanth J. Morton, Patrick Number Theory The periodic points of the algebraic function defined by the equation $g(x,y) = x^3(4y^2+2y+1)-y(y^2-y+1)$ are shown to be expressible in terms of Ramanujan's cubic continued fraction $c(τ)$ with arguments in an imaginary quadratic field in which the prime $3$ splits. If $w = (a+\sqrt{-d})/2$ lies in an order of conductor $f$ in $K$ and $9 \mid N_{K/\mathbb{Q}}(w)$, then one of these periodic points is $c(w/3)$, which is shown to generate the ring class field of conductor $2f$ over $K$. |
| title | On Ramanujan's cubic continued fraction |
| topic | Number Theory |
| url | https://arxiv.org/abs/2311.06591 |