Geometry and Transcendence of the Hexponential
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910434749054976 |
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| author | Schmieding, Scott Simon, Christopher-Lloyd |
| author_facet | Schmieding, Scott Simon, Christopher-Lloyd |
| contents | The modular group $\operatorname{PSL}_2(\mathbb{Z})$ acts on the upper-half plane $\mathbb{HP}$ with quotient the modular orbifold, uniformized by the function $\mathfrak{j} \colon \mathbb{HP}\to \mathbb{C}$. We first show that second derived subgroup $\operatorname{PSL}_2(\mathbb{Z})''$ corresponds to a $\mathbb{Z}^2\rtimes \mathbb{Z}/6$ Galois cover of the modular orbifold by a hexpunctured plane, uniformized by the hexponential map $\operatorname{hexp} \colon \mathbb{HP} \to \mathbb{C} \setminus (ω_0\mathbb{Z}[j])$, which is a primitive of $Cη^4$ where $ω_0\in i\mathbb{R}$ and $C\in \mathbb{R}$ are explicit constants and $η$ is Dedekind eta function. We describe the values of the cusp-compactification $\partial \operatorname{hexp}\colon \mathbb{QP}^1\to ω_0 \mathbb{Z}[j]$. After defining the radial-compactification $\operatorname{Shexp} \colon \mathscr{R} \to \mathbb{R}/(2π\mathbb{Z})$, we construct a simple section $\operatorname{InSh} \colon \mathbb{R}/(2π\mathbb{Z}) \to \mathscr{S} \bmod{\operatorname{PSL}_2(\mathbb{Z})'}$ where $\mathscr{S} \subset \mathbb{RP}^1$ is a set of numbers whose continued fraction expansions arise from Sturmian sequences, which contains the set $\mathscr{M}$ of Markov quadratic irrationals as those numbers arising from periodic Sturmian sequences. We will show that the values of $\operatorname{InSh}$ are either Markov quadratic irrationals or transcendental. Finally we provide a continued fraction expansion for $\operatorname{hexp}$, and discuss its monodromy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17628 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometry and Transcendence of the Hexponential Schmieding, Scott Simon, Christopher-Lloyd Number Theory Dynamical Systems 30F35, 11A55, 37B10, 11F06, 11F20, 11F30, 11L05, 11L07, 11J06 The modular group $\operatorname{PSL}_2(\mathbb{Z})$ acts on the upper-half plane $\mathbb{HP}$ with quotient the modular orbifold, uniformized by the function $\mathfrak{j} \colon \mathbb{HP}\to \mathbb{C}$. We first show that second derived subgroup $\operatorname{PSL}_2(\mathbb{Z})''$ corresponds to a $\mathbb{Z}^2\rtimes \mathbb{Z}/6$ Galois cover of the modular orbifold by a hexpunctured plane, uniformized by the hexponential map $\operatorname{hexp} \colon \mathbb{HP} \to \mathbb{C} \setminus (ω_0\mathbb{Z}[j])$, which is a primitive of $Cη^4$ where $ω_0\in i\mathbb{R}$ and $C\in \mathbb{R}$ are explicit constants and $η$ is Dedekind eta function. We describe the values of the cusp-compactification $\partial \operatorname{hexp}\colon \mathbb{QP}^1\to ω_0 \mathbb{Z}[j]$. After defining the radial-compactification $\operatorname{Shexp} \colon \mathscr{R} \to \mathbb{R}/(2π\mathbb{Z})$, we construct a simple section $\operatorname{InSh} \colon \mathbb{R}/(2π\mathbb{Z}) \to \mathscr{S} \bmod{\operatorname{PSL}_2(\mathbb{Z})'}$ where $\mathscr{S} \subset \mathbb{RP}^1$ is a set of numbers whose continued fraction expansions arise from Sturmian sequences, which contains the set $\mathscr{M}$ of Markov quadratic irrationals as those numbers arising from periodic Sturmian sequences. We will show that the values of $\operatorname{InSh}$ are either Markov quadratic irrationals or transcendental. Finally we provide a continued fraction expansion for $\operatorname{hexp}$, and discuss its monodromy. |
| title | Geometry and Transcendence of the Hexponential |
| topic | Number Theory Dynamical Systems 30F35, 11A55, 37B10, 11F06, 11F20, 11F30, 11L05, 11L07, 11J06 |
| url | https://arxiv.org/abs/2402.17628 |