2024/02/27 by Schmieding, Scott, Simon, Christopher-Lloyd
#11A55 #11F06 #11F20 #11F30 #11J06 #11L05 #11L07 #30F35 #37B10 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2402.17628
The modular group PSL2(ℤ) acts on the upper-half plane \mathbbHP with quotient the modular orbifold, uniformized by the function \mathfrakj \colon \mathbbHP→ ℂ. We first show that second derived subgroup PSL2(ℤ)'' corresponds to a ℤ2\rtimes ℤ/6 Galois cover of the modular orbifold by a hexpunctured plane, uniformized by the hexponential map hexp \colon \mathbbHP → ℂ ∖ (ω0ℤ[j]), which is a primitive of Cη4 where ω0∈ iℝ and C∈ ℝ are explicit constants and η is Dedekind eta function. We describe the values of the cusp-compactification ∂ hexp\colon \mathbbQP1→ ω0 ℤ[j]. After defining the radial-compactification Shexp \colon \mathscrR → ℝ/(2πℤ), we construct a simple section InSh \colon ℝ/(2πℤ) → \mathscrS \bmodPSL2(ℤ)' where \mathscrS ⊂ \mathbbRP1 is a set of numbers whose continued fraction expansions arise from Sturmian sequences, which contains the set \mathscrM of Markov quadratic irrationals as those numbers arising from periodic Sturmian sequences. We will show that the values of InSh are either Markov quadratic irrationals or transcendental. Finally we provide a continued fraction expansion for hexp, and discuss its monodromy.