Special Seminar Group Theory/Geometry 

The modular group $\operatorname{PSL}_2(\mathbb{Z})$ acts on the upper-half plane $\mathbb{HP}$ with quotient the modular orbifold $\mathbb{M}$. We will recall why the second derived subgroup $\operatorname{PSL}_2(\mathbb{Z})''$ corresponds to a $\mathbb{Z}^2 \rtimes \mathbb{Z}/6$ Galois cover of $\mathbb{M}$ by a hexpunctured plane $\mathbb{M}''$, whose uniformization $\operatorname{hexp} \colon \mathbb{HP} \to \mathbb{M}''$ is given by integrating the fourth power of the Dedekind $\eta$ function. Our aim is to investigate the arithmetic of the Fourier coefficients $\psi(n)$ of $\eta^4$ and the transcendental properties of $\operatorname{hexp}$.

 First, we will describe its cusp-compactification $\operatorname{Shexp} \colon \mathbb{QP}^1 \to \omega_0 \mathbb{Z}[j]$ and its special values, refining the central values of certain twisted $L$-series associated to $\eta^4$ which converge conditionally. These can be interpreted as the correlations of $\psi(n)$ with circle rotations of rational angle.

 Then, we will find a partially defined radial-compactification $\operatorname{Shexp} \colon \mathbb{RP}^1 \to \mathbb{R}/(2\pi\mathbb{Z})$, and construct a ``simple'' section $\operatorname{InSh} \colon \mathbb{R}/(2\pi\mathbb{Z}) \to \mathbb{RP}^1$ whose values will consist of Sturmian numbers, namely those whose continued fraction expansions are Sturmian sequences over $\{1,2\}$, of which the periodic sequences correspond to the Markov quadratic irrationals. This yields the correlations of $\psi(n)$ with circle rotations of Stumian angle.

 Finally, we explain why Sturmian numbers are either Markov quadratic or transcendent, as a first manifestation of the transcendence of $\operatorname{Shexp}$. This is analog to Schneider's theorem about the algebraic values of the $J$-function at algebraic numbers. Moreover we give a continued fraction expansion for $\operatorname{hexp}$, relying on its representation by hypergeometric series, with monodromy group $\mathbb{Z}^2$.

This is joint work with Scott Schmieding.