Algebraic integers as special values of modular units

Abstract

Let $\varphi(\tau)=\eta((\tau+1)/2)^2/\sqrt{2\pi}e^\frac{\pi i}{4}\eta(\tau+1)$ where $\eta(\tau)$ is the Dedekind eta-function. We show that if $\tau_0$ is an imaginary quadratic number with $\mathrm{Im}(\tau_0)>0$ and $m$ is an odd integer, then $\sqrt{m}\varphi(m\tau_0)/\varphi(\tau_0)$ is an algebraic integer dividing $\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and $\theta_K$ be an element of $K$ with $\mathrm{Im}(\theta_K)>0$ which generators the ring of integers of $K$ over $\mathbb{Z}$. We develop a sufficient condition of $m$ for $\sqrt{m}\varphi(m\theta_K)/\varphi(\theta_K)$ to become a unit.