Algebraic integers as special values of modular units

Abstract

Let $\phi (\tau )=\eta ((\tau +1)/2{)}^2/\sqrt{2\pi }{e}^{\frac{\pi i}{4}}\eta (\tau +1)$ 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 $\sqrt{m}\phi (m{\tau }_0)/\phi ({\tau }_0)$ 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 $\sqrt{m}\phi (m{\theta }_K)/\phi ({\theta }_K)$ to become a unit.