The lambda invariants at CM points

Abstract

In the paper, we show that $\hat{I}»(z_1)-\hat{I}»(z_2)$, $\hat{I}»(z_1)$ and $1-\hat{I}»(z_1)$ are all Borcherds products in $X(2)\times X(2)$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $\hat{I}»(\frac{d+\sqrt{d}}{2})$, $1-\hat{I}»(\frac{d+\sqrt{d}}{2})$, and $\hat{I}»(\frac{d_1+\sqrt{d_1}}{2})-\hat{I}»(\frac{d_2+\sqrt{d_2}}{2})$, with the latter under the condition $(d_1,d_2)=1$. Finally, we use these results to show that $\hat{I}»(\frac{d+\sqrt{d}}{2})$ is always an algebraic integer and can be easily used to construct units in the ray class field of $\mathbb{Q}(\sqrt{d})$ of modulus $2$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.