The lambda invariants at CM points

Abstract

In the paper, we show that $λ(z_1) -λ(z_2)$, $λ(z_1)$ and $1-λ(z_1)$ 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 $λ(\frac{d+\sqrt d}2)$, $1-λ(\frac{d+\sqrt d}2)$, and $λ(\frac{d_1+\sqrt{d_1}}2) -λ(\frac{d_2+\sqrt{d_2}}2)$, with the latter under the condition $(d_1, d_2)=1$. Finally, we use these results to show that $λ(\frac{d+\sqrt d}2)$ 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.