The Lambda Invariants at CM Points

Abstract

Abstract In this paper, we show that $\lambda (z_1) -\lambda (z_2)$, $\lambda (z_1)$, and $1-\lambda (z_1)$ are all Borcherds products on $X(2) \times X(2)$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $\lambda (\frac{d+\sqrt d}2)$, $1-\lambda (\frac{d+\sqrt d}2)$, and $\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$, with the latter under the condition $(d_1, d_2)=1$. Finally, we use these results to show that $\lambda (\frac{d+\sqrt d}2)$ 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.