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.
 Award ID(s):
 1601946
 NSFPAR ID:
 10351906
 Editor(s):
 Balakrishnan, Jennifer; Elkies, Noam; Hassett, Brendan; Poonen, Bjorn; Sutherland, Andrew; Voight, John
 Date Published:
 Journal Name:
 Arithmetic Geometry, Number Theory, and Computation
 Page Range / eLocation ID:
 583  587
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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