The lambda invariants at CM points

TL;DR

This paper investigates the properties of lambda invariants at CM points, showing they are Borcherds products, deriving explicit factorization formulas for their norms, and demonstrating their role in constructing units in ray class fields.

Contribution

It introduces new explicit formulas for lambda invariants at CM points and connects them to Borcherds products and class field theory, providing tools for algebraic and number-theoretic applications.

Findings

Lambda differences and values are Borcherds products.

Explicit factorization formulas for norms of lambda invariants.

Lambda invariants are algebraic integers used to construct units in ray class fields.

Abstract

In the paper, we show that $\lambda(z_1) -\lambda(z_2)$, $\lambda(z_1)$ and $1-\lambda(z_1)$ 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 $\lambda(\frac{d+\sqrt d}2)$, $1-\lambda(\frac{d+\sqrt d}2)$, 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.