The Chowla-Selberg formula for quartic Abelian CM fields

Abstract

We provide explicit evaluations of the Chowla-Selberg formula for quartic abelian CM fields due to Barquero-Sanchez and Masri. These identities relate values of a Hilbert modular function at CM points to values of Euler’s gamma function Γ \Gamma and an analogous function Γ 2 \Gamma _2 at rational numbers. Our work consists of two main parts. First, we implement an algorithm in SageMath to compute the CM points. Second, we exhibit families of quartic abelian CM fields for which the part of the formula involving values of Γ \Gamma and Γ 2 \Gamma _2 takes a particularly simple form.