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The Chowla-Selberg formula for quartic Abelian CM fields


Author: Robert Cass
Journal: Proc. Amer. Math. Soc. 144 (2016), 2753-2769
MSC (2010): Primary 11F41; Secondary 11R42
DOI: https://doi.org/10.1090/proc/12935
Published electronically: March 22, 2016
MathSciNet review: 3487212
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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 $ \Gamma $ and an analogous function $ \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 $ \Gamma $ and $ \Gamma _2$ takes a particularly simple form.


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Robert Cass
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: robert.cass@uky.edu

DOI: https://doi.org/10.1090/proc/12935
Received by editor(s): June 9, 2015
Published electronically: March 22, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society