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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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  • [1] A. Barquero-Sanchez and R. Masri, The Chowla-Selberg formula for abelian CM fields and Faltings heights, Compos. Math., to appear.
  • [2] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Computational algebra and number theory (London, 1993). MR 1484478, https://doi.org/10.1006/jsco.1996.0125
  • [3] Jan Hendrik Bruinier and Tonghai Yang, CM-values of Hilbert modular functions, Invent. Math. 163 (2006), no. 2, 229-288. MR 2207018 (2008b:11053), https://doi.org/10.1007/s00222-005-0459-7
  • [4] S. Chowla and A. Selberg, On Epstein's zeta function. I, Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 371-374. MR 0030997 (11,84d)
  • [5] Atle Selberg and S. Chowla, On Epstein's zeta-function, J. Reine Angew. Math. 227 (1967), 86-110. MR 0215797 (35 #6632)
  • [6] Christopher Deninger, On the analogue of the formula of Chowla and Selberg for real quadratic fields, J. Reine Angew. Math. 351 (1984), 171-191. MR 749681 (86f:11085), https://doi.org/10.1515/crll.1984.351.171
  • [7] Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR 1070716 (92e:11001)
  • [8] Kazuya Kato, Nobushige Kurokawa, and Takeshi Saito, Number theory. 2, Translations of Mathematical Monographs, vol. 240, American Mathematical Society, Providence, RI, 2011. Introduction to class field theory; Translated from the 1998 Japanese original by Masato Kuwata and Katsumi Nomizu; Iwanami Series in Modern Mathematics. MR 2817199 (2012f:11001)
  • [9] Serge Lang, Algebraic number theory, 2nd ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. MR 1282723 (95f:11085)
  • [10] M. Lerch, Sur quelques formules relatives au nombre des classes, Bull. Sci. Math. 21 (1897) 290-304.
  • [11] Blair K. Spearman and Kenneth S. Williams, The conductor of a cyclic quartic field using Gauss sums, Czechoslovak Math. J. 47(122) (1997), no. 3, 453-462. MR 1461424 (98d:11132), https://doi.org/10.1023/A:1022407300351
  • [12] W. A. Stein et al., Sage Mathematics Software (Version 6.7). The Sage Development Team, 2015. http://www.sagemath.org
  • [13] M. Streng, Complex multiplication of abelian surfaces, Ph.D. Thesis, Universiteit Leiden, 2010.

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Additional Information

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

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