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Transcendental values of class group $ L$-functions, II


Authors: M. Ram Murty and V. Kumar Murty
Journal: Proc. Amer. Math. Soc. 140 (2012), 3041-3047
MSC (2010): Primary 11J81; Secondary 11M32
DOI: https://doi.org/10.1090/S0002-9939-2012-11201-8
Published electronically: January 30, 2012
MathSciNet review: 2917077
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Abstract: Let $ K$ be an imaginary quadratic field and $ \mathfrak{f}$ an integral ideal. Denote by $ Cl(\mathfrak{f})$ the ray class group of $ \mathfrak{f}$. For every non-trivial character $ \chi $ of $ Cl(\mathfrak{f})$, we show that $ L(1,\chi )/\pi $ is transcendental. If $ \mathfrak{f} = \overline {\mathfrak{f}}$, then complex conjugation acts on the character group of $ Cl(\mathfrak{f})$. Denoting by $ \widehat {Cl(f)}^+$ the orbits of the group of characters, we show that the values $ L(1,\chi )$ as $ \chi $ ranges over elements of $ \widehat {Cl(\mathfrak{f})}^+$ are linearly independent over $ \overline {\mathbb{Q}}$. We give applications of this result to the study of transcendental values of Petersson inner products and certain special values of Artin $ L$-series attached to dihedral extensions.


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Additional Information

M. Ram Murty
Affiliation: Department of Mathematics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada
Email: murty@mast.queensu.ca

V. Kumar Murty
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada
Email: murty@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9939-2012-11201-8
Keywords: Class group $L$-functions, transcendental values, Petersson inner products, Artin $L$-series.
Received by editor(s): July 23, 2010
Received by editor(s) in revised form: March 30, 2011
Published electronically: January 30, 2012
Additional Notes: The research of both authors was partially supported by NSERC Discovery grants.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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