Transcendental values of class group -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

Published electronically:
January 30, 2012

MathSciNet review:
2917077

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an imaginary quadratic field and an integral ideal. Denote by the ray class group of . For every non-trivial character of , we show that is transcendental. If , then complex conjugation acts on the character group of . Denoting by the orbits of the group of characters, we show that the values as ranges over elements of are linearly independent over . We give applications of this result to the study of transcendental values of Petersson inner products and certain special values of Artin -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:
http://dx.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.