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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Counting cusps of subgroups of $ \mathrm{PSL}_2(\mathcal{O}_K)$


Author: Kathleen L. Petersen
Journal: Proc. Amer. Math. Soc. 136 (2008), 2387-2393
MSC (2000): Primary 11F23, 22E40, 11A07
Published electronically: March 14, 2008
MathSciNet review: 2390505
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Abstract: Let $ K$ be a number field with $ r$ real places and $ s$ complex places, and let $ \mathcal{O}_K$ be the ring of integers of $ K$. The quotient $ [\mathbb{H}^2]^r\times [\mathbb{H}^3]^s/\mathrm{PSL}_2(\mathcal{O}_K)$ has $ h_K$ cusps, where $ h_K$ is the class number of $ K$. We show that under the assumption of the generalized Riemann hypothesis that if $ K$ is not $ \mathbb{Q}$ or an imaginary quadratic field and if $ i \not \in K$, then $ \mathrm{PSL}_2(\mathcal{O}_K)$ has infinitely many maximal subgroups with $ h_K $ cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.


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Additional Information

Kathleen L. Petersen
Affiliation: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada
Email: petersen@mast.queensu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09262-9
PII: S 0002-9939(08)09262-9
Received by editor(s): June 5, 2006
Received by editor(s) in revised form: July 12, 2006, November 28, 2006, and June 11, 2007
Published electronically: March 14, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.