Counting cusps of subgroups of PSL 2(OK)

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Let K be a number field with r real places and s complex places, and let O K be the ring of integers of K. The quotient [ℍ 2] r x [ℍ 3] s/PSL 2(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 ℚ or an imaginary quadratic field and if i ∉ K, then PSL 2(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.

Original languageEnglish (US)
Pages (from-to)2387-2393
Number of pages7
JournalProceedings of the American Mathematical Society
Issue number7
StatePublished - Jul 2008
Externally publishedYes


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