The Group Structure of the Normalizer of Γ₀(N) After Atkin–Lehner

Abstract

We determine the group structure of the normalizer of Γ₀(N) in SL ₂(ℝ) modulo Γ₀(N). These results correct the Atkin–Lehner statement (Atkin and Lehner, 1970, Theorem 8).

Key Words:

• Atkin-Lehner involutions
• Normalizer of modular groups

2000 Mathematics Subject Classification:

• 20H05
• 19B37
• 11G18

ACKNOWLEDGMENTS

Work partially supported by MTM2006-11391.

After the article were accepted we learned that a similar result was obtained by M. Akbas and D. Singerman in The normalizer of Γ₀(N) in PSL(2, R) with a different proof. See the postcript. We thank John McKay for drawing the references in the Postcript to our attention.

Notes

Communicated by J. L. Gomez Pardo.