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**Congruence subgroups of \(\text{PSL}(2,\mathbb{Z}\)) of genus less than or equal to 24.**
*(English)*
Zbl 1060.11021

The article under review reports on recent work in which the authors “have computed a complete list of congruence groups up to genus 24”. Since the full tables are too extensive to be included in the paper the authors provide only “tables containing a full list of the congruence groups up to genus 3”. The full tables are available online at http://www.uncg.edu/mat/faculty/pauli/congruence/.

The authors’ calculations are based upon two important inequalities – that of Cox and Parry relating the genus of a congruence subgroup to its level and that of Zograf and Kim and Sarnak relating the genus to the index.

This interesting paper has its genesis in a deep fact: there exist only finitely many congruence subgroups in \(\text{SL}(2,\mathbb{Z})\) of fixed genus. This theorem has resulted from extensive work by several mathematicians (most notably D. McQuillan, J. B. Dennin jun., J. Thompson and D. Cox/W. R. Parry) during the period 1965–1984, work inspired by the conjecture of Hans Rademacher for the case of genus 0: there exist only a finite number of congruence subgroups of genus 0. This was ultimately proved by J. B. Dennin jun. [Ill. J. Math. 16, 502–518 (1972; Zbl 0255.10029)], who settled the case of general genus as well [Proc. Am. Math. Soc. 51, 282–288 (1975; Zbl 0313.10022)]. Further proofs were given by J. G. Thompson [Proc. Symp. Pure Math. 37, 533–555 (1980; Zbl 0448.20044)] and D. A. Cox and W. R. Parry [J. Reine Angew. Math. 351, 66–112 (1984; Zbl 0531.10028)].

The authors’ calculations are based upon two important inequalities – that of Cox and Parry relating the genus of a congruence subgroup to its level and that of Zograf and Kim and Sarnak relating the genus to the index.

This interesting paper has its genesis in a deep fact: there exist only finitely many congruence subgroups in \(\text{SL}(2,\mathbb{Z})\) of fixed genus. This theorem has resulted from extensive work by several mathematicians (most notably D. McQuillan, J. B. Dennin jun., J. Thompson and D. Cox/W. R. Parry) during the period 1965–1984, work inspired by the conjecture of Hans Rademacher for the case of genus 0: there exist only a finite number of congruence subgroups of genus 0. This was ultimately proved by J. B. Dennin jun. [Ill. J. Math. 16, 502–518 (1972; Zbl 0255.10029)], who settled the case of general genus as well [Proc. Am. Math. Soc. 51, 282–288 (1975; Zbl 0313.10022)]. Further proofs were given by J. G. Thompson [Proc. Symp. Pure Math. 37, 533–555 (1980; Zbl 0448.20044)] and D. A. Cox and W. R. Parry [J. Reine Angew. Math. 351, 66–112 (1984; Zbl 0531.10028)].

Reviewer: Marvin I. Knopp (Philadelphia)

### MSC:

11F06 | Structure of modular groups and generalizations; arithmetic groups |

11-04 | Software, source code, etc. for problems pertaining to number theory |

11F22 | Relationship to Lie algebras and finite simple groups |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

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\textit{C. J. Cummins} and \textit{S. Pauli}, Exp. Math. 12, No. 2, 243--255 (2003; Zbl 1060.11021)

### Online Encyclopedia of Integer Sequences:

Number of congruence subgroups of PSL(2,Z) of genus n, up to conjugacy in PSL(2,Z).Total number of congruence subgroups of PSL(2,Z) of genus n.

Number of congruence subgroups of PSL(2,Z) of genus n, up to conjugacy in PGL(2,Z).

Maximal level of a congruence subgroup of PSL(2,Z) of genus n.

Maximal index of a congruence subgroup of PSL(2,Z) of genus n.

Total number of torsion-free congruence subgroups of PSL(2,Z) of genus n.

Number of torsion-free congruence subgroups of PSL(2,Z) of genus n, up to conjugacy in PSL(2,Z).

Number of torsion-free congruence subgroups of PSL(2,Z) of genus n, up to conjugacy in PGL(2,Z).

Maximal level of a torsion-free congruence subgroup of PSL(2,Z) of genus n.

Maximal index of a torsion-free congruence subgroup of PSL(2,Z) of genus n.

### References:

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[15] | DOI: 10.2307/2373033 · Zbl 0122.03703 · doi:10.2307/2373033 |

[16] | DOI: 10.1090/S0002-9939-1965-0181618-X · doi:10.1090/S0002-9939-1965-0181618-X |

[17] | Petersson H., J. Reine Angew. Math. 250 pp 182– (1971) |

[18] | DOI: 10.1090/S0002-9939-01-06176-7 · Zbl 0981.20038 · doi:10.1090/S0002-9939-01-06176-7 |

[19] | Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions (1971) · Zbl 0221.10029 |

[20] | Thompson J. G., Santa Cruz Conference on Finite Groups pp 533– (1980) |

[21] | Zograf P., J. Reine Angew. Math. 414 pp 113– (1991) |

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