Congruence subgroups associated to the Monster. (English) Zbl 1099.11021

Let \(\Delta\) denote the set of all genus zero subgroups \(G\) of \(\Lambda=\text{PSL}_2(\mathbb{R})\) for which \[ \Gamma_0(m) \leq G \leq N(\Gamma_0(m)), \] for some positive integer \(m\), where \(N(\Gamma_0(m))\) is the normalizer of \(\Gamma_0(m)\) in \(\Lambda\). (The genus of a (discrete) subgroup \(S\) of \(\Lambda\) is the genus of the Riemann surface \(\mathbb{H}^2/S\), where \(\mathbb{H}^2\) is the hyperbolic \(2\)-space.)
In this paper the authors determine all such \(m\) and, for each such \(m\), they list all the members of \(\Delta\). In the process they list also all those members of \(\Delta\) of width \(1\) at \(\infty\). The proofs involve computations using GAP.
Their results answer an earlier question of Conway and Norton from “moonshine theory”.
The authors discuss the overlap between their results and those of C. J. Cummins in “Congruence subgroups of groups commensurable with \(\text{PSL}_2(\mathbb{Z})\) of genus \(0\) and \(1\)” [Exp. Math. 13, No. 3, 361–382 (2004; Zbl 1099.11022)].


11F06 Structure of modular groups and generalizations; arithmetic groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
11F22 Relationship to Lie algebras and finite simple groups


Zbl 1099.11022
Full Text: DOI Euclid EuDML


[1] DOI: 10.1007/BF01359701 · Zbl 0177.34901
[2] DOI: 10.1017/S001708950000940X · Zbl 0709.20025
[3] Akbas M., London Mathematical Society Lecture Notes Series 165 pp 77– (1992)
[4] DOI: 10.1515/9783110893106.327
[5] DOI: 10.1112/blms/11.3.308 · Zbl 0424.20010
[6] Cummins C., Exp. Math. 13 (3) pp 361– (2004) · Zbl 1099.11022
[7] Cummins C., Exp. Math. 12 (2) pp 243– (2003)
[8] DOI: 10.1215/S0012-7094-93-07202-X · Zbl 0828.20016
[9] Harada K., Modular Functions, Modular Forms and Finite Groups (1987)
[10] DOI: 10.1007/BF01112194 · Zbl 0143.30601
[11] Kluit P. G., PhD diss., in: ”Hecke Operators on G*(N) and their Trace.” (1979)
[12] DOI: 10.1006/jabr.2001.8756 · Zbl 0987.11031
[13] DOI: 10.1017/S0308210500017352 · Zbl 0467.10018
[14] DOI: 10.1215/S0012-7094-01-11028-4 · Zbl 1012.11031
[15] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions. (1971) · Zbl 0221.10029
[16] Thompson J. G., Proc. Symp. Pure Math. AMS 37 pp 533– (1980)
[17] Zograf P., J. reine angew. Math. 414 pp 113– (1991)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.