1. On Rademacher's Conjecture : Congruence Subgroups of Genus Zero of the Modular Group
2. Congruence Subgroups Associated to the Monster

Congruence Subgroups of Genus Zero associated to the Monster Simple Group


Let m be a positive integer and let h be the largest divisor of 24 such that m = nh^2. By results of Conway and Norton (Monstrous Moonshine, Bull. London Math Soc, 11 (1979)), the normaliser of Gamma_0(m) is given by N(Gamma_0(m)) = Gamma_0^+(nh|h) = [[1/h,0],[0,1]]Gamma_0^+(n) [[h,0],[0,1]], where Gamma_0^+(n) is the group generated by Gamma_0(n) and all its Atkin-Lehner involutions.

A complete list of all intermediate subgroups of Gamma_0(m) < N(Gamma_0(m)) of genus zero is given in the files attached below. A brief explanation of the notations used in the files can be found in the following.

(i) Intermediate groups of genus 0 of Gamma_0(m) < N(Gamma_0(m)) = Gamma_0^+(nh|h) form t conjugacy classes. They are listed as n.h.1, n.h.2, ... , n.h.t. Suppose that the r-th conjugacy class possesses k groups. Then these groups are listed as n.h.r.1, n.h.r.2, ... , n.h.r.k.

(ii) Class with * (n.h.r*) in $\Gamma_0^+(nh|h)$ possesses a single member only. As a consequence, this group is normal in Gamma_0^+(nh|h). Take 1.2.1* for example, the * means that the class 1.2.1 possesses a single member 1.2.1.1 only and the group 1.2.1.1 is a normal subgroup of Gamma_0^+(2|2).

(iii) "Idx" gives the index of G = n.h.r.l in N(Gamma_0(m)). "W" gives the width of infinity. Note that the width is not invariant under conjugation.

(iv) "Generators" gives a set of generators of G modulo Gamma_0(m). The tuple [a,b,c,d] represents the matrix [[a,b],[c,d]]/(ad-bc)^{1/2}, where [a,b] is the first row, [c,d] is the second, and (ad-bc)^{1/2} is the square root of ad-bc.

Further, a complete list of all intermediate subgroups of Gamma_0(m) < N(Gamma_0(m)) of genus zero, width 1 (at infinity) can also be found in "List of groups of width 1 (at infinity)". In the case "n.h.r.k" does not appear in our list, it means that the r-class possesses no groups of width 1 (at infinity).

List of groups for n between 1-2
List of groups for n between 3-4
List of groups for n between 5-8
List of groups for n between 9-16
List of groups for n between 17-38
List of groups for n between 39-119
List of groups of width 1 at infinity

Number of intermediate subgroups of genus zero

The table below gives the number of conjugacy classes of intermediate subgroups of genus zero. The entry 52(177,25) for n = 2, h = 12 (m = 2 .12^2 = 288) means that there are 52 conjugacy classes of intermediate subgroups of Gamma_0(288) < N(Gamma_0(288)) of genus zero and there are altogether 177 intermediate subgroups (25 of them have width 1 at infinity). The remaining entries can be read similarly.



n|h 1 2 3 4 6 8 12 24
1 1(1,1) 4(6,4) 5(10,8) 11(30,21) 20(84,56) 17(70,32) 34(166,42) 41(209,2)
2 2(2,2) 8(10,7) 11(30,26) 20(42,27) 34(119,67) 25(60,16) 52(177,25) 57(195,0)
3 2(2,2) 10(16,12) 7(12,9) 21(53,31) 24(55,22) 24(68,10) 37(99,6) 40(114,0)
4 ** ** ** ** ** 18(30,10) ** 32(65,1)
5 2(2,2) 6(12,8) 6(13,10) 11(29,14) 12(30,6) 12(32,2) 17(47,0) 18(50,0)
6 5(5,5) 19(27,19) 12(30,21) 32(59,27) 32(66,14) 35(69,10) 48(111,13) 51(121,0)
7 2(2,2) 7(9,6) 4(6,4) 8(12,2) 11(18,4) 8(12,0) 12(21,0) 12(21,0)
8 ** ** ** ** ** 7(9,1) ** 8(13,0)
9 ** ** 5(7,5) ** 10(18,1) ** 11(21,0) 11(21,0)
10 5(5,5) 14(16,10) 8(12,7) 19(24,8) 20(30,7) 19(24,0) 25(38,0) 25(38,0)
11 1(1,1) 3(5,3) 2(5,3) 5(12,6) 4(9,0) 5(12,0) 6(16,0) 6(16,0)
12 ** ** ** ** ** 17(21,0) ** 20(25,0)
13 2(2,2) 3(3,1) 3(3,1) 3(3,0) 4(4,0) 3(3,0) 4(4,0) 4(4,0)
14 3(3,3) 8(10,6) 6(10,7) 11(15,5) 11(17,0) 11(15,0) 14(22,0) 14(22,0)
15 3(3,3) 9(15,10) 5(7,4) 10(18,2) 11(19,0) 10(18,0) 12(22,0) 12(22,0)
16 ** ** ** ** ** 3(3,0) ** 3(3,0)
17 1(1,1) 2(2,1) 1(1,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
18 ** ** 7(9,4) ** 10(14,0) ** 10(14,0) 10(14,0)
19 1(1,1) 2(2,1) 2(2,1) 2(2,0) 3(3,0) 2(2,0) 3(3,0) 3(3,0)
20 ** ** ** ** ** 7(7,0) ** 7(7,0)
21 3(3,3) 6(6,3) 4(4,1) 6(6,0) 8(8,1) 6(6,0) 8(8,0) 8(8,0)
22 2(2,2) 5(5,3) 2(2,0) 5(5,0) 5(5,0) 5(5,0) 5(5,0) 5(5,0)
23 1(1,1) 2(4,2) 1(1,0) 2(4,0) 2(4,0) 2(4,0) 2(4,0) 2(4,0)
24 ** ** ** ** ** 4(4,0) ** 4(4,0)
25 2(2,2) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
26 2(2,2) 5(5,3) 2(5,0) 6(6,1) 5(5,0) 6(6,0) 6(6,0) 6(6,0)
27 ** ** 1(1,0) ** 1(1,0) ** 1(1,0) 1(1,0)
29 1(1,1) 2(2,1) 1(1,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
30 6(6,6) 11(13,6) 8(10,4) 11(13,0) 13(17,0) 11(13,0) 13(17,0) 13(17,0)
31 1(1,1) 1(1,0) 2(2,1) 1(1,0) 2(2,0) 1(1,0) 2(2,0) 2(2,0)
32 ** ** ** ** ** 1(1,0) ** 1(1,0)
33 2(2,2) 3(3,1) 2(2,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0)
34 1(1,1) 2(2,1) 1(1,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
35 2(2,2) 3(3,1) 2(2,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0)
36 ** ** ** ** ** ** ** 4(4,0)
38 1(1,1) 2(2,1) 1(1,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
39 2(2,2) 2(2,0) 3(3,1) 2(2,0) 3(3,0) 2(2,0) 3(3,0) 3(3,0)
41 1(1,1) 2(2,1) 1(1,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
42 3(3,3) 6(6,3) 3(3,0) 6(6,0) 6(6,0) 6(6,0) 6(6,0) 6(6,0)
44 ** ** ** ** ** 2(2,0) ** 2(2,0)
45 ** ** 1(1,0) ** 1(1,0) ** 1(1,0) 1(1,0)
46 2(2,2) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
47 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
49 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
50 2(2,2) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
51 1(1,1) 2(2,1) 1(1,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
54 ** ** 1(1,0) ** 1(1,0) ** 1(1,0) 1(1,0)
55 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
56 ** ** ** ** ** 1(1,0) ** 1(1,0)
59 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
60 ** ** ** ** ** 4(4,0) ** 4(4,0)
62 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
66 2(2,2) 3(3,1) 2(2,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0)
69 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
70 2(2,2) 3(3,1) 2(2,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0) 3(3,0)
71 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
78 2(2,2) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0) 2(2,0)
87 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
92 ** ** ** ** ** 1(1,0) ** 1(1,0)
94 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
95 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
105 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
110 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)
119 1(1,1) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0) 1(1,0)


Kok Seng Chua and Mong Lung Lang computed with GAP 4.3

 

Lang Mong Lung
Department of Mathematics
National University of Singapore
Singapore 117543
Republic of Singapore

matlml@math.nus.edu.sg


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Last Modified: Nov. 13 2003