modular functions arising from some finite groups...mathematics of computation volume 37, number 156...

MATHEMATICS OF COMPUTATIONVOLUME 37, NUMBER 156OCTOBER 1981

Modular Functions Arising From Some Finite Groups

By Larissa Queen

Abstract In [2] Conway and Norton have assigned a "Thompson series" of the form

q~x + Htq + Hjq2 + ...

to each element m of the Fischer-Griess "Monster" group M and conjectured that these

functions are Hauptmoduls for certain genus-zero modular groups. We have found for a

large number of values of N, all the genus-zero groups between r0(jV) and PSH.2, R) that

have Hauptmoduls of the above form, and this provides the necessary verification that the

series assigned in [2] to particular elements of M really are such Hauptmoduls. (Atkin and

Fong [1] have recently verified that H„(m) really is a character of M for all n.) We compute

Thompson series for various finite groups and discuss the differences between these groups

and M. We find that the resulting Thompson series are all Hauptmoduls for suitable

genus-zero subgroups of PSLQ., R).

1. Summary. Some remarkable connections between the Fischer-Griess "Mon-

ster" group M and modular functions have recently been reported in [2] and [5]. It

has been noticed that the coefficients in the ^-series fory

00

jiz)= 2 c„q" = q~x + 744+ 196884? + 21493760?2 + ...,n--l

where j is the fundamental modular function on T = PSL(2, Z) and q = e2mz, are

linear combinations of the character degrees dn of M. By considering J = j — 744,

i.e., disregarding the constant term, and replacing the coefficients c„ in the ^-series

for J by the corresponding representations of M, one obtains a formal power series

[5]

H_xq~x +0 + Hxq + H2q2 + H3q3 + ...,

where q = e2mz and Hn is a representation of degree c„ known as a head represen-

tation. Replacing H„ by its character value Hn(m) for various elements m e M, we

obtain other functions [5]:

Tm(z) - q~x + Hx(m)q + H2im)q2 + H3(m)q3 + ...,

which are called the Thompson series in [2]. The H¡im) are called head characters.

Conway and Norton [2] have computed the functions Fm(z) for all elements m of

M, and they conjectured that, for every m £ M, Tmi¿) is a Hauptmodul for a

group F(m) between T^N) for some N and its normalizer in PSLÍ2, R), i.e., Fm(z)

generates a genus-zero function field invariant under Firn). In this paper we

describe certain related calculations. The detailed working can be found in [4]. We

have explicitly verified that the modular functions assigned to various m by

Received June 16, 1980.1980 Mathematics Subject Classification. Primary 20C15, 10D05.

547

©1981 American Mathematical Society

0025-5718/81/OO00-O172/$O9.5O

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

548 LARISSA QUEEN

Conway and Norton [2] are actually Hauptmoduls for the groups they mention.

Atkin and Fong [1] have recently verified that Hn(m) really is a character of M for

all«.

We have also considered other finite groups, usually derived from centralizers of

elements of M, and computed their head character tables. In particular, we

consider

.0 = 2.(.l)

E

3.2.SUZ

2.HJ

F

2.A7

H

MX2

for which we have

.1 is the Conway simple group

Thompson's simple group

Suz is Suzuki's sporadic simple group

HJ is the Hall-Janko simple group

Harada-Norton simple group

Schur double cover of A7

Held's simple group

Mathieu simple group

elements of M

2B

35

3C

5A

5B

1A

IB

UA

centralizer in M

21+24-G

31 + 12-G

3 X £

5 X F

51+6-G

7X H

71+4-G

G = .l

G = 2.Suz

G = 2.HJ

G - 2.A7

11 x M12

2. Notation and Terminology. As usual, for a positive integer JV, we define

r(*0 ={(a b)&T:a=d=l (mod N), b = c = 0 (mod N)\

r0(JV)={(" *)er:c = 0(modiV)}.

and

We also define

which is a subgroup of PSL(2, R) and is conjugate to ro(/t). In [2] this group is

denoted by T0inh\h); however, we prefer to reserve this name to denote a subgroup

of ro(n: h) of index h which has Hauptmodul T^h (see below). Thus, adapting the

rest of the notation from [2], we always have the same name for the genus-zero

subgroup of PSLÍ2, R) and the corresponding Hauptmodul in the canonical form

(i.e. beginning q ~x + 0 + aq + . . .). Also, this notation provides a natural way of

enumerating all discrete subgroups of PSL(2, R) containing T0(N) for a given N;

see [4].


MODULAR FUNCTIONS ARISING FROM SOME FINITE GROUPS 549

Thus, we write

r0(n: h) + e,f,.. . for <ro(n: h), we, wf, . . . ),

r0(/t: h) + when all we for T0(/j: h) are present,

r0(n: /i) — or ro(n: h) when no wc ̂ 1 is present,

where

w. = I ( ' \: e divides n exactly, and ade2 — ben — e > 0• Wchn de I J

is a single coset of r0(/i: h). The we are called the Atkin-Lehner involutions for

ro(n: h) [2]. The corresponding Hauptmoduls (when they exist) are denoted by

tn+ej,...C17)' tn+(^lz) and '«-C^) or 'nC^X respectively. Tr/w) and its Hauptmodul

t„(z) are a particular case when h = 1.

r.+„¿...(**) = tH+t/i...(hz)- constant term

is the canonical Hauptmodul for r0(« : h) + e, f, . . . .

If F is a subgroup of ro(n: h) + e,f,. . . of index h, with Hauptmodul T and

(T(z))h = Tn(hz) + K, where AT is a constant, we denote F by T^nh | h) +

e,f, ... &ndTbyTnhlh+e¿ .

In this work we also define

where g is such that g | 24 and (g,f) = 1. The corresponding functions are

and t„(j/g)+(hz), respectively. These are used to label a wide class of groups and

functions arising from various finite groups G considered in this paper.

From [2] we quote certain identities which are called there replication formulae:

\{T2 - r(2)(2z)} = {H2q + H4q2 +...}+ Hx (duplication),

\{ T3 - F(3)(3z)} = {H3q + H6q2 +...} + HXT+ H2 (triplication),

\{ T5 - F(5)(5z)} = {H5q + Hxoq2 +...} + HXT3 + H2T2

+ (H3-H2)T + (H4-H2HX),

where T = Tm(z), F(n) = Tmn and Hr = Hr(m), for m £ M. Many identities are

obtained by comparing powers of q.

3. Discussion of Calculations and Results. When computing their Thompson

series, the groups mentioned above fall naturally into two classes. The first one

consists of G2 = .0, G3 = 3.2.Suz, G5 = 2.HJ and G7 = 2.A7. Each of these groups

has a central element — 1 and two algebraically conjugate representations [d+] and

[d_] of degree d = 24/(p — 1), where/» is the order of the element of M whose


550 LARISSA QUEEN

centrahzer involves Gp for p = 2, 3, 5, 7. If G is one of these groups, then for every

g G G the Thompson series tg is given by the formula

tg(z) = q~x II (1 - el9")(l - e2<7") • • • (1 - eaq")

Ar*

= $"' + H0ig) + Hxig)q + H2ig)q2 + ...

(q = e2wiz), where the e's are the eigenvalues of g in the representation [d^„/p)\, and

in/p) is the Legendre symbol. The H ¡ig) is a generalized character of G, while it

can be shown that H¡i — g) is a proper character; see [2]. However, our calculations

show that the replication order of tg is always less than or equal to the replication

order of t_g, and so they cannot be interchanged. We also observe that, as was not

the case for M, the constant term is significant. In fact, HQi — g) is the character of

G corresponding to the representation [d+].

The second class includes E, F, H and Mx2. To compute the Thompson series for

g G G, where G is one of the groups above, we have to find, by trial and error,

linear combinations of irreducible characters of G that work, i.e., such that the

resulting Thompson series can be identified as modular functions for certain

discrete subgroups of PSL(2, R). Of course, such linear combinations do not have

to exist, and in fact it is quite amazing that they do. For these groups, H ¡ig) is a

proper character of G and the constant term is immaterial.

Tables I-VIII give values of head characters for these groups, together with the

decomposition of the H¡ig) into the irreducible characters of G. We found that to

every element g G G, where G is one of the groups above, there corresponds a

function

tgiz) = q~x + H0ig) + Hxig)q + H2ig)q2 +...,

which can be identified as a Hauptmodul for a discrete subgroup F = Fig) of

PSLÍ2, R) containing T0(N) for some N and such that FB = <(¿ ])>, where F^ is

the stabilizer of the cusp at z = ico. However, Fig) is not necessarily contained in

the normalizer of ro(Af) in PSL(2, R), as was the case for M. Tables I-VIII also

include the corresponding ro(/v*) and the type t (i.e., the name of the fixing group

and the corresponding Hauptmodul) of tg(z) for every g G G.

Some observations resulting from this work, in particular some necessary altera-

tions in the replication formulae and the more general form of fixing group F(g),

have already been reported upon in [2].

Let G be one of the groups considered above. Let p be the order of the element

of M from whose centralizer G was derived. If (n/p) = -1, HH(g) are rational for

all g G G, and the /i-tuplication formulae are used with algebraic conjugation.

If g G G of order s such that (s,p) ■» 1, then its Thompson series T is the same

as Tm for some m G M, i.e., we can obtain new functions only from the elements of

G whose order is divisible by p.

Consider T = Tg, where g £ M or g G G for one of the groups G discussed

above. The replication formulae [2] define a function F(n), called the n-tuplicate of

T. If T - Tm for m G M,



For an arbitrary Thompson series T, we say that T has replication order n if

*W = J-

We note that / is a self-replicating function, and it assumes the role of the identity.

Let G and/? be as above and let T — Tg for g G G. If (/t, p) = 1,

^(») = Tg-

Hence, if g G G and o(g) = s such that is,p) = 1,

r(i) = T1A(G)>

where 1A(G) is the identity element of G, and T has replication order ps. On the

other hand, if (n,p) ¥= 1,

T(n) = Tm,

for some m G M. For example, we found in F that T5A has replication order 5,

TXQB has replication order 10, TX5A has replication order 15, T20c has replication

order 20 and T25A, T25B have replication order 25, where 5A, 105, 15.4, 20C, 25A,

and 255 are elements of order 5, 10, 15, 20, and 25, respectively.

We would like to note that since G = 3.2.Suz contains a central element w of

order 3, for every tg(z) given in Table II we also have

'„(*) - «>'«(* + 1/3)

and

<bM = «*,(* + 2/3) = CW'where * denotes algebraic conjugation.

In our calculations we used the character tables from [3].

Department of Pure Mathematics and Mathematical Statistics

University of Cambridge

Cambridge CB2 1SB, England

1. A. O. L. Atkin & P. Fong, A communication at the A. M. S. Conference on Finite Simple Groups,

Santa Cruz, 1979.2. J. H. Conway & S. P. Norton, "Monstrous moonshine," Bull. London Math. Soc., v. 11, 1979, pp.

308-340.3. J. H. Conway, R. T. Curtis, S. P. Norton & R. A. Parker, An Atlas of Finite Groups. (In

preparation.)4. L. Queen, Some Relations Between Finite Groups, Lie Groups and Modular Functions, Ph.D.

Dissertation, Cambridge, April 1980.5. J. G. Thompson, "Some numerology between the Fischer-Griess monster and the elliptic modular

function," Bull. London Math. Soc., v. 11, 1979, pp. 352-353.


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LA CM+ cd

LA VOTí —

Tí

LA+LATÍ ++ CM ,QCd il VO

VO LA VO VO —— tí — — +

+ + + + CM

cd

— Oi o — cMtATíLAvot— ooen —


modular functions arising from some finite groups...mathematics of computation volume 37, number 156...

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