C. R. Acad. Sci. Paris, t. 327, Serie I, p. 343-348, 1998Analyse complexe/Complex Analysis(Theorie des groupeslGroup Theory>

Fuchsian groups, Schwarzians, and theta functions

John McKAY, AI)(lellah SEBBAR

Department of Mathematics and CICMA, Concordia University, 1455 de Maisonneuve Blvd. West,Montreal, Quebec II3G 1M8, CanadaE-mail : mckay@cs.concordia.ca.sehhar@cicma.concordia.ca

(Reeu et accepte Ie 20 juillet 1998)

Abstract. We describe a family of Fuchsian groups of genus zero, for which the Schwarzians oftheir Hauptmoduls are holomorphic automorphic forms of weight 4, which coincidewith theta functions of certain root lattices of rank 8. © Academie des Sciences/Elsevier,Paris

Groupes fuchsiens, scluoarziens et fonctions theta

Resume. On decrit une famille de groupes fuchsiens de genre zero .. les derivees schwarziennesde leurs « Hauptmoduls » sont des formes automorphes holomorphes de poids 4, etcoincident avec les fonctions theta de certains reseaux de rang 8. © Academic desSciences/Elsevier, Paris

Version Irenceise abregee

Pour une fonetion 1 meromorphe dans un domaine du plan eomplexe, on definit sa deriveeschwarzienne par :

(1") , ( 1") 2 1'" (1" ) 2{f,z}==2 T - T ==27'-3 T

L'expression {I, z} reste invariante si I'on transforme 1 par une fraction lineaire. Si w == w(z) estune fonetion de z, on a la regle suivante :

{f, z} == {I,w}(dwjdz)2 + {w,z} .

En partieulier, si w'(zo) :f. 0, alors dans un voisinage de Zo, on a {z ,w} == -{w,z}(dzjdw)2.Si 1 est une fonction automorphe pour un groupe fuehsien G, alors {I,T} est une forme automorphe

de poids 4 pour G. Et si G est de genre zero, i.e. si la eompaetifieation de la surface de Riemann G\f:Jest de genre zero, et si 1 est un « Hauptmodul » pour G, alors {T,f} est une fonetion rationnellede 1. On peut montrer :

Note presentee par Jean-Pierre SERRE.

0764-4442/98/03270343 © Academie des ScienceslElsevier, Paris 343

,. McKay, A. Sebbar

PROPOSITION 1. - Soient G un groupe fuchsien de genre zero et I un « Hauptmodul » pour G;alors {f, r} est une forme holomorphe de poids 4 pour le normalisateur de G dans PSL2(R).

Reciproquemeni, tout element de G qui laisse {I, r} invariant appartient au normalisateur de G.

Tout groupe G de genre zero dans lequel les translations sont engendrees par r I--t r + 1 admetun « Hauptmodul » normalise de la forme :

I(q) = ~ +L anqn, q = exp(21rir), an E C.q n~l

Pour une telle expression, on peut montrer que, pour tout entier n ~ 1, it existe un unique polynomeunitaire Pn de degre n dont les coefficients dependent de ceux de I et tel que Pn(J(q)) - l/qn soitune serie entiere en q sans terme constant.

Si 1'0n ecrit Pn(J(q)) - l/qn = n l:rn~1 hrn,nqrn, alors hrn,n = hn,rn et hn,1 = an; de plus:

PROPOSITION 2. - On a

4~2 {f, r} = 1 + 12 L mnhrn,nqm+n.m,n2::1

Une etude analytique locale de {f, r} donne:

PROPOSITION 3. - La forme {f, r} a un pole double en chaque point fixe elliptique de G, et estholomorphe partout, y compris aux points paraboliques.

Ceci nous amene aexaminer les groupes de genre zero sans elements elliptiques. Plus precisement,pour des raisons de finitude (voir [5]), on considere les groupes G qui satisfont :

(1) G est de genre zero. sans element elliptique, contient un certain fo(n) avec indice fini ett: H r + 1 engendre le stabilisateur a l'infini.

Les n pour lesquels fo(n) satisfait a cette condition sont n = 4, 6, 8, 9, 12, 16, 18. Si Inest Ie « Hauptmodul » normalise pour fo(n), alors 4;2 {fn' r} est egale a une fonction theta d'unreseau de rang 8; plus precisement, pour n = 4, 9, 16, on obtient E4 ( Vii r), OU E4 est la seried'Eisenstein normalisee de poids 4 qui n'est autre que la fonction theta pour Ie reseau de type E 8 .

Pour n = 6,8,12,18, on obtient respectivement (}c2 0 F4(r), (}D4EllD4(2r) (}4A2(4r) et OD4EllD4

(3r ).Chacun des fo(n) ci-dessus admet un unique conjugue qui satisfait a la condition (1). Pour cesgroupes, la derivee schwarzienne s'obtient apartir des fonctions theta ci-dessus, en ajoutant 1/2 aleur argument. De plus, il y a trois groupes de genre zero satisfaisant la condition (I) et qui ne sontconjugues achacun fo(n) : Ie groupe note G2713 qui est Ie groupe d'invariance du « Hauptmodul »h713(r) = ",(27r)/",(3r), OU '" est la fonction de Dedekind, Ie groupe G2713 qui est l'unique conjuguede G2713 (par r H r + 2) qui satisfait (1), et Ie groupe G3218 qui est Ie groupe d'invariance du« Hauptmodul » h218(r) = ",(32r)h(8r). Leurs derivees schwarziennes ne sont pas des fonctionstheta, mais elles sont donnees respectivement par:

8 ",(r)6",(9r)6 21 ( 1) 16E4(3r) - 48",(3r) - 216 ",(3r)4 et 5E4 8r + 2" - 5E4(16r).

On conjecture que la liste des 17 groupes qui satisfont (1) telle qu'elle est donnee ci-dessus estcomplete.

1. Introduction

This paper is motivated by the discovery of the remarkable connection between Hauptmoduls andmonstrous moonshine, an aspect of the representation theory of the monster sporadic simple group,

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M [1]. A prototype is the elliptic modular function, j, whose Fourier coefficients provide degrees ofrepresentations of M, for which the classical Heeke action is described by a polynomial Pn(j). Weapply the Schwarzian differential operator to each Hauptmodul f of a genus zero Fuchsian group G.This yields a weight 4 automorphic form {f, r} for the normalizer of G in PSL2(R), where r is inthe upper half-plane SJ. This form is expressed in Proposition 2.2 completely in terms of the Fouriercoefficients of P; (1) (see Section 2).

The automorphic form {f, r} is holomorphic at the cusps and has poles of order 2 at elliptic fixedpoints in SJ. This leads us to consider only genus zero Fuchsian groups which have no elliptic element.Following [5], we consider only those groups which contain some fo(n) with finite index and suchthat the stabilizer of 00 is generated by r H r + 1.

In other words, these groups are canonically generated by parabolic elements and the width ofthe cusp at 00 is 1.

We determine all n such that fo(n), or a conjugate, satisfies these conditions. There are 14 suchgroups and it appears that there are only 3 more groups which are not fo(n) or a conjugate.

The Schwarzian of Hauptmoduls for such groups, being holomorphic on SJ and at the cups, iscompletely determined in terms of canonical weight 4 automorphic forms. For 14 groups, these formsare theta functions of variously normed rank 8 lattices [2], and for the 3 remaining cases they aresimple linear combinations of Eisenstein series and known cusp forms. The theta functions arise onlywhen the groups are, up to conjugacy, fo(n). The significance of the lattices of the theta functionsinvolved will be a subject of a later note.

2. Genus zero Fuchsian groups and Schwarzians

Let f be a nonconstant meromorphic function on a domain of the complex plane, we define theSchwarzian of f by:

_ (1")' _(fll

) 2 _ fill _ (1")2{f, z} - 2 f' l' - 2 i' 3 f'

Any linear fractional transformation of f leaves {f, z} invariant. For w = w (z), we have

{f,z} = {f,w}(dw/dz)2 + {w,z}, (2.2)

and {f, z} = 0 if and only if f is a linear fractional transform of z. It follows that in a neighbourhoodof Zo where w'(zo) =I 0 we have {z,w} = -{w,z}(dz/dw)2.

Let G be a Fuchsian group for the upper half-plane SJ, and let f be an automorphic form of weightk(k ~ 0) with respect to G, that is, a meromorphic function on SJ satisfying:

( k. ar + b (a b )fg·r)=(cr+d) f(r), with g'r=-- for rE~ and d E<5,cr + d c

with some growth conditions at the cusps. We say that the elements of Gleave f invariant even whenk =I o. Then {f, r} (r E SJ) is an automorphic form of weight 4, while {r, f} is an automorphicfunction. For G of genus zero, in the sense that the compactification of the Riemann surface G\SJ isof genus zero, and for f a Hauptmodul for G, we have {r, f} is a rational function of f; moreover:

PROPOSITION 2.1. - Let G be a genus zero Fuchsian group and fa Hauptmodulfor G. Then {f,r}is a weight 4 automorphic form for the normalizer of G in PSL2 (R). Conversely, any element of Gwhich leaves if, r} invariant normalizes G.

Proof. - Let 9 be an element of PSL2(R) which normalizes G. The function f(9 . T) defines anautomorphic function for G which is a linear fractional transform of f. It follows that

{f(g' r),g· r} = {f(T),g' r} = (cr + d)4{f,r},

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and so {j,T} is an automorphic form of weight 4 for the normalizer of G in PSL2(R). For theconverse, any element of G which leaves {j,T} invariant normalizes the invariance group of f whichis G. 0

Any genus zero group G with translations generated by T t-+ T + 1 has a normalized Hauptmodulof the form

f(q) = ~ + L anqn, q =exp(27riT), an E C.q n~l

One can show that for each positive integer n, there is a unique monic polynomial Pn of degree nwhose coefficients depend on those of f such that P,,(f(q)) - l/qn is a power series in q withno constant term (see [3]). As an example, the elliptic modular function j normalized to have theform (2.3) satisfies Pn(j) = nTn(j) for all n ~ I, where the Tn are classical Heeke operators.

If we write

Pn(f(q)) - : = n L hm,,,qm,q m~l

then ti-«,« = hn,m and hn,l = an and we have (see [3]).

PROPOSITION 2.2. - Let f be given by (2.3) then

4:2 {j,T} = 1 + 12 L mnhm,nqm+".,n,n~ l

(2.4)

3. Expansion at cusps

Let G be a genus zero Fuchsian group and f a normalized Hauptmodul for G. We assume that Gis finitely generated, or, equivalently, that a Dirichlet region 7) for G has a finite number of sides. Ifa vertex of V corresponds to an elliptic fixed point which is not a pole for f and if f(TO) = ao and nis the order of the transformation fixing TO , then in a neighbourhood of TO we have

{T, j} = (1 - :2) (f ~ a )2 + f ~oao '

with Co being a constant. If TO is a finite cusp which is not a pole, then f is a power series in thevariable w = exp (21fi ) for some constant c. Using (2.2), we have

C T-TO

1 C{T,j} = (f-a)2+ f-a +"', a=f(To),

It follows that (T - TO)4{f,T} = (T - To)4(f/)2{T,j} is a power series in w. This is just the growthcondition for the holomorphy of the weight 4 automorphic form at the cusp TO. The holomorphy at avertex which is a pole is clear. In a neighbourhood of an elliptic point TO, f (T) = a+al (T - TO) n +.. "where a = f( TO), al i 0, and n is the order of the elliptic element fixing TO. Hence {j,T} has apole of order 2 coming from 1'2/(f - a)2. We have proved :

PROPOSITION 3.1. - The weight 4 form {j,T} has a pole of order two at elliptic points, and isholomorphic elsewhere including parabolic points.

4. Congruence groups with no elliptic element

According to the previous section, the Schwarzian of a Hauptmodul for a genus zero group withno elliptic elements is holomorphic everywhere, including the cusps. We examine the finitely manygroups G (see [5]), satisfying the following conditions:

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I . G is a genus zero group ;2. G contains some fo(n) with finite index;3. the stabilizer of the cusp at 00 is generated by r 1-+ r + 1;4. G contains no elliptic elements.It is known that fo(n) is of genus zero if and only if n = 1, ..., 10, 12, 13, 16, 18, 25. A trace

argument shows that fo(n) contains no elliptic element if and only if neither -1 nor -3 is a squaredifferent from 1, modulo n. This implies that fo(n) satisfies (1)-(4) if and only if n = 4, 6, 8, 9, 12,16 and 18. We denote the normalized Hauptmodul for fo(n) by In, then:

14(r ) = ( 1](r))8 , ( ) = 1](r)51](3r) f ( ) = 1](4r) 12 , ( ) = ( 1](r) )31](4r) , J6 r 1](2r)1](6r)5' 8 r 1](2r)41](8r)8' J9 r 1](9r) '

1](4r)41](6r)2 1](8r)6 1](6r)1](9r)3!I2(r) = 1](2r)21](12r)4' !I6(r) = 1](4r)21](16r)4' 118(r) = 1](3r)1](18r)3'

For n = 4, 9, 16, the groups fo(n) and I'( y'n) are conjugate, and these are the only ones whichsatisfy this property, except n = 36 for which f o(36) (or [(6)) is not of genus zero. Since thenormalizer of I'( y'n) is the fuJI modular group I' and the space of holomorphic weight 4 forms for I'is l-dimensional, using Proposition 2.1 we deduce:

PROPOSITION 4.1. - Let In be a Hauptmodulfor the group f o(n) where n = 4,9,16. Then1

41r2{fn ,r} = E4(y'nr ),

with E4 (r) being the normalized Eisenstein series of weight 4.

As a special case, for the classical eJliptic modular function>' on [(2), we have1

"2P,r} = E4 (r ).1r

Notice that E4 (r ) is the theta function of the root lattice E g • For the rest of the groups fo(n)satisfying the conditions (1)-(4), the normalizers are as follows : f o(3)+ for f o(12), f o(6)+ for f o(6)and the normalizers of f o(8) and f o(18) are both conjugate to f o(2)+, where fo(n)+ is the groupobtained by adjoining to I'o(n) all its Atkin-Lehner involutions. Now, the dimensions of the spaces ofweight 4 automorphic forms for the groups f o(2)+, f o(3)+ and f o(6)+ are 1, 1, and 2 respectively(see [4D, and we obtain:

PROPOSITION 4.2. - We have:1 1

41r2{f6 ,r} = t1G20F4(r), 41r2{fs ,r} = t12D4(2r),

1 141r2{f12 ,r} = B4A2(4r ), 41r2{!Ig ,r} = B2D. (3r ).

For n = 4, 6, 8, 9, 12, 16, 18, the group fo(n) has a single conjugate which satisfies theconditions (1)-(4), we denote such group by I'o(n). The conjugating maps are r 1-+ r +1/2 for n = 6,9, 18; r 1-+ r + 1/4 for n = 4, 8, 12, and r 1-+ r + 1/8 for n = 16. The normalized Hauptmodulsfor these conjugate groups are denoted by 7n and are given by:

_ (7/(r+~))8 _ 1](r)417(4r)417(6r)4 _ 17(2r)414(r) = 17(4r) , 16(r) = 1](2r)417(3r)417(12r)4 ' 18(r) = 17(8r) 4'

- 1](2r)91](9r)317(36r)3 - 17(2r)217(8r)217(12r)219(r) = 1](r)37/(4r)317(18r)9 ' Idr) = 17(4r)217(6r)217(24r)2 '

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The Schwarzians of these Hauptmoduls are given by the theta functions in the above propositionwith the argument shifted by 1/2.

There are three more groups satisfying the conditions (1)-(4) but not conjugate to any ro(n).These are: the group that we denote by G2713' which is conjugate to a subgroup of index 3 in r o(9)(containing r o(81) ; its conjugate G;713 via the map T t--t T + 1/2; and the third group G321s•conjugate to a subgroup of index 8 in r o(4). We have:

where 12713, and 13218 denote the normalized Hauptmoduls for the groups G2713 and G321s respectively.The formulas for the Schwarzians are obtained by a dimension argument, knowing that the normalizerof G2713 is r o(9)+ and the normalizer of G3218 is r o(4)+.

Our tabulations lead us to believe that the above 17 groups are the only ones satisfyingconditions (1)-(4).

We thank Oliver Atkin and Simon Norton for their help.

References

[1] Conway J.H.. Norton S.P., Monstrous moonshine, Bull. London Math. Soc. II (1979) 308-339.[2] Conway J.H., Sloane NJ.A., Sphere packings, lattices and groups, Second edition, Grundlehren Math. Wiss., Springer-Verlag,

New York, 1993.[3] McKay J., Sebbar A., Fuchsian groups, Schwarzians, and Lattices (to appear).[4] Skoruppa N.-P., Zagier D., Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988) 113-146.[5] Thompson J.G., A finiteness theorem for subgroups of PSL 2(R) which are commensurable with PSL 2(l), Proc. Sympos.

Pure Math. 37, Amer. Math. Soc., 1980, pp. 533-555.

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