JOURNAL OF ALGEBRA 179, 808᎐837Ž. 1996 ARTICLE NO. 0038
Replication Formulae for nh< -Type Hauptmoduls
Charles R. Ferenbaugh
E-Systems, Garland Di¨ision, Dallas, Texas 75266-0023 Communicated by George Glauberman
Received October 18, 1994
1. INTRODUCTION
In the ‘‘Monstrous Moonshine’’ paperwx CoN Conway and Norton ob- served empirically that a modular function could be assigned to each conjugacy class of M in such a way that the Fourier coefficients of these functions were apparently characters of M. They also noticed that the power maps of Monster elements seemed to correspond to certain identi- ties relating these functions; these identities came to be known as replica- tion formulae. Later it was discovered that similar replication relations and power maps seemed to hold for a wider class of modular functions. At the time, all of these correspondences were conjectural. In this paper we shall give a proof of the replication formulae and related results for a class of modular functions which we call nh< -type; this class includes all of the ‘‘Monstrous Moonshine’’ functions and many others as well. Connections to other results. There is some overlap between the present paper and several other recent papers; here is a brief list of the related papers. After I proved the results in this paper, I received a copy of Koike’s preprintwx Koi , which proves many of the same results. He proves that nh< -type functions are completely replicable, using a method of coset decompositions which can be shown equivalent to the lattice method used here. He also proves several of the same subsidiary results such as the compression formulae. Unfortunately, Koike’s preprint does not seem to have been published even though it came out several years ago.Ž Inwx Fer2 , I claimed that Koike’s proof was incomplete. I now withdraw that claim; I had misread part of his preprint..
The lattice presentation of the standard Hecke operator Tn described here is already well established in the literature; seewxwx Ser , Kob for more details.
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0021-8693r96 $12.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. REPLICATION FORMULAE 809
A complete list of nh< -type groups of genus zero can be found inwx Fer1 . This list also contains a variant sort of group in which some of the qe’s are replaced by qe’s. These groups are also completely replicable, and satisfy suitably modified power-map rules and compression formulae; this can be proved by minor modifications of the proofs given here. In the preprintwx CuN , Cummins and Norton prove that all Hauptmoduls of a certain type are replicableŽ though not necessarily completely replica- ble. . Their proof works for a much wider class of functions than this proof; however, they do not specify what the replicates are. For this reason, their proof cannot be used in Borcherds’ construction of the Monster Lie algebra, as described below. Finally, we mention the proof of the original Moonshine conjectures, which follows from Borcherds’ construction inwx Bor of a Lie algebra acted on naturally by the Monster. The traces of these actions are some set of completely replicable functions, since they satisfy Norton’s bivarial form of the replication formulae. Borcherds then uses the result of this paper or that ofwx Koi that the modular functions JzmŽ., as listed explicitly inwx CoN , are completely replicable. He verifies that the coefficients a123, a , a , and a5 of the two sets of functions are identical. As observed inwx ACMS , any replicable function is completely determined by the values of a123, a , a , and a5 for itself and its replicates. So this is sufficient to prove that all of the an are identical, that is, that the Monster Lie algebra has as traces the functions JzmŽ.. ŽNote that this method works only if the replicates are known. Other- wise, coefficients up to a23 would need to be checked; see the appendix to wxCuN .. O¨er¨iew. In Section 1, we summarize some background material, and prove a few simple results concerning modular groups. In Section 2, we work with Hauptmoduls of groups ⌫0Ž.N , defining and proving the replica- tion formulae in terms of the twisted Hecke operator Tˆm. We then extend these methods to general nh< -type groups in Section 3.
1.1. Modular Groups and Hauptmoduls We begin with a brief review of some notation and results fromwx Fer1 . We consider subgroups of PSL2Ž.R of the form
abrh ⌫0Ž.nh
where the we’s are Atkin᎐Lehner involutions
ae brh 2 we s a, b, c, d g Z, ade y bcnrh s e ½5ž/cn de Ä4 Ž. and e012s 1, e , e , . . . is a subgroup of Ex nrh , the group of exact, or 2 Hall, divisors of nrh under the operation e) eЈ s eeЈrŽ.e, eЈ . When Ž. Ž. hs1, we simplify the notation by writing ⌫00N or ⌫ N y for the base group, and WE for the Atkin᎐Lehner involutions. We write ⌫ to denote Ž. Ž . the group ⌫021 s PSL Z . We shall restrict our attention to groups for which h <24; in this case Ž .Ž.Ž ⌫01nh< qe,e2, . . . normalizes ⌫0N , where N s nh. Seewx CoN for more details on this.. Ž . Next, we define a homomorphism : ⌫01nh< qe,e2,...ªC*as follows: Ž. 1. s 1 for elements of ⌫0 N , where N s nh; Ž.Ž 2. s 1 for all the Atkin᎐Lehner involutions WE of ⌫0 N except when E has a prime divisor not dividing nrh.; y2 i r h Ž.11rh 3. s e for the coset containing01 ;
"2irh Ž.10 4. s e for the coset containingn 1 , where the sign is q if z ¬ y1rNz is present, y if not. Ž.Ž . The kernel of is a normal subgroup of index h, denoted 1rh ⌫0nhmodular group G to be a $ meromorphic bijection f: HqrGª Cˆ. Such an f is a generator for the field of functions invariant under G. Furthermore, the groups we en- counter will always contain the map z ¬ z q 1. This implies that fzŽ.can 2 iz be written as a Fourier series in terms of q s e , and that it can be normalized to the form
y1 2 q q 0 q aq12qaq qиии . We shall call such a function a normalized Hauptmodul. In general, we shall use the symbol JzŽ.to denote the n < hqe12, e ,... Ž.Ž . normalized Hauptmodul of 1rh ⌫01nh 1.2. Moonshine and Replication Inwx CoN , Conway and Norton made the following conjecture: PROPOSITION 1.1Ž. ‘‘Monstrous Moonshine’’ conjecture . To each ele- ment m of the Monster M we may assign a certain modular function JmŽ. z , in REPLICATION FORMULAE 811 Ž. such a way that the Fourier coefficients ak m gi¨en by y1 2 2 iz JzmŽ.sqq0qamq12Ž.qamq Ž. qиии Ž.q s e are in fact characters of M. Furthermore, the function JmŽ. z is always the Ž.Ž . normalized Hauptmodul for some modular group 1rh ⌫01nh The functions JzmŽ.are given by an explicit list inwxwx CoN or Fer1 . As noted in the Introduction, this conjecture was eventually proved by Borcherds inwx Bor . Using these correspondences, Conway and Norton then observed empir- ically that the power maps for elements of M have the following simple expression: Ž. PROPOSITION 1.2 power-map rule . If Jm is of type n< h q e12, e ,..., XX Ž. Ž. then Jm a is of type nЈ 1.3. In¨ariance Groups of Replicates We note here some useful properties of the fixing groups under consid- eration. These will be used later in the proof of the replication formulae. First we recall the following lemmaŽ for a proof seewx Fer1. : Ž. Žabrh . Ž. LEMMA 1.3. Let m, n s 1, and letcn d be in ⌫0 nh<.Then: 1. There exist a1111, b , c , d such that m< c1 and ab11rh abrh ' Ž.mod ⌫0Ž.N . ž/cn11 d ž/cn d 812 CHARLES R. FERENBAUGH 2. Similarly, but with m<< b11 instead of m c . Let Ff1Ž.denote the subgroup of FfŽ.containing all elements with determinant 1. Then we can prove the following theorem: THEOREM 1.4. Suppose f is a Hauptmodul of type n< h q e12, e ,... . 1. If p ¦ n, and if p< c, then abrh abprh gFf11Ž.m gFfŽ.. ž/cn d ž/Ž.crpn d 2. If p< h, abhp abrh rŽ.r Žp. gFf11Ž.« gFfŽ.. ž/cn d ž/cnŽ.rpd 3. If p< n and p ¦ h, abph abrh r Žp. gFf11Ž.m gFfŽ.. ž/cn d ž/cnŽ.rpd Proof. Note that, in all three cases, the second matrix mentioned is Ž.p 0 simply the conjugate of the first by01 . Let Hp denote this homomor- phism. Let be the homomorphism used to define FfŽ., and let s e2 i r h. In case 1, we have Hp ⌫00Ž.np<< h ª ⌫ Ž.nhªCh 1 11rh 1prh p sy y ž/ž/01¬¬ 01 10 "p 10 "1 s ž/np 1 ¬¬ž/n 1 s 1 ⌫00Ž.Np ¬ ⌫ ŽN, p .¬ 1. ŽŽ..Ž.p So for the matrices ␥ in the left-hand column, we have Hp ␥ s ␥ 2 Žnoting that, since p ¦ h and h <24, we have p ' 1Ž.. mod h . But these matrices generate ⌫00Žnp< h.Ž, so this equality holds on all of ⌫ np< h.. ŽŽ.. Ž. Thus Hp ␥ s 1 if and only if ␥ s1, and case 1 is proved. ŽNote: by Lemma 1.3 we may assume that pc< .. REPLICATION FORMULAE 813 Similarly in case 2: Hp Ј ⌫<<⌫ 00Ž.nhª Ž. Ž.Ž.nrphrpªChrp 1 11rh 11rŽ.hrp p sy y ž/01¬¬ž/01 "1 10 10 "p s ž/n1¬¬ž/nrp1 s1 ⌫00Ž.N¬⌫ ŽNrp,p .¬1, where Ј is the defining homomorphism for FfŽŽp... This shows, similarly ŽŽ..Ž.p Ž. Ž Ž.. to case 1, that Ј Hpp␥ s ␥ and thus ␥ s1 implies Ј H ␥ s 1 Žthough in this case, unlike the preceding one, the reverse is not always true. . This proves case 2. Finally, in case 3 we have Hp Ј ⌫00Ž.nh<<ª⌫Ž. Ž.nrphªCh 1 11rh 1prh p sy y ž/ž/01¬¬ 01 "1 10 10 "1 s ž/n1¬¬ž/nrp1 s1 ⌫00Ž.N¬⌫ ŽNrp,p .¬1. ŽŽ.. Ž. We assert that Ј Hp ␥ equals 1 exactly when ␥ does. If h s 1or2 this is trivial. If h G 3, there are two possibilities: 1. p ' 1Ž. mod h and the same sign is taken at both choices. Then ЈŽŽ..Hp ␥and ␥ Ž.are always equal. Ž. ŽŽ.. 2. p ' y1 mod h and opposite signs are taken. Then Ј Hp ␥ and ␥Ž.are complex conjugates. Tables 1.1 and 1.2 ofwx Fer1 contain a complete list of Hauptmoduls of nh< -type. By inspection, each of the h G 3 groups listed satisfies one of these two cases. This completes the proof. Note that property 3 of this theorem does not hold for arbitrary nh<<-type groups, such as 24 4y with p s 3. So our reference to the list of Hauptmoduls inwx Fer1 is a necessary part of the proof. By combining properties 1 and 3 of the above result, we have: Ž. ŽŽp.. COROLLARY 1.5. If p ¦ h, then F11 f : Ff . In the ph ŽŽ.. Ž11 . LEMMA 1.6. With notation as abo¨e, if p< h, then Hp F1 f and 01 Žp. generate F1Ž f .. 814 CHARLES R. FERENBAUGH Ž Ž p.. Proof. Suppose ␥ g Ff1 . By Lemma 1.3, we can find a  such that Ž. Ž. ␥sHp . Now Ј ␥ s 1. We saw in the proof of Theorem 1.4 that ŽŽ..Ž.p Ž. ЈHp s ,so must be a pth root of unity. Let 11rp ␦s . ž/01 2i p k Then ␦Ž.sey r, so we may pick an integer k such that ␦Žy .s1. yk Ž. Then ␦gFf1 and k 11 HŽ.␦yk ␥. ž/01 p s Since ␥ was arbitrary this completes the proof. 2. REPLICATION ON ⌫0Ž.N In this section we will state the replication formulae, and develop the tools necessary to prove these formulae in the case where f is the ⌫ Ž. Hauptmodul JNy for 0 N . In some cases this can be done easily using the familiar Hecke operators Tm from the theory of modular functions, which restate certain sums of f ’s in terms of functions on lattices. We then develop a natural definition for the twisted Hecke operator Tˆm, which can be used to express and prove the replication formulae in all cases. Some of this material is patterned after Section VII.4 inwx Ser ; also compare Section III.5 ofwx Kob . A more expository presentation of these results can be found inwx Fer2 . 2.1. Lattices and Modular Functions: Some Preliminaries We begin by characterizing ⌫0Ž.N in terms of lattices: PROPOSITION 2.1. Gi¨en any lattice L11; C with basis , 2, let LNs ²:N12,.Then ab ⌫Ž.Na,b,c,dZ,²:ab,cdL, 01s½ž/cd gq21q2s1 ²: NaŽ.1212qb ,cqd sLN4. This motivates the following definition: DEFINITION 2.2. For any positive integer N,alattice pair of type N is Ž. an ordered pair of lattices L1, LNNsuch that L : L11and L rLNN( C . Ž. A basis for L1, LN is a pair of elements 12, g C such that L 1s REPLICATION FORMULAE 815 ²: ² : 12,and LNs N12, , and such that 1r 2is in the upper halfplane Hq. Such a basis can always be chosen, by the structure theorem for finitely generated abelian groups; and the Hq condition can be met by replacing 11by y if necessary. DEFINITION 2.3. Given a lattice pair Ž.L1, LN of type N, for any kN< Ž. we define Ind k L1, LNkNkto be the unique lattice L such that L : L : L1 and wxL1, Lk s k. PROPOSITION 2.4. There is a one-to-one correspondence between: 1. complex-¨alued functions F on lattice pairs of type N which ha¨e the scalar multiplication property FŽ.Ž.L1, LNs FL1,LN Ž. for all lattice pairs L1, LN and all g C*; and q Ž. 2. functions f: H ª C in¨ariant under ⌫0 N . Sketch of proof. Suppose we are given a function fzŽ.invariant under ⌫01Ž.N. Then, for any lattice pair ŽL , LN .of type N, we can define 1 FLŽ.,L f , 1Nsž/ 2 where 12, is a basis for Ž.L1, LN. Then F is independent of our choice of basis, since f is invariant under ⌫0Ž.N . Ž. Ž . Ž² : ² :. Conversely, given FL1,LN , we define fzsFz,1 , Nz, 1 . Then fcan be shown invariant under ⌫0Ž.N , using the fact that f depends only on L1 and LN . For more details seewx Kob . 2.2. Hecke Operators Given any positive integer N, define the Q-vector space XN to be the set of all formal Q-linear combinations of lattice pairs of type N, and define XˆNns X . nN[< Then any linear operator on XNNor Xˆcan be defined by giving its values at each lattice pair. DEFINITION 2.5. For any m g Zq, the twistedŽ. or generalized Hecke operator TˆˆmNis defined on X by 1 TLˆmŽ.1,LNNs Ý ŽLЈ,LЈlL .. m wxL1:LЈsm 816 CHARLES R. FERENBAUGH q The twisted homothety operator Rˆl for l g Z is defined by 1 RLˆlŽ.Ž1,LNs lL11, lL l LN .. l2 Ž. Note that, for any lattice LЈ ; L1, we have LЈr LЈ l LNn( C for some nN<,soTˆˆmland R are in X ˆ N. Note also that, if we add the condition Ž. LЈrLЈlLNN(Cto the sum in Tˆm, the definition becomes that of the standard Hecke operator Tm as given inwx Kob . We can then state the following proposition, which is a generalization of proposition VII.5.10 ofwx Ser .Ž The proof is analogous to Serre’s, and will not be repeated here.. PROPOSITION 2.6. Our operators Rˆˆln and T satisfy the identities RRˆˆlmsR ˆ lm for l, m g Z RTˆˆlnsTR ˆˆ n l for l, n g Z TTˆˆmnsT ˆ mn if Ž.m, n s 1 TTˆˆppnnsT ˆ pq1qpT ˆ pny1 R ˆ p for p prime, n G 1. Equivalently, we can express later Tˆm in terms of earlier ones: COROLLARY 2.7. Suppose p< m for some prime p. Then TTˆˆ ifp5 m ¡ pmrp Tˆms~ 2 TTˆˆpT ˆ2 R ˆ if p< m. ¢pmrpmy rpp ÄŽa.4 Now suppose, for a function f s JNy, we define its replicates f according to the power-map rule. We can then easily verify the following two properties: Žn. Ž. PROPOSITION 2.8. 1. For any n< N, fisin¨ariant under ⌫0 Nrn . Ža. ŽgcdŽa, N .. Ža. 2. For any a, f s f . In other words, we ha¨ef sf for all a relati¨ely prime to N, and similarly for the replicates of f. By property 1, for any nN< , we may use f Žn. to define a corresponding function F on lattice pairs of type Nrn, and by Q-linearity this extends to define F on all of XˆN . Given such an f and its corresponding F, let A be any linear operator on XˆN . Then we define Ž.fA< Ž. zsFAŽ. Ž²:²: z,1 , Nz,1. . REPLICATION FORMULAE 817 We can now find equivalent definitions for fT<<ˆˆmland fRwhich do not involve lattices. It is easily verified that 2 ŽgcdŽl, N .. Žl. Ž.lf< RˆlŽ. z sfz Ž.sfz Ž.. The expression for Tˆm is derived using the following lemma, quoted from Section VII.5.2 ofwx Ser : ²: LEMMA 2.9. Gi¨en a lattice L s 12, and an integer m G 1, any ²: sublattice of index m in L can be expressed as a122q b , d , where a,b,d are integers with ad s m and 0 F b - d. Each set of ¨alues of a, b, d gi¨es a distinct sublattice. Using this lemma, we find that this definition of Tˆm is equivalent to that proposed inwx ACMS : PROPOSITION 2.10. az b Ža. q Ž.mf< TˆmŽ. z s Ýf . ž/d adsm 0Fb-d The standard Hecke operator TmŽ. L1, LN includes only those terms of TLˆmŽ.1,LN which are lattice pairs of type N. In the language of modular functions, Tmmcontains only the terms of Tˆ that mention f itself, rather Ža.Ža. than some different f . Since f s f exactly when Ž.a, N s 1, we have: COROLLARY 2.11. az q b Ž.mf< TmŽ. z s Ýf . ž/d adsm, Ž.a, N s1 0Fb-d In particular, we note the following two special cases: Ž. COROLLARY 2.12. If m, N s 1, then f<< Tmms fTˆ . COROLLARY 2.13. If p is a prime with p< N, then zzq1 zqŽ.py1 Ž.pf< TpŽ. z s f q f q иии qf , ž/ppž/ ž/ p and ˆ Ž p. ž/pf<< TppŽ. z s fpz Ž .qŽ.pf T Ž. z . It is clear that FTŽŽm L1,LNm ..and FT ŽŽˆ L1,LN ..are invariant under any transformation which preserves L1 and LN . Restating this in terms of f gives: Ž. PROPOSITION 2.14. 1. For any m, and any f in¨ariant under ⌫0 N , the Ž. function f< Tm is in¨ariant under ⌫0 N . 818 CHARLES R. FERENBAUGH 2. For any m, and any f satisfying Proposition 2.8, the function f< Tˆm is Ž. in¨ariant under ⌫0 N . 2.3. Replication for ⌫ The property of replicability can be expressed in several different ways. We use here the definition given inwx ACMS . 1 DEFINITION 2.15. Given a Hauptmodul f with Fourier series qy q Ýϱ i Ž. is1 aqim, we define Pfto be the unique polynomial of degree m such that ym PfzmŽ.Ž.(q Ž.mod qZ wxq . ym In this case we say PfmŽ.has leading term q . DEFINITION 2.16. A function f is called replicable if there exist func- tions Ä f Ža.4, called replicates of f, such that the m-plication formula mf< Tˆmms PfŽ. holds for all positive integers m. We shall refer to all such formulae collectively as the replication formulae, and we call a function completely replicable if it and all its replicates are replicable. If f is a normalized Hauptmodul JNy, we define the replicates of f by the power map rule given in Proposition 1.2. We shall show later that these Ž. replicates satisfy the replication formulae. Also, if g s Pfm , we shall Ža. Ža. define the replicate g to be the corresponding polynomial PfmŽ .. We can now prove the statement which was used inwx CoN as a prototype for the replication formulae: Ža. PROPOSITION 2.17. The function J is completely replicable, with J s J for all a G 1. Proof. First we recall from Proposition 2.14 that JT< ˆm is invariant under ⌫ for all m. This shows that ŽmJ< Tˆm.Ž z . is some rational function of JzŽ.. Next we calculate the leading coefficients of the Fourier series for < ˆ ym Ýϱ i mJ Tmi, and find it to have the form q q s1aqi. Next we observe that J has no poles in the upper half-plane, and the same is true for each of the summands in mJ< Tˆm. Finally, we note that the fundamental region for ⌫ has no cusps on the real line. So the only poles of mJ< TˆmŽ. z , Ž .Ž . Ž . modulo ⌫, are at z s ϱ. Thus mJ< Tˆm z is a polynomial in Jz; and, from its Fourier series above, we see that it must be PJzmŽŽ... Now suppose we replace J by JNy. The first few steps in the above proof still work: mf< Tˆm has the correct invariance group; its Fourier series has the proper form; and it has no poles in Hq. But when N ) 1, the REPLICATION FORMULAE 819 fundamental region for ⌫0Ž.N has cusps on the real line, and the sum- mands of mf< Tˆm have poles on the real line as well. So we are forced to check whether these poles and cusps coincide. We will need additional techniques to do this checking; these techniques will be developed in the next few sections. 2.4. Variable Substitutions Suppose we want to express the transformation z ¬ Ž.Ž.az q b r cz q d Ž.ab in terms of lattices. We can do this by letting the matrix A s cd act on a basis 12, for a lattice pair Ž.L1, LN. This defines a lattice homomor- phism ␣: L1 ¬ LЈ for some lattice LЈ. In general, different choices of 12, will give different homomorphisms ␣. However: THEOREM 2.18. LetŽ. L1, LN be a lattice pair of type N, and let A be a matrix. Then the action of A onŽ. L1, LN is independent of the chosen basis exactly when A is in the normalizer of ⌫0Ž.N . Proof. Let 12, be a basis for Ž.L1, LN, and let ␣ be the action of A XX on Ž.L1, LN using that basis. Similarly let 12, be another basis for Ž.L1,LN , and let ␣ Ј be the action of A using that basis. Now we compute the matrix AЈ for ␣ Ј in terms of the basis 12, . We find that AЈ s y1 TAT, where T is the transformation matrix taking the basis 12, to X X the basis 12, . Since both are bases for the same lattice pair, we have Ž. Tg⌫0N. Now we want to require that ␣ and ␣ Ј have the same effect when applied to Ž.L1, LN . We notice that ␣ Ž.Ž.L1, LNs ␣ Ј L1, LNm A ' AЈ Ž.mod ⌫0Ž.N y1y1 mTATA g ⌫0Ž.N y1 mATA g ⌫0Ž.N . X X The above computation was done for a specific 12, . Now we find that XX when we let 12, range over all possible bases of Ž.L1, LN, then T ranges over all elements of ⌫0Ž.N . So the above equation holds for every Ž. y1 Ž. Ž. basis of L1, LN exactly when ATA is in ⌫00N for all T g ⌫ N , that is, exactly when A normalizes ⌫0Ž.N . One type of substitution we shall be using frequently is the Atkin᎐Lehner involution We. This is always in the normalizer of ⌫0Ž.N ; so the preceding theorem applies, and the reader may verify that: Ž. PROPOSITION 2.19. Gi¨en a matrix We and a lattice pair L1, LN , WLeŽ.1,LNeNsŽ.L,eL re. 820 CHARLES R. FERENBAUGH Ž. Ž. Furthermore, if k< N and Lkks Ind L1, LN, let k s lm, where l, e s 1 and m< e. Then WLeksmL lerm. 2.5. Fourier Series of Hauptmoduls We state here a few lemmas about Fourier series, which we shall use later in computing poles. ϱ k LEMMA 2.20. If fŽ. z Ý aq,then s kskk0 az b ϱ q kakrd f sÝaqk , ž/d ksk0 where is the root of unity e2 ibrd. Ž.The reader can easily verify this. LEMMA 2.21. Let f be a Hauptmodul for some modular group G, and let ␣ normalize G. Then AfŽ. z q B fŽ.␣zs CfŽ. z q D for some A, B, C, D g C. Proof. Since f is a Hauptmodul for G, we have the equivalence fzŽ.sfw Ž .mzs␥w for some ␥ g G. Then we must also have the equivalences f Ž.␣ z s f Ž␣ w .m ␣ z s ␥␣w for some ␥ g G m␣zs␣␥ Јw for some ␥ Ј g G m z s ␥ Јw. So fŽ.␣ z is invariant under G, and therefore it is a rational function of fzŽ.. But f Ž␣ z .has only one zero and one pole Ž modulo G ., so it must take the form shown. If ␣ϱ \ ϱ Ž.mod G , this shows that there is no pole of f Ž.␣ z at z s ϱ.In fact: Ž. ŽŽ.. COROLLARY 2.22. With f, G, ␣ as abo¨e, let g z s Pfzm .Then, if ␣ϱ\ϱŽmod G .Ž., g ␣ z has no pole at z s ϱ; that is, gŽ.␣ z has leading term 0. REPLICATION FORMULAE 821 2.6. Locating Poles and Cusps We can now address the problem which arose a few sections ago when we tried to prove the replication formulae for JNy. We want to compare the poles of Žmf< Tˆm.Ž z . with the cusps of the fundamental region for ⌫0Ž.N, to see whether they coincide. We begin by quoting the following two lemmas fromwx Fer1 : LEMMA 2.23. The images of ϱ on the real line under ⌫0Ž.N are a Ž.a, cN s 1; a, c g Z; c / 0. ½5cN LEMMA 2.24. A fundamental region for ⌫0Ž.N can be chosen so that its cusps on the real line are k yNN Ž.k,N)1, F k F . ½5N22 EMMA Ž. L 2.25. Suppose f s PJmNy.Then: ˆ 1. If p ¦ N, none of the summands of f< Tp ha¨e poles at any real cusp of ⌫0Ž.N . 2. If p<< N, none of the summands of f Tˆp ha¨e poles at any real cusp Ž. Ž. of ⌫0 N , except possibly at the cusps of the form krN, where k, N s p. Proof. First suppose p ¦ N. The only poles of f are at ϱ and its images under ⌫0Ž.N , so by Lemma 2.23 we have a Ä4poles of fpzŽ.s Ža,cN .s 1; a, c g Z; c / 0 ½5cNp z q japqjcN poles of f s Ž.a, cN s 1; a, c g Z; c / 0. ½5ž/pc½5N Now we wonder whether any of these poles can take the form krN for some k with Ž.k, N ) 1. Since an N is present in all of the denominators already, it suffices to show that the numerators are all relatively prime to N. First we observe that Ž.a, N s 1, which rules out the poles of fpzŽ.; and Ž.Ž.ap q jcN, N s ap, N s 1, which rules out the poles of fzŽŽq j..rp. This proves case 1. Next suppose pN< . The poles of fzŽŽqj .rp . are as before. But now Ž p. Ž. fhas invariance group ⌫0 Nrp , so we have a Ž p. Ä4poles of fpzŽ.s Ž.a,cNŽ.rp s1; a, c g Z; c / 0. ½5cN 822 CHARLES R. FERENBAUGH We check the numerators as before. For the poles of fpzŽ p.Ž., we find Ža, N .s Ža, p .; and for the poles of fz ŽŽqj .rp ., we find Žap q jcN, N.Žsap, N .s p. Since in either case the only possible common factor is p, this proves case 2. So the only cases we need to check are those with pN< , at cusps of the form krN where Ž.k, N s p. In these cases, several of the summands of pf< Tˆp will have poles there, and we shall show that the residues cancel. HEOREM Ž. Ž . 5 T 2.26. Let f z s PJmNy,and let p N. Let m be any element Ž. Ž. of ⌫00Nrp not in ⌫ N . Then 0if p ¦ m pf< Tˆ Ž. Mz has leading term ž/p ½pqymrp if p< m. ²: Proof. Let be the action of M on the basis z, 1. Let Lk s kz,1 X and Lkks L for all kN< . Then pF< TˆpŽ.L1, LN X spF< TˆpŽ. L1, LN X sÝFLŽ.Љ,LЉlLN wxL1:LЉsp X X Љ sFLŽ.ŽpN,L qFL p,pL NrpN .q ÝFL Ž,pL rp . wxL1:LЉsp X LЉ/Lpp,L X X Љ sFLŽ.pN,L qFW pŽ. L1, LNNq ÝFLŽ.,pL rp, wxL1:LЉsp X LЉ/Lpp,L X X noting that, by the definition of M, we have LNN/ L but L NpNs L p. X rX r Together these imply Lpp/ L . Also, we note that, for any LЉ / Lp,we have Љ X Љ X X X X L l LNpNs L l L l L rpNs pL l L rpNs pL rp. Now the sublattices of index p in L1 are ²:² : ² : ² : z,p;zq1, p ;..., zqŽ.py1,p; and pz,1 sLp. X²: So Lp s z q k, p for some k, and the above formula becomes z kzj Žp.q q pf< TˆppŽ. Mz s f q fWŽ.Ž. z q Ýf ; ž/ ž/pp ž/ 0Fj-p j/k REPLICATION FORMULAE 823 and, evaluating leading terms of Fourier series using Lemma 2.20 and Corollary 2.22, we find 0ifpm km m pjmmp ¦ qy r q0qÝqyrs ½pqymrpif pm<, 0Fj-p j/k 2i p where s e r. HEOREM Ž. Ž Ž.. T 2.27. Let f z s PJmNy z.Let p be a prime such that 2 Ž. pN<.Let M be any element of ⌫0 Nrp . Then 0 if p ¦ m pf< Tˆ Ž. Mz has leading term ž/p ½pqymrp if p< m. Ž p. Ž. Proof. Since pN< ,f has invariance group ⌫0 Nrp . Applying Lemma 2.23 we have a Ž p. Ä4poles of fpzŽ.s Ž.a,cNŽ.rp s1, a, c g Z, c / 0. ½5cN Since pN< rp, the conditions ŽŽa, cNrp ..s1 and Ža, cN .s 1 are equiva- lent. So none of these poles can satisfy Ž.a, N ) 1, and none of them can exist at a cusp of fzŽ.. So we may restrict our attention to pf< Tp. But by Corollary 3.10 in the next section we have Ž.pf<< TppŽ. Mz s Ž.pf T Ž. z zzq1 zqŽ.py1 sfqf qиии qf , ž/ppž/ ž/ p and on evaluating Fourier series, we find the sum either has no poleŽ if m p p¦m.Ž, or has leading term pqy r if pm<., as claimed. 2.7. Replication for ⌫0Ž.N Now that we have resolved the problem of finding poles and cusps, we can proceed to prove the replication formulae for Hauptmoduls JNy. HEOREM T 2.28. E¨ery Hauptmodul JNy is completely replicable, with replicates as gi¨en by the power-map rule. < ˆ Ž. Proof. We want to show that mf Tmms Pffor any f s JNyand for any m. We proceed by double induction, first on N and then on m. In the 824 CHARLES R. FERENBAUGH base cases, we observe that N s 1 has already been done as Proposition Ž. 2.17, while m s 1 is trivial since fT< ˆ11sPfsf. In the general case, we may assume the statement holds for all NЈ - N, and for all mЈ - m for the current N. By Proposition 2.14, mf< Tˆm has ym invariance group ⌫0Ž.N , and its Fourier series has leading term q .So the only difficulty is to show that mf< Tˆm has no polesŽŽ.. modulo ⌫0 N except at ϱ. Since m ) 1, we may pick a prime p such that pm< , and let l s mrp. Then, by Corollary 2.7 and the induction hypotheses, mf< Tˆm <<5ˆˆ ˆ ¡mf Tlp T s pPlŽ. f T p if pm s~ mf<< Tˆˆ T mpf R ˆ T ˆ pPŽ. f < Tˆ pP f Žp. if pm2<. ¢ lpy plrplpls y rpŽ. Now there are four cases. Ž.ˆ 1. If p ¦ N and pm5 , then pPlp f< T has no poles at any real cusp of ⌫0Ž.N by Lemma 2.25. 2 < Ž.<ˆ Ž.Ž p. 2. If p ¦ N and pm, then pPlpl f T y pPrp f has no poles at any real cusp of ⌫0Ž.N : we handle pPlpŽ. f< Tˆas in case 1, and we ŽŽp.. Žp. observe directly that pPlr pN f has no such poles since f s f s J y. 3. If pN<5and pm, then by Lemma 2.25 we need only check the cusps of the form krN, where Ž.k, N s p. For any such cusp, there exist a, b such that ak q bN s p. Then the matrix ab Ms ž/yNrpkrp Ž. Ž. maps krN to ϱ, and M g ⌫00Nrp y ⌫ N . We can then use either Lemma 2.26 or Lemma 2.27, together with the fact that p ¦ l, to conclude that mf< Tˆm has no poles at that cusp. 4. If pN< and pm2 < , then as in case 3, the only cusps that need checking are of the form krN with Ž.k, N s p. These can be moved to ϱ Ž. Ž. by some M g ⌫00Nrp y ⌫ N . Again we refer to either Lemma 2.26 or Lemma 2.27, this time with pl<, to find that ŽŽ.pPlp f< Tˆ.Ž Mz . has leading ylrp Ž. term pq . Now since M is in ⌫0 Nrp , which is the invariance group of Ž p.Žp.Ž. Žp.Ž. Ž Žp.Ž.. f, we have fMzsfz,so pPlr p f Mz has leading term yl r p pq . These two quantities cancel, so the sum Žmf< Tˆm.Ž Mz . has no poles at that cusp. REPLICATION FORMULAE 825 3. REPLICATION ON nh< -TYPE GROUPS In the preceding section, we proved replication for functions whose invariance groups were exactly ⌫0Ž.N . We now extend these methods to deal with larger groups, which contain normalizer elements of ⌫0Ž.N . Many of these proofs are similar toŽ. though more tedious than those of the preceding section, so some details will be omitted. 3.1. Hecke Operators and Normalizer Elements Consider the Hauptmodul f J of the general nh< -type s n < hqe12, e ,... Ž.Ž . group 1rh ⌫01nh PROPOSITION 3.1. Suppose f JorPJŽŽ ..,with s n < hqe12, e ,... mn Ža. XX Proof. We begin by noting that f is of type nЈ < hЈ q e12, e , . . . , with nЈsnrŽ.a,nand hЈ s hr Ž.a, h . Define NЈ s NrŽ.a, N . We see that Ž.a,nh<Ž.Ž. a, na,h, so that n h nh N nЈhЈ s и ssNЈ. Ž.Ž.Ž.Ža,na,ha,nh a, N . Ža. So ⌫00Ž.NЈ is contained in ⌫ Ž.nЈhЈ , which is in the fixing group of f . If aN<, we have NЈ s Nra, and this proves property 1. If a ¦ N, we can X replace a by Ž.a, N ; this will leave NЈ, nЈ, hЈ, and the ei’s unchanged, which proves property 2. So Proposition 2.14 applies to f, and fT< ˆm is invariant under ⌫0Ž.N . We now wish to show that fT< ˆm has a larger invariance group, namely, the same invariance group as f. To do this we investigate how lattice pairs interact with normalizer elements. First we prove the following general lemma: LEMMA 3.2. Let ␣ be any lattice isomorphism ␣: L1 ª L, and let LЈ and LN be sublattices of L1. Then aTˆˆmŽ.Ž L1, LNms T ␣L1, ␣LN . ␣TLmŽ.Ž1,LNmsT␣L1,␣LN .. 826 CHARLES R. FERENBAUGH Proof. First we note that 1 ␣TLˆmŽ.1,LNNs Ý␣ ŽLЈ,LЈlL . m wxL1:LЈsm 1 sÝŽ.␣LЈ,␣LЈl␣LN m wxL1:LЈsm Tˆˆm␣Ž.ŽL1,LNmsT␣L1,␣LN . 1 sÝŽ.LЉ,LЉl␣LN. m wx␣L1:LЉsm Now each ␣LЈ is a sublattice of index m in ␣L, and so must be one of the LЉ. And conversely, each LЉ must be an ␣LЈ for some LЈ; so the two sums are equal. This proves the first equation. The second is proved in the same Ž. manner, with the extra condition LЈr LЈ l LNN( C placed on the first sum, and similar conditions placed on the other sums. We can now apply the above lemma to the specific cases of normalizer elements. Ž. Ž.abrh THEOREM 3.3. Let L1, LN be a lattice pair of type N, and A s cn d Ž . an element of ⌫0 nh< .If p is a prime such that p< c and p ¦ h, then TALpŽ.1,LNpsAЈTL Ž.1,LN TALˆˆpŽ.1,LNpsAЈTL Ž.1,LN Ž.abprh where AЈ s cnrpd. Proof. Let 12, be a basis for Ž.L1, LN, and let ␣ be the isomor- phism of L whose action with respect to 12, is given by the matrix A. Then, by Lemma 3.2, Tp␣ Ž.Ž.L1, LNps ␣TL1,LN Tˆˆp␣Ž.Ž.L1,LNps␣TL1,LN. So now we need to find a matrix which expresses the action of ␣ on a sublattice LЈ of index p in L1. For the moment, we will assume that if pN< then LЈ / Lp. By the structure theorem we may pick a new basis XX Ž. ²XX:Ž. 12,for L 1, LNsuch that LЈ s 12, p . Since A normalizes ⌫ 0N REPLICATION FORMULAE 827 X X we may assume that A expresses the action of ␣ with respect to 12, . X X Then the matrix of ␣ with respect to 12, p is 10 abrh10 abprh s , ž/ž/ž/01rpcn d 0p ž/cnrpd which is how we defined AЈ above. We now consider the special case when pN< and LЈ s Lp. In this case ²: we know that LЈ s p12, , and that the matrix of ␣ with respect to p12,is 1rp 0 abrhp0 abrhp s . ž/ž/ž/01cn d 01 ž/cnp d ŽNote that, by Lemma 1.3, we may assume without loss of generality that 2 pb< .. But we now observe that, since p ¦ h and h < 24, we have p ' 1 Ž.mod h . This implies that brp ' bp Ž.mod h and cp ' crp Ž.mod h ,so that this matrix is equivalent modulo ⌫0Ž.N to AЈ, and thus AЈ can equally well be used to give the action of ␣ with respect to p12, . Ž. It remains to show that AЈ can be substituted for ␣. Since AЈ g ⌫0nh<, we know that AЈ normalizes ⌫0Ž.N ; and since all lattice pairs in TLpŽ.1,LN are of type N, the substitution is valid in the Tppcase. In the Tˆcase, there may also be a lattice pair of type Nrp. If this is so, pN<, and since p ¦ h Ž .ŽŽ.. we have pn< . Now since AЈ g ⌫00nh< ;⌫ nrph<, it normalizes Ž.Ž. ⌫00nhrp s ⌫ Nrp . So we can still substitute AЈ for ␣, and the proof is done. When A is an element of the fixing group Ff1Ž., we know by Theorem 1.4 Ž. that AЈ g Ff1 as well. So we have: COROLLARY 3.4. If f PJŽ.,and if p h, then f< Tˆ is s mn THEOREM 3.5. LetŽ. L1, LN be a lattice pair of type N, and let E5 N and p ¦ E. Then WTEpŽ. L1,LNpEsTW Ž. L1,LN WTEpˆˆŽ. L1,LNpEsTW Ž. L1,LN. Ž.aE b Ž. Proof. Let A s cN dE be a WE matrix for ⌫0 N . Without loss of 2 generality we may assume pb< 3.2 we have ␣TLpŽ.Ž.1,LNpsT␣L1,LN ␣TLˆˆpŽ.Ž.1,LNpsT␣L1,LN. Ž.aE bp Now the matrix for ␣ on a sublattice of index p is eithercNrpdE or aE brp Ž.cpN dE , by a conjugation argument similar to that of Theorem 3.3. Either of these is a WE matrix for ⌫01Ž.N . We also note that Ž.L , LNis a lattice pair of type N, as are all of the summands in TLpŽ.1,LN by definition of Tp. So in the first equation we can replace every ␣ by WE unambiguously, by Theorem 2.18. For the second equation we use the additional assumption that p ¦ E. Ž. Every summand in TLˆp 1,LN is either of type N or type Nrp. Since Ž. p¦E, we must have EN5 rp. It then follows that WE normalizes Ž. ⌫0 Nrp, and again we can replace every ␣ by WE. Applying this to functions, it clearly follows that: COROLLARY 3.6. If f PJŽ.,and if p is a prime such that s mn 3.2. Compression Formulae Equation 1 of the following theorem was observed inwx CoN and called the ‘‘compression formula’’; several variants, including Eqs. 2 and 3, were proved inwx Koi . THEOREM 3.7Ž. Compression Formulae . Let f J , where s np< hqe12,e ,... p¦h.Then: 1. If p is one of the ei, then zz1 zŽ.p1 Žp. q qy fzŽ.sfz Ž.qf qf qиии qf . ž/ppž/ ž/ p 2. If p is not one of the ei, but ep is, for some e with p ¦ e, then zz1 zŽ.p1 Žp. q qy fWzŽ.epsfWzŽ.qf qf qиии qf . ž/ppž/ ž/ p 3. If ep is one of the ei, for some e with p< e, then zz1 zŽ.p1 Žp. q qy fWzŽ.e sf qf qиии qf . ž/ppž/ ž/ p REPLICATION FORMULAE 829 To prove these we first define a new operator: DEFINITION 3.8. For any prime p, we define the compression operator Cˆp by the formula 1 CLˆpŽ.1,LNNs Ý ŽLЈ,pL .. p wxL1:LЈsp LЈrpLNpN(C Ž. Ž. THEOREM 3.9. For any lattice pair L1, LpN let L Ns Ind N L1, LpN . Then pTpŽ.Ž. L1, LpNq WL p 1,LifppN ¦N pCˆ Ž. L , L p1Ns < ½pTpŽ. L1, LifpNpN . Ž. Proof. Let Lpps Ind L1, LpN . In the p ¦ N case we have pCˆpŽ. L1, LNNpNs Ý ŽLЈ, pL .q Ž.L , pL wxL1:LЈsp X L1/Lp sÝŽ.Ž.LЈ,LpN lLЈqLp,pL N wxL1:LЈsp X L1/Lp spTpŽ.Ž. L1, LpNq WL p 1,LpN . The pN< case is similar, except that LpNrpL is isomorphic to Cp= C N, which is not isomorphic to CpN. So the last term on the right-hand side is omitted throughout. COROLLARY 3.10. LetŽ. L1, LN be a lattice pair of type N, and let be a Ž. mapping which fixes L1 and LNr pNs Ind rpL1, LN, where p is prime and pN2<.Then TpŽ.Ž.L1, LNps TL1,LN. ˆŽ. Proof. By Theorem 3.9 both are equal to CLp 1,LNrp. COROLLARY 3.11. LetŽ. L1, LN be a lattice pair, and let p be prime. Then: 1. If p ¦ e, CWˆˆpeŽ. L1,LNepsWC Ž. L1,LN. Ž.abrh Ž<. < 2. If A s cn d is an element of ⌫0 n h such that p c and p ¦ h, 830 CHARLES R. FERENBAUGH then CALˆˆpŽ.1,LNpsAЈCL Ž.1,LN, Ž.abprh where AЈ s cnrpd. Proof. We use the expressions for Cˆp given by Theorem 3.9. To prove statement 1, we note that Wepis known to commute with both T Žby Theorem 3.5.Ž and Wp by Theorem 2.7 ofwx Fer1. . Similarly, statement 2 is proved by combining Theorem 3.3 with Lemma 2.6 fromwx Fer1 . Proof of Compression Formulae. We prove formula 1 only; the others are proved similarly. Let F be the function on lattice pairs corresponding q to the function f on H , and let G be the composition of F and Cˆp. Then, by Theorem 3.9, we have pGŽ. L1, LNps pFŽ.Ž. TŽ. L1, LpNq FW p Ž. L1,LpN , or equivalently, zzq1 zqŽ.py1 gzŽ.sf qf qиии qf q fWzŽ.p, ž/ppž/ ž/ p where the function g corresponding to G is invariant under ⌫0Ž.N . By Corollary 3.11, if p e and f is invariant under W ,sois g. Also, by ¦ iei Žp. Corollaries 1.5 and 3.11, g must be invariant under Ff1Ž.,sowe see that g has the full invariance group of f Ž p.. Ž.Ž. Now we know that fWzp sfz. So all that remains is to show that Ž p. gsf , and we do this by examining the poles of gzŽ.in the sum above. By adding the Fourier series of the f ’s we see that g has a Fourier series with leading term qy1, so we need only show that it has no poles elsewhere. Since f is holomorphic on Hq it suffices to check the real line. We quote the following results fromwx Fer1 : Ž.Ž . LEMMA 3.12. The fundamental region for 1rh ⌫01nh kn ynn k,/ei᭙i,FkF. ½5nhž/ 22 Ž.Ž . LEMMA 3.13. The images of ϱ on the real line under 1rh ⌫0nh XX Žp. LetÄ 1, e12, e , . . .4 be the set of ei’s under which f is invariant. Then X X every eiiiis either an e or equal to e p for some i. With this notation we list the poles of the f ’s: aeЈ np 2 Ä4poles of fzŽ.s aeЈ , c s eЈ D½5cnpž/ h eЈ aeЈpnp2 jaeŽ.Јp ,c seЈp D½5cnp ž/h eЈ z jaeЈjcn np q q2 poles of f s aeЈ , c s eЈ ½5ž/pcD ½5nhž/ eЈ aeЈp q jcn2 np j D½5ž/aeŽ.Јp ,c seЈp. eЈ cn h Ž.Ž . XX Now the fundamental region for 1rh ⌫01nh COROLLARY 3.14. With f as abo e, let g PJŽ..Then: ¨ s mn 3.3. Expanding the In¨ariance Group We now use the results of the preceding two sections to show that certain values of fT< ˆp have the appropriate invariance group. THEOREM 3.15. Let f PJŽ.,and let e be one of the e ’s. s mn 2. If p<< m, then the quantity pf Tˆ pPŽ. JŽ p. is in ariant pmy rpn 1. If pe5 , let ⑀ s erp. Then we have py1 z j Ž p. q pf< TˆpŽ. z s fpz Ž .qÝf ž/ ž/p js0 Žp. Ž p. sfŽ.pz q f Ž.Ž.Wz⑀yfWz⑀, where this substitution comes from either the first compression formulaŽ if Wp is present.Ž. , or from the second otherwise . The last term in this equation is clearly invariant under W⑀p, since W⑀p commutes with W⑀.So we need to show that the sum of the first two terms is invariant as well. ²: We rewrite the above equation in lattice form. Let L1 s z, 1 and ²: Ž. LpN spNz, 1 , and for kN ⑀ FLŽ.ppN,L qFW⑀Ž. L1, LNppNs FLŽ.Ž.,L qFL⑀, LNr⑀. Ž.Ž.⑀ Now WL⑀ p 1,LpN sL⑀pN,pL r⑀⑀, so replacing F by FW pgives ⑀ ⑀ ⑀ FpLŽ.Ž.⑀,pLNr⑀q F LppN, L , which is equal to the preceding quantity, by the scalar multiplication property of F. So we are done. 2 2. If pe< , let ⑀ s erp as before. Then py1 z k Ž p. q pf< TˆpŽ. z s fpz Ž .qÝf ž/ ž/p ks0 Žp. Ž p. sfŽ.pz q f ŽWz⑀ ., this time using the third compression formula. This quantity is shown to be invariant under W⑀ p, using the same proof as above. THEOREM 3.16. Let f PJŽ.,and let p be a prime not s mn Proof. We first note that fT< ˆp is invariant under ⌫0Ž.N by Proposition 2.14. Next, we observe that ⌫01Ž.nh< 2. Atkin᎐Lehner involutions WE , which are covered by Corollary 3.6 if p ¦ E, or by Theorem 3.15 if pE< . 3.4. Poles and Cusps in the n< h Case We now develop results regarding poles and cusps of fT< ˆp for general nh< -type Hauptmoduls f. The method is similar to that of Section 2.6. LEMMA 3.17. Suppose f PJŽ.. s mn THEOREM 3.18. Let fŽ. z PJ Ž .Ž. z,where w is not present, s mn 0if p ¦ m pf< Tˆ Ž. Mz has leading term ž/p ½pqymrp if p< m. Ž.abrh Proof. Suppose M s cnŽ.rpd. By Lemma 1.3 we may pick aЈ,bЈ,cЈ,dЈsuch that pc<Јand abrhaЈbЈrh ' Ž.mod ⌫0Ž.Nrp . ž/ž/cnŽ.rpd cЈ Ž.nrpdЈ 834 CHARLES R. FERENBAUGH Ž p. This new matrix, which we shall call MЈ,isin Ff1Ž ., because M and MЈ Ž. are congruent modulo ⌫0 Nrp ; and since pc<Ј, by Corollary 1.5 we see Ž. Ž. that MЈ is also in Ff11. So we can write M s MЈM with MЈ g Ff1and Ž. Ž. M10g⌫Nrpy⌫ 0N. We also have, by Theorem 3.3, fTM<<ˆˆˆp ЈM1sfMЉTMp 1sfTM THEOREM 3.19. Let fŽ. z PJ Ž Ž.. z.Let p be a prime such s mn 0if p ¦ m pf< Tˆ Ž. Mz has leading term ž/p ½pqymrp if p< m. Ž. Ž. Proof. First, we note that it suffices to consider M g ⌫00Nrp y ⌫ N , by the reasoning in Theorem 3.18. Ž p. Next we show that fpMzŽŽ ..has no pole at z s ϱ. Applying Lemma 3.13 to f Ž p. we have ae Ž p. 2 poles of fpzŽ.s½5Ž.ae , cnŽ.rhp s e; a, c g Z; c / 0. De cn Suppose My1ϱ was a pole of fpzŽ p.Ž.. We know that My1ϱ has the form aЈrcЈŽ.Nrp, where ŽaЈ, cЈNrp .s 1 and p ¦ cЈ Žotherwise M would be in Ž.. ⌫0 N. So to express this fraction in the form aercn, we must multiply both numerator and denominator by p. Then, since pn< Ž.rhp , we must Ž 2 Ž.. Žp. Ž. XX have pae< ,cnrhp . But f is of type nrph 3.5. The Polynomial Rule In their explicit constructions of the Monstrous Moonshine Haupt- moduls, Conway and Norton expressed the nh< -type Hauptmoduls with h ) 1as hth roots of their replicates. This happens in general, as we shall now prove; as a result, the values of fT< ˆp can be computed directly whenever ph< . REPLICATION FORMULAE 835 Throughout this section we will suppose that f J , and that s n < hqe12, e ,... pis a prime which divides h. k PROPOSITION 3.20. Let k be any integer. Then fŽ. z q krh s yfzŽ., 2 i h where s e r . Ž.1 krh Proof. Since01 normalizes the invariance group of f, by Lemma 2.21 we have kAfzŽ.qB fzqs ž/hCfzŽ.qD for some A, B, C, D g C. But the pole of fzŽ.qkrhis at ϱ,soCmust equal 0. So D must be nonzero, and by rescaling we may assume D s 1. Then we note that the Fourier series of fzŽ.qkrh has leading term k 1 k y qy and constant term 0. So we must have A s y and B s 0. Žp. 1p LEMMA 3.21. fzŽ.s ŽfpzŽ.qc .r for some constant c. p Proof. We define the function gzŽ.sfz Žrp .. The invariance group ŽŽ.. Ž.p 0 of g obviously contains HFfpp, where H denotes conjugation by 01 as in Lemma 1.6. Also, using Proposition 3.20 we can see that gzŽ.q1s gzŽ., so that g is invariant under the map z ¬ z q 1. By Lemma 1.6, this proves that g is invariant under FfŽ Ž p... Next we check for poles. We note that fzŽ.rp has no poles modulo HFfpŽŽ..except at ϱ, by definition of f; therefore the same is true of gzŽ.. Also, if g has a pole at z0 , by Proposition 3.20 it also has one at z0 q 1. Again by Lemma 1.6, this shows that g has no poles modulo FfŽ Ž p.. except at ϱ. Finally, since g is invariant under z ¬ z q 1, it has a Fourier series in terms of q, and using the Fourier series of f we find that 1 gzŽ.sqy qcqpositive powers of q, for some constant c.So g is, up to a constant, the Hauptmodul fzŽp.Ž., and we have z p Ž p. fzŽ.sf yc. ž/p We now replace z by pz and solve for fzŽ.to give the desired result. Ž p. OROLLARY < Ž Ž ..Ž . Ž Ž ..Ž . C 3.22. Suppose p m. Then Pmm f z s Pfrp pz. Proof. First we suppose m s p. From the preceding lemma we know that p Ž p. fzŽ.ycsfpz Ž .. 836 CHARLES R. FERENBAUGH The left-hand side of this equation is clearly a polynomial in fzŽ., while 2 iz the right-hand side, when expressed in terms of q s e , has leading yp Ž Ž ..Ž . term q . So this quantity is in fact Pfp z, which proves the m s p case. ŽŽŽp...Ž . ym In general, note that Pfm r p pzhas leading term q . This quantity must be a polynomial in fzŽ., since we know from the m s p case that fpzŽ p.Ž.is a polynomial in fzŽ.. So by definition it must equal ŽPfm Ž ..Ž z .. Ž. THEOREM 3.23. Suppose g s Pfm .Then 0 if p ¦ m pg< T p s pP fŽ p. if p< m ½ mrpŽ. . Proof. We observe that z q k Ž.pg< TpŽ. z s Ýg ž/p 0Fk-p z km 0ifp¦m s Ý g s ž/p ½pgŽ. z p if pm< . 0Fk-p r But in the pm Žp. gzŽ.sŽ.Pfmm Ž.Ž. zsŽ.PfrpŽ.Ž.pz , and using this identity on the last line completes the proof. Ž. COROLLARY 3.24. Suppose g s Pfm .Then PmpŽ. f if p ¦ m pg< Tˆp s Pf pP fŽ p. if p< m ½ mpŽ.qmrpŽ. . Proof. Add ŽPfzmp Ž ..Ž . to the right side of Theorem 3.23, and Žp. ŽPfm Ž ..Ž pz . to the left side.Ž By Corollary 3.22 these are equal.. Ž.Ž . 3.6. Replication for 1rh ⌫01nh THEOREM 3.25. E ery n< h-type Hauptmodul J is completely ¨ n < hqe12, e ,... replicable, with replicates as gi¨en by the power-map rule. REPLICATION FORMULAE 837 Proof. We want to show that mf< Tˆ PfŽ.for f J and mms s n <<5ˆˆ ˆ ¡mf Tlp T s pPlŽ. f T p if pm s~ mf<< Tˆˆ T mpf R ˆ T ˆ pPŽ. f < Tˆ pP f Žp. if pm2<. ¢ lpy plrplpls y rpŽ. If ph<, this equals PfmŽ.directly, using Corollary 3.24. Ž ˆ .Ž . If p ¦ h, we note that the Fourier series of mf< Tm z has leading term qym , as desired; and by Theorem 3.16, it has invariance group FfŽ..Soit ŽŽ.Ž.. remains to show that it has no poles modulo 1rh ⌫01nh REFERENCES wxACMS D. Alexander, C. Cummins, J. McKay, and C. Simons, Completely replicable functions, in ‘‘Groups, Combinatorics, and Geometry, Durham, 1990,’’ pp. 87᎐98, Cambridge Univ. Press, Cambridge, UK, 1992. wxBor R. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, In¨ent. Math. 109 Ž.1992 , 405᎐444. wxCoN J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 Ž.1979 , 308᎐339. wxCuN C. Cummins and S. Norton, Rational Hauptmoduls are replicable, preprint. wxFer1 C. Ferenbaugh, The Genus Zero Problem for nh<-type groups, Duke Math. J. 72 Ž.1993 , 31᎐63. wxFer2 C. Ferenbaugh, Lattices and generalized Hecke operators, in ‘‘Proceedings, Mon- ster Conference, Ohio State University, May 1993,’’ to appear. wxFMN D. Ford, J. McKay, and S. Norton, More on replicable functions, Comm. Algebra 22 Ž.1994 , 5175᎐5193. wxKob N. Koblitz, ‘‘Introduction to Elliptic Curves and Modular Forms,’’ Springer-Verlag, BerlinrNew York, 1984. wxKoi M. Koike, On replication formula and Hecke operators, preprint. wxSer J. P. Serre, ‘‘A Course in Arithmetic,’’ Springer-Verlag, BerlinrNew York, 1977.