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Replication Formulae for N 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. 808 0021-8693r96 $12.00 Copyright Q 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 G0Ž.N , defining and proving the replica- tion formulae in terms of the twisted Hecke operator TÃm. 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 G0Ž.nh<s a,b,c,dgZ,ad y bcnrh s 1 ½5ž/cn d and G Ž.nh< e,e,... ²G Ž.nh< ,w,w,...: , 01q 20s ee12 810 CHARLES R. FERENBAUGH 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) e9 s ee9rŽ.e, e9 . When Ž. Ž. hs1, we simplify the notation by writing G00N or G N y for the base group, and WE for the Atkin]Lehner involutions. We write G to denote Ž. Ž . the group G021 s PSL Z . We shall restrict our attention to groups for which h <24; in this case Ž .Ž.Ž G01nh< qe,e2, . normalizes G0N , where N s nh. Seewx CoN for more details on this.. Ž . Next, we define a homomorphism l: G01nh< qe,e2,...ªC*as follows: Ž. 1. l s 1 for elements of G0 N , where N s nh; Ž.Ž 2. l s 1 for all the Atkin]Lehner involutions WE of G0 N except when E has a prime divisor not dividing nrh.; y2 p i r h Ž.11rh 3. l s e for the coset containing01 ; "2pirh Ž.10 4. l 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 l is a normal subgroup of index h, denoted 1rh G0nh<q e12,e, . We call such modular groups nh< -type groups. Finally, we define a Hauptmodul of a modular group G to be a $ meromorphic bijection f: HqrGª CÃ. 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 2p 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 q??? . 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 G01nh<qe,e2, . For any function f invariant under this group, we shall say f is of type n< h q e12, e ,... 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 2p iz JzmŽ.sqq0qamq12Ž.qamq Ž. q??? 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 G01nh<qe,e2,... of genus zero, where n is the order of the element m. We call this group the fixing group FŽ. m . 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 n9 <h9 q e12, e ,..., where n9 s nr n, a ; h9 s hr h, a ; X X and e12, e ,... are the di¨isors of n9rh9 among the numbers e12, e ,... a Note in particular that FmŽ.Ž.sFm whenever Ž.n, a s 1. For general Monster elements m, Conway and Norton found replication formulae which seemed to relate JzmmŽ.to its various powers Jza Ž.for all integers a. Later, inwxwx ACMS , FMN , it was discovered that many other modular functions fzŽ.could be assigned replicates fŽa.Ž. z which seemed to satisfy these formulae. Inwx FMN it was noted that many of the non-Monstrous replicable functions could be described using the nh< notation, and for these the same ``power-map'' rule held between the fixing groups FfŽ.and Ff ŽŽa... A complete list of all Hauptmoduls of these nh< -type, or Monster-like, groups can be found inwx Fer1 . 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 G0 nh<.Then: 1. There exist a1111, b , c , d such that m< c1 and ab11rh abrh ' Ž.mod G0Ž.N . ž/cn11 d ž/cn d 812 CHARLES R. FERENBAUGH 2. Similarly, but with m<< b11 instead of m c .
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