Kashiwa Lectures on

New Approaches to the Monster

John McKay1

edited and annotated by Yang-Hui He2,3,4,5

1 | CICMA & Department of Mathematics and Statistics, Concordia University,
---|---
| 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada
2 | London Institute for Mathematical Sciences, Royal Institution of Great Britain,
| 21 Albemarle Street, Mayfair, London W1S 4BS, UK;
3 | Merton College, University of Oxford, OX14JD, UK;
4 | Department of Mathematics, City, University of London, EC1V 0HB, UK;
5 | School of Physics, NanKai University, Tianjin, 300071, P.R. China

<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

These notes stem from lectures given by the first author (JM) at the 2008
“Moonshine Conference in Kashiwa” 111Organized by the Institute for the
Physics and Mathematics of the Universe (IPMU) under the support of the
Graduate School of Mathematical Sciences, the University of Tokyo. and contain
a number of new perspectives and observations on Monstrous Moonshine. Because
many new points have not appeared anywhere in print, it is thought expedient
to update, annotate and clarify them (as footnotes), an editorial task which
the second author (YHH) is more than delighted to undertake. We hope the
various puzzles and correspondences, delivered in a personal and casual
manner, will serve as diversions intriguing to the community.

###### Contents

  1. 1 Introduction
    1. 1.1 Resources
    2. 1.2 Talk Outline
    3. 1.3 Where to Start ?
  2. 2 Monstrous Moonshine
    1. 2.1 Primes in the Monster’s Order
    2. 2.2 Balance
    3. 2.3 Conjugacy Classes
    4. 2.4 Frame Shape
    5. 2.5 Faber Polynomials
    6. 2.6 Grunsky Coefficients
  3. 3 Hecke operators and Faber Polynomials
    1. 3.1 Replicable Functions: Norton’s Basis
    2. 3.2 Elastica
    3. 3.3 Faber Polynomials and Symmetric Functions
    4. 3.4 Norton’s Conjecture on Replicable Functions
    5. 3.5 Mahler’s Recurrence Relations

##  1 Introduction

I am very honoured to be able to attend and participate in this conference
which I would very much appreciate to be a relevantly informal business where
some mathematics gets done. It is said that there’s a very short time between
being the youngest member of a conference, and becoming the oldest. I don’t
know quite about Harada-San, but I’m 68, so I’m probably there.

My background is computer science, basically. I started off computing
character tables because of a remark by a professor that computing character
tables was more of an art than a science, and I thought that that should not
be the case, and I was very fortunate in starting off in the 60s, at about the
same time as the discovery of the modern sporadics. Janko’s first group was
discovered in 1964.

I’m sure you will have some delightful word that would express the contents of
moonshine, which means something dubious, among other things, and moonshine,
of course, is illicitly produced liquor 222The Kanji, or Chinese characters,
for the word is , literally meaning “secretly made alcohol”. Of course, the
word does not quite capture the sense of “madness” in English which Conway
originally used to express the incredible nature of the Moonshine Conjectures.
However, in classical Chinese poetry, numerous allusions are made to drinking
accompanied by moon-light. The great poet Li Po (701-762) supposedly drowned
himself, in his habitual state of inebriation, trying to grasp the reflections
of moonlight in a lake. Thus perhaps , or “moon-light liquor” is a more
fitting translation..

###  1.1 Resources

#### Books on Moonshine:

There is a book by Mark Ronan, the very popular book [71], and there is a
series of four fifteen minute talks by him on the BBC3 radio in Europe, at the
beginning of this month. They have been recorded. If you want to know more
about this recording I can tell you 333 Since its incipience [14] and proof
[9] (of course, there remains many more things to be understood, including
even Ogg’s initial mysterious observation of the supersingular primes [69] –
q.v. recent accounts in [27, 72]), Moonshine has developed into a vast field.
The reader is referred, for example, to Ronan’s book and interview [71], du
Sautoy’s recent account [28], as well as nice technical progress reports of
[36] and [23]. In parallel, there has been much activity in the physics
community extending Moonshine to beyond the Monster, with special focus on
Mathieu 24 and its relation to the elliptic genus of K3 surfaces [29, 20, 18,
35, 19, 44, 41, 24], to matrix models [39], dessins d’enfants [40, 77] and to
exceptional Lie algebras [43]. .

#### Using the Web and Moonshine’s web page:

One of the things that I would like to emphasize is the use of the computer
and the web today (Internet). I was giving a talk recently and several people
come up to me and said: how do you manage to make these astonishing
connections between things? Well, it’s not that difficult, really. You have
immense resources available on the web, and they grow all the time, and what
happened to me, one of the connections I made, was after having got knocked
down by a car, maybe again because I was thinking too much about problems. I
was in bed for two months, and I just searched on the web for phrases in
papers, and providing there are not more than a 100 or so papers you can
actually go through these papers and see where they are relevant to your
interests; a very effective and quite successful way of finding things.

If you really want to know how to get information if you are a graduate
student, there is something in the literature: there’s Bruce Reznick [70] who
has written an article on extracting information, and there are some
techniques which don’t seem to be too well known.

We have had web pages in the past on moonshine groups. One of them was started
up by Chi-Han Sah from Stony Brook, and we had quite a nice little group on
that, but he died after an operation, and the whole business folded after
that. Then Chris Cummins, who is a colleague of mine, put up a moonshine page
(this was several years ago) with the latest papers and things, and that seems
to have disappeared. So maybe the time is right to start up something again,
where we can discuss things 444 With the growth of blogs, especially in
mathematics, there are several emergent sites which are useful [8]. However,
it would be useful to consolidate these resources, collect comments and have
them maintained professionally; much in the spirit of the PolyMath projects
[68]. .

###  1.2 Talk Outline

Now, I have three ideas which I think worth pursuing, to the extent that you
can show that they are not worth pursuing if necessary, and I will talk about
them and explain what little there is to be said about them.

I particularly want to emphasize a few things which are not as well known as
they should be. One of them is the action of the Hecke operator and its
connection with some very classical objects called _Faber polynomials_. Faber
was, I believe, a numerical analyst (an analyst), and in 1903, in
Mathematische Annalen, he wrote a paper on solving an approximation problem
which was of interest to some fairly eminent people, including Hilbert, and
that’s how they started. But in fact that’s not quite true. They go back to a
man called Francesco Faà di Bruno, and he was an Italian. Probably the only
Italian mathematical saint. He was beatified in 1988. He died in March 27,
1888.

So these polynomials are of some interest, and one can now look at them from a
rather different light. Using these polynomials we can define what we call
_replicable functions_. This is a finite class of functions of about several
hundred of them, amongst which the 171 functions which arise in the Monster’s
context as which we call monstrous moonshine today. These Faber polynomials
describe the Hecke action, and that’s part of the game.

I spoke to an eminent number theorist a year or two back and he told me that
everything was over and we didn’t need to think any more about moonshine and
we understood everything about it. But that’s very far from being the case, in
my view. I think there are several things which are worth thinking about. One
of them is Witten’s idea that there might be some 24-dimensional manifold
which would explain this moonshine by looking at the action of the Monster
group on the free loop space of the manifold. I can’t find much by Witten on
this, but maybe he’s written something. That would be a very nice goal to
either establish the existence of the manifold and the Monster’s action, or to
show that such a thing does not exist. I think Borcherds, for example, doesn’t
think that an action on $\mathbb{M}$ exists, but I don’t think you should
necessarily take much notice of experts; my experience has been rather
negative in that respect.

Then finally, as a sort of dream, it would be whether one could gather
together all the finite simple groups. Let’s initially see whether we can put
the Monster within a better framework, presumably generalizing the Chevalley
work in the Tohoku Journal in 1955 [11], and maybe one can pick up the Monster
by generalizing, and it’s conceivable that you might be able to pick up other
groups; the other six pariahs ($J_{1},\ J_{3},\ J_{4}$, Lyons, O’Nan,
Rudvalis) in this way. John Duncan has found some more moonshine attached to
two of the pariahs.

Let me make another remark. I think that it’s quite useful if one finds
mathematical objects in other contexts, to find whether there is a connection
between them, and I’ll say a little bit more about that later. I’ll give an
example of it shortly.

###  1.3 Where to Start ?

Galois 1832

As of a starting point of the talk, one can start with Galois, who died in
1832, and Galois’ work in recognizing the notion of simplicity of a group,
normal subgroups, and the other result that $\rm{PSL}_{2}(p)$ is realizable on
the cosets of a subgroup of index $p$ providing that $p$ is not bigger than
$11$. So there are certain cases of that, which at least initially, were
believed to be related to the Monster (see [14]).

Mathieu 1861, 1873

One can start with Mathieu in 1861. He wrote a paper in 1861 [55] in which he
said he had found five new groups as transitive extensions of classical linear
groups 555 We recall that $k$-transitive means the following. Let $G$ be a
permutation group on $n$ points and $\\{a_{1},a_{2},\ldots,a_{k}\\}$,
$\\{b_{1},b_{2},\ldots,b_{k}\\}$ are two sets of points with $a_{i}$ distinct
and $b_{i}$ distinct. If there is an element $g\in G$ mapping each $a_{i}$ to
$b_{i}$ for $i=1,\ldots,k$, then $G$ is $k$-transitive. The only 4-transitive
groups are the symmetric group $S_{k\geq 4}$, the alternating group $A_{k\geq
6}$, and the Mathieu groups $M_{24}$, $M_{23}$, $M_{12}$ and $M_{11}$ [10]. .
In the 1861 paper he describes the smaller Mathieu groups $M_{11}$ and
$M_{12}$, and he says that others exist. Then in a paper 12 years later, he
writes that his friends had a bit of trouble seeing how to construct his big
groups, and so in 1873 he gives a description of the big groups $M_{24}$,
$M_{23}$ and $M_{22}$.

Now, it’s notable, and this is true throughout, that the Schur multiplier 666
We recall that for a finite group $G$, the Schur multiplier is the finite
Abelian group whose exponent - the LCM of the order of all elements - divides
$|G|$. More generally, the Schur multiplier of a group $G$ is the second group
homology $H_{2}(G;\mathbb{Z})$.  of groups associated with the sporadic groups
is larger than one would expect. The Schur multiplier of $\rm{PSL}_{3}(4)$ is
an exceptionally big group; a group of order $48$. Now, $\rm{PSL}_{3}(4)$ is a
group of size $2^{6}\cdot 3^{2}\cdot 5\cdot 7=20160$ and is the Mathieu group
$M_{21}$. So $M_{21}$ is not a sporadic group, but it is a classical group
which starts the chain of sporadic groups, $M_{21},\ M_{22},\ M_{23},\
M_{24}$.

I don’t know how much skepticism there was when Mathieu wrote this, but there
is a paper 25 years after the second Mathieu group by a man called G. A.
Miller, who delighted in writing about the problems of other people’s work,
and his paper attempts to show that $M_{24}$ doesn’t exist. In 1900 he wrote a
paper [65] in French correcting himself. So that’s a history of the Mathieu
groups, and we’ll come back to the Mathieu groups later.

Janko 1964

We could start with the Janko groups. Janko was a hard worker. He did an
enormous amount of work attempting to find sporadic groups, and he ended up
with four groups, which are called $J_{1}$, $J_{2}$, $J_{3}$ and $J_{4}$. So
he worked very hard, and the first successful outcome was in 1964. This is the
start of modern era for the sporadic groups.

Plato 400 B.C.

I could start with around the Plato’s date, around 400 B.C., with the
description of the Platonic solids (cf. part (a) of Figure 1). Why are we
interested in them? There’s this very curious bijection between the Platonic
solids and their symmetry groups inside $\rm{SU}(2)$, and the $A-$, $D-$ and
$E-$type Lie structures [59]. So that’s the reason for that. Predating Plato
there’s an interesting guy called Empedocles 777 Empedocles (circa 490-430
BC), pre-Socratic Greek philosopher, known as the originator of the cosmogenic
theory of the four elements. . I’ll say perhaps a bit more at the end about
him. He was interesting because he forecast and predicted the finite speed of
light, which I think was quite good for about 500 B.C.

Skara Brae 3000 B.C.

And then there is something less well-known: Skara Brae. Skara Brae is a
settlement in the Orkneys 888Orkney Islands, northern Scottland, GB. which was
discovered in about 1850. It’s called _late neolithic_ (that’s a cultural
date), but it is about 3200 B.C. This settlement, in a very isolated part of
the world, contained some carved stones, and I’ll show you some pictures of
them (cf. part (b) of Figure 1). Now these stones are all about the same size,
and nobody has any idea what they were for. The belief is they were not
weapons because they’re not damaged, and the possibility is that they gave the
opportunity or permission to speak if you held one of these balls.I don’t know
whether the dodecahedron was above the cube or not, but anyhow there are these
things around. And if you go to the Ashmolean Museum in Oxford 999 Ashmolean
collection AN1927.2727-2731, Oxford University, q.v.
.http://www.ashmolean.org/ash/britarch/highlights/stone-balls.html , they have
them there. What was rather fun was that I mentioned this to Nigel Hitchin,
the (emeritus) Savillian Professor of Geometry at Oxford, and is about 400
yards from these things, and he’d never heard of them.

(a)  (b)

Figure 1: (a) The 5 Platonic solids, the tetrahedron (T), cubic (C),
octahedron (O), dodecahedron (D) and icosahedron (I); C-O and D-I are graph
duals and T is self-dual. (b) The Neolithic carved stones from Skara Brae,
Scotland (in the Ashmolean Museum, Oxford), circa 3200BC.

##  2 Monstrous Moonshine

What I would like to do is make some remarks, and see where we get going from
here. Conway and Norton’s paper [14] was published at the end of October 1979.
And the story behind that you’ve probably all heard, Fischer was visiting me
in Montreal, I wrote a letter to Thompson saying that one of the coefficients
of the elliptic modular function $j$ was $1$ larger than the dimension of the
smallest faithful representation of the Monster. Fischer took that back to
Princeton. I think they all laughed at the concept of there being any
connections, but there are. Borcherds has reminded me that what had happened
was that I was reading a paper by Swinnerton-Dyer and Oliver Atkin [5], and in
that paper they give the $q$-expansion for the $j$-function.

Oliver Atkin used to be a next door neighbour of mine in the ATLAS computer
lab, and he is a number theorist. I was working on these big finite groups,
like the Janko group whose order is $175,560$, and he was working on groups
associated with two-by-two matrices. I was sure they would be much simpler
things than I was working on, but I turned out to be wrong, in retrospect.

This is the order of the monster 101010 The Monster, largest of the 26
sporadic finite simple groups, is a 2-generated group, according to the ATLAS
[6]:  $M=\langle
a,b|a^{2}=b^{3}=(ab)^{29}=u^{50}=(au^{25})^{5}=(ab^{2}(b^{2}a)^{5}b(ab)^{5}b)^{34}=1;u:=(ab)^{4}(abb)^{2}\rangle\
.$ (1)

$|\mathbb{M}|=2^{46}.3^{20}.5^{9}.7^{6}.11^{2}.13^{3}.17.19.23.29.31.41.47.59.71\
.$ (2)

There are $15$ primes there. I don’t quite know what name they should be
given, but anyhow these are the _Monstrous primes_ , or the _Monstrous
supersingular primes_. We will return to these primes shortly. Thompson makes
the remark. He says the order of a finite group is a very strong invariant.
These primes appear elsewhere 111111 There are marvelous recent expositions on
how these 15 primes appear in 5 different contexts by Sankaran [72], as well
as how they can be explained from Moonshine [27].  in Erdenberger [30] (see
later).

That is indeed true, and if you’re trying to construct the groups, as we were
in the early days, when these sporadic groups were sprouting so that every few
weeks there would be a new one, one of the first things was getting hold of
the order, and then using Sylow’s Theorem to build up some structure, and
perhaps guessing a subgroup, and then using that subgroup and the character
table, then using characters building up the character table for the group,
and announcing that, and then someone would come along and say the group
doesn’t exist because the character table doesn’t satisfy some property or
other. Then that property was eventually corrected. You had a correct table as
far as you knew, and the question was trying to construct the group from the
character table, and if that could be done, that was usually done by computer
coset enumeration, and then using some technique to prove that the subgroup
that you had made exists on the basis of the character table, did indeed
exist.

###  2.1 Primes in the Monster’s Order

It would be very useful to know more about these primes. I don’t think there
is so much that can be said about them, except for a remark that Ogg made,
when he was attending at talk by Serre, I believe, at the Collège de France
[69]. This would be in the early 70’s. One takes

$\Gamma_{0}(p)=\left\\{\left(\begin{matrix}*&*\\\
0&*\end{matrix}\right)\bmod\,p\right\\}\subset PSL(2;\mathbb{Z})\ ,$ (3)

together with the Fricke involution
$\alpha_{p}:={\tiny\left(\begin{matrix}0&1\\\ -p&0\end{matrix}\right)}$.
Consider the group

$\Gamma_{0}(p)^{+}=\left<{\Gamma}_{0}(p),\ \alpha_{p}\right>\ ,$ (4)

and think of it as acting on the upper-half plane 121212 Indeed, a classical
fact is that the upper-half plane, ${\mathfrak{H}}:=\\{z\in\mathbb{C}:{\rm
Im}(z)>0\\}$, when adjoining appropriate compactification points known as
cusps which live in $\mathbb{Q}\cup\infty$, quotients the full modular group
to give the Riemann sphere, of genus 0. . Then the genus of the Riemann
surface ${\Gamma}_{0}(p)^{+}\backslash\mathbb{H}$ is zero precisely when $p$
is one of the 15 supersingular primes that appear in the Monster’s order.
That’s one number theoretic characterization of these primes 131313 Genus 0
congruence subgroups are very rare. For example, there are only 33 which are
torsion free [74] and the relation of these with elliptic surfaces, especially
with K3 surfaces, is discussed in [75, 61, 40, 44]. . We still don’t know why,
and Thompson regarded that as one of the major questions to be answered in
connection with the Monster.

There is another way of saying it: for elliptic curves defined in
characteristic $p$, then all the supersingular $j$-invariants of these curves
(being a priori in $\mathbb{F}_{p^{2}}$) are lying in the base field
$\mathbb{F}_{p}$, rather than in $\mathbb{F}_{p^{2}}$ , precisely if $p$ is
one of the above 15 primes.

Now, rummaging through the contents of the preprints on the arXiv.org every
weekday you look through and see if there’s anything of interest. We found a
paper by Cord Erdenberger [30], who is a student of Klaus Hulek from Hannover.
And these 15 primes come up in his work. His title is “The Kodaira Dimension
of Certain Moduli Spaces of Abelian Surfaces” (MR20923323 (2004)). He
considers Abelian surfaces $(1,p)$ polarized ($p$ a prime), and uses Jacobi
cusp forms of weight $2$ and level $p$, and these apparently exist just when
$p$ does not divide the order of the Monster. So in a sense, one might say
that they are related to the existence of this Monster group. This is one
connection which needs some explanation. Here you are working with a subgroup
of the symplectic group rather than the modular group.

I’ve contacted Erdenberger and his supervisor Hulek, and nobody seems to know
quite whether this is really saying something new, or whether it can be
interpreted in terms of these supersingular elliptic curves that I mentioned
earlier. It’s something that should be pursued, at least try to find out
whether there is a connection or not.

I would suggest if you want to follow this up, look in Math review for the
Math Review number I have given above. The reason being that Sankaran reviewed
it, and he does mention this in the review. You won’t find any paper about it,
and certainly Erdenberger was not aware of it. I don’t know whether anyone is
pursuing this; I don’t know of any pursuit of this fact 141414 The reader is
referred to the recent works of [27] and [72] for various explanations. .

###  2.2 Balance

This is typical of the sort of fact that you can gather, and one can formalize
it as something (I don’t know if it’s a great thing to do so) maybe the words
are “retro-syntactic retrieval”, or something like that, but the game is very
simple 151515 This was mentioned earlier in the introduction about how one
could retrieve information and establish correspondences. . You have a bunch
of people working on different subjects, and then if you study the phrases
that are used in common by these people, or you find people who use a common
phrase, there’s a good chance that there is some related activity going on
between the people that use this phrase. I didn’t do that in this case. I
think I was just looking through the arXiv.org and found it.

Now let me say something about balance. If we go back to the first paper, the
word used is “seminal”, certainly it was the only paper for a long time on the
subject by Conway and Norton [14]. Conway is here, and Simon Norton is not
here. I don’t think he should be forgotten. He is very often the motivating
force between a lot of activity, some of which never gets published.

In this paper there is a list of observations which are introductory to the
business of moonshine and one of them is that elements of the group $M_{24}$
are _balanced_. So, what does this mean, and what is its significance? If you
take a permutation in terms of disjoint cycle lengths you have a bunch of
numbers which form a partition of the degree. A permutation is _balanced_ if
the product of the lengths of pairs from the outside-in is constant. Here is
an example 161616 Thus, an element of the permutation group $S_{24}$ of degree
24, would have cycle notation
$(a_{1})(a_{2},a_{3})(a_{4},\ldots,a_{10})(a_{11},\ldots,a_{24})$, which is
indeed the shape of one of the $1575$ conjugacy classes of $S_{24}$ (1575 is
the number of unrestricted partitions of 24). , a permutation with cycle
lengths $1$, $2$, $7$ and $14$ is balanced, since $N=1.14=2.7=14$. The number
$N$ is called the _balance number_ , if it exists.

In 1980 or thereabouts, we had a conference called “The Coming of Age of the
Finite Groups”, and some of the people were here then. Dummit and Kisilevsky
and myself classified all permutations of degree $24$ that are balanced [25].
Why choose 24? Well, we’re going to replace $k$ by $\eta(q^{k}),\ q=e^{2\pi
i\tau}$, for each cycle of length $k$, and thus form the product, we call it
_eta products_.

We found all the $\eta-$products which are weakly multiplicative in the
coefficients171717 The simplest case is the famous $\Delta(q)=\eta(q)^{24}$,
which is the modular discriminant function, with q-expansion
$\Delta(q)=\sum\limits_{n=1}^{\infty}\tau(n)q^{n}$ with $\tau(n)$ being the
Ramanujan tau-function. This is weakly multiplicative in the sense that
$\tau(m\ n)=\tau(m)\tau(n)$ if $\gcd(m,n)=1$. The multiplicative eta-products
appear in physics, especially in partition functions in string theory and are
discussed in [37, 20, 41]. . There are exactly $30$ of them, and all the
permutations in the Mathieu group $M_{24}$ are balanced. And being balanced
and weakly multiplicative is of the same thing as a theorem of Bryan Birch and
Morris Newman on that [64]. And you have a cusp form for each balanced
permutation, with what’s called a Grossen-character, whose weight is half the
number of parts.

Now we can generalize this to eta-quotients instead of eta-products by writing
fractions. Here is an example: $2^{24}/1^{24}$ which means
$\eta(2\tau)^{24}/\eta(\tau)^{24}$. The multiplicative $\eta$-quotients have
been classified too 181818 Cf. also [47, 54] for relations to elliptic curves.
by Yves Martin [53]. These appear in [3].

The products are straight-forward because there are only finitely many
partitions to look at, so you just go through them, find out the ones which
look like they’re multiplicative by looking at the first few coefficients and
checking, and then filter them out and then you have to prove something. But
for the quotients that’s a different matter. The quotients are much more
difficult. There is potentially infinitely many of them. The guy who had done
a paper on them is Yves Martin. He hasn’t completely done it. He made an
assumption that both the eta-quotient and the eta-quotient with the above
Fricke involution action on it are weakly multiplicative and that’s not asked
for. So, the general question about what eta-quotients are multiplicative is
not known.

Being multiplicative means that you have some Euler product through the
(inverse) Mellin transform 191919 That is, we can form the Dirichlet series
for the coefficients $c_{n}$ of the q-expansion $\sum\limits_{n}c_{n}q^{n}$
for these eta-products to give $\eta_{g}(s)=\sum\limits_{n}c_{n}n^{-s}$. This
can then be taken as product over primes as in (5). , and here is the Euler
product (we use the subscript $g$ to identify the particular eta-product):

$\eta_{g}(s)=\prod_{p}\left(1-\frac{a_{p}(g)}{p^{s}}+\frac{b_{p}(g)}{p^{2s}}\right)^{-1}$
(5)

with $a_{p}$ and $b_{p}$ integers; we have

$a_{p}^{2}-a_{p^{2}}=b_{p}=\chi\left(\frac{\cdot}{p}\right)p^{w_{g}-1}\ ,$ (6)

where $\chi$ is some character and $w_{g}$ is the weight of the eta-quotient,
which is half the number of parts. In particular, on the identity, the
partition is $1^{24}$ and the weight is $w_{g}=24/2=12$, which yields
$p^{11}$. We have a degree $p^{11}$ generalized character. These are proper
characters which can be checked this directly. There is some anomalous
behavior for $p=3$, but other than that everything is clear.

###  2.3 Conjugacy Classes

There are curious connections with physics and conjugacy classes, and I think
someone raised this the other day at the EWM meeting 202020 Encounters with
Mathematics, Chuo University, May, 2008,
http://www.math.chuo-u.ac.jp/ENCwMATH/45.shtml. , that in studying elliptic
genera you look at commuting pairs of elements, and that’s related to the
number of conjugacy classes in the group when working with a finite group. And
I just make the passing remark, the class number - the number of conjugacy
classes- of $M_{24}$ is 26, which should ring a bell with some physicists
212121 In string theory, the critical space-time dimension of the bosonic
string is 26 and that of type II super-string is 10. . You have 5 quadratic
boxes of irrationalities, so there are 21 classes of cyclic subgroups, or if
you like, 21 rationally irreducible representations. The group $M_{12}$,
believe it or not, has $10$ instead of $26$.

If you want to know more about this remark about conjugacy classes, there is
recent publication, a large book in fact called ”From Number Theory to
Physics”, in Les Houches proceedings of 2002, containing a paper with Sebbar
and myself [62]. More recently, in a paper in a conference proceedings on
things to do with moonshine, there’s a paper by Anda Degeratu and Katrin
Wendland [22], and they look again at one of these situations that you
discover by reading and saying “my goodness, it’s the same number there.” They
are looking at a situation where the appropriate number replacing 26 is 194,
and 194 is the number of conjugacy classes in the Monster 222222 In other
words, as remarked in [62], the number of conjugacy classes of $M_{12}$ and
$M_{24}$ are respectively 10 and 26, the critical dimension of the
supersymmetric and the bosonic string theories. Moreover, 194, the number of
conjugacy classes of the Monster, is the Picard number of the base of an
elliptically fibred Calabi-Yau threefold in an extremal case of heterotic-F-
theory duality as studied in [4]. Furthermore, of these 194 classes,
considered as column-vectors in the character table, only 163 are linearly
independent; and of course, 163 is a famous Heegner number where the
exponential assumes an almost-integer value: $\exp(\pi\sqrt{163})\sim
640320^{3}+744$. . So they are looking at a situation that is interesting if
anything comes out of it.

###  2.4 Frame Shape

Now, these shapes we’ve been talking about – the partition of $n$ where $n$ is
the degree, or the more general situation when you divide one term by another
is called _Frame shapes_. Now “Frame” is the name of a person, J. S. Frame,
not an abstract notion, and he was an interesting man. His thesis was on
character tables in the 1930’s [33]. He did character tables like other people
do crosswords. So that’s who Frame was.

Basically speaking, what you are doing is that you are describing the
eigenvalues within an orthogonal group. So if you have a fraction, you want to
make sure that the eigenvalues that you take away from the denominator are
already in the numerator, to make any sense. There’s a paper by Takeshi Kondo
[50], who wrote a very nice paper about the Frame shapes of elements in the
automorphism group of the Leech lattice, Aut$(L)$. If you go down from there
to Conway’s group $\cdot 1$, which is this group modulo the action on
diameters - i.e., Aut$(L)/\left<\pm 1\right>$ \- you get a mixture of the
functions that describe the elements on the various classes and they are not
as consistent as those for the monster $\mathbb{M}$.

By the way, there’s a very nice survey by Masao Koike [49], I will say more
about him later on. Again this was in Sugaku, in Japanese that has been
translated into English by the AMS translations $\\#160$. This is one of the
few early surveys on moonshine, so that’s a useful paper too.

###  2.5 Faber Polynomials

If we take a Riemann map 232323 This approach of looking at Moonshine from the
perspective of geometric function theory, in terms of the shape of the
analytic functional form of $j(q)$ and generalizations, is very much the
spirit of the current lecture notes, and is also summarized in [60].  from the
exterior of some region in the complex plane containing two points at least,
by the Riemann mapping theorem we can map the exterior of this region with
some conditions at infinity to the exterior of a disk of radius $d$, and we
can normalize this

$z=\phi(w)=dw+d_{0}+\sum_{k\geq 1}d_{k}w^{-k}\ .$ (7)

If you are doing analysis you don’t have to worry about the constant term and
if you are doing moonshine you put it equal to 0 and the radius equal to 1. We
get an inverse of the same forms as (7)

$w=\phi^{-1}(z)=z/d+g_{0}+\sum_{k\geq 1}g_{k}z^{-k},$ (8)

and similarly we can take $d=1$ and $g_{0}=0$. The Faber polynomial $F_{n}$ is
the part of $(\phi^{-1}(z))^{n}$ with non-negative powers of $z$. So you’re
picking up the polynomial part of this series $(\phi^{-1}(z))^{n}$.

You may not be familiar with this, but if you’re looking at pseudo-
differential operators that’s a standard procedure to pick the plus part of
the operator. That’s what the Faber polynomial does for you, and that how it’s
used and I’ll say quite a lot more about that 242424 In other words, we
consider a meromorphic function and its inverse with Laurent expansion of the
form (7) and (9). This is clearly inspired by the form of the q-expansion of
the $j$-invariant, as we shall shortly see. We emphasize that the constant
term is 0, so henceforth, by the $j$-invariant, we mean the normalized one
$j(q)-744$. In [39], this shape of a Laurent series was interpreted as the
master-field of a large N matrix model, whereby giving a modular matrix model.
.

What we do is we compose with the map $z\mapsto 1/q=e^{-2\pi iz}$, to get

$f(z)=q^{-1}+\sum_{k\geq 1}a_{k}q^{k},\ \quad\Im(z)>0\ .$ (9)

The $a_{k}$’s are general coefficients and we’ll take them to be integers, but
they need not be integers, and Simon Norton has classified the functions we
are interested in (the replicable ones to come later) even when they have
complex coefficients. If these coefficients are not integers, they do lie in a
field whose Galois group is an elementary $2$-group over the rationals. In
other words, the $a_{k}$’s, lie in a composite of quadratic fields. I don’t
think we really know which quadratic fields and why, but anyhow that’s where
they lie when we’re talking about replicable functions, that’s the ones we are
interested in.

But for our purposes and for all the stuff here we work with the $a_{k}$’s
being integers. That avoids any problems with Galois theory and is convenient.
The functions of the form (9) are typical functions we shall study, and we are
going to study them first of all slightly more generally than the connection
with the Monster, and then specialize to functions that are attached to
$\mathbb{M}$.

###  2.6 Grunsky Coefficients

We can define the elliptic modular function $j$ by the above property,
because, if this holds for some other function $f(z)$ for all positive
integers $n$, the level of this function $f(z)$ must be equal to $1$, and is
therefore a rational function of $j(z)$, and so, using the fact that $j$ is
normalized at infinity as in (9), we have that $f(z)=j(z)$.

For $f$ as in (9) we write

$F_{n}(f)=q^{-n}+n\sum_{m\geq 1}h_{m,n}(f)q^{m}\ .$ (10)

the coefficients $h_{m,n}$ are called _Grunsky coefficients_ , see [38, 67].
These are in fact symmetric in the $m,n$ indices 252525 We can in fact define
the Grunsky coefficients and Faber polynomials in the following way. We will
encounter some of the ensuing expressions in due course. Let $g(z)$ be a
holomorphic univalent (i.e., one-to-one on the open set) function on the unit
disk $|z|<1$, normalized so that $g(0)=0,\ g^{\prime}(0)=1$. Then the function
$f(z)=g(1/z)^{-1}$ is a non-vanishing univalent function outside the unit disk
with simple pole at $\infty$ with residue 1; that is,
$f(z)=z+a_{0}+a_{1}z^{-1}+a_{2}z^{-2}+\ldots\ .$ The expansion coefficients
$c_{nm}$ of
$\log\frac{f(z)-f(w)}{z-w}=-\sum\limits_{m,n>0}c_{nm}z^{-m}w^{-n}$ (11) are
the Grunsky coefficients. Definition (11) implies, upon
$z\frac{\partial}{\partial z}$ on both sides, that
$\frac{zg^{\prime}(z)}{f(z)-f(w)}-\frac{z}{z-w}=\sum\limits_{m,n>0}mc_{nm}z^{-m}w^{-n}$.
Thus we define the Faber polynomials $F_{n}(w)$, as
$\frac{zg^{\prime}(z)}{f(z)-w}=\sum\limits_{n\geq 0}F_{n}(w)z^{-n}\ .$ (12) It
is non-trivial that, thus defined, $F_{n}(w)$ are monic polynomials of degree
$n$. In fact, $F_{n}$ are themselves polynomials in the coefficients $a_{i}$
in the definition of $f(z)$. This is seen as follows. Definition (12) implies,
upon applying $\int_{0}^{\infty}dz\frac{1}{z}$ on both sides, that
$\log\frac{f(z)-w}{z}=-\sum\limits_{n\geq 1}\frac{1}{n}F_{n}(w)z^{-n}$.
Expanding out $f(z)$ order by order and comparing with (12) then gives the
recursion  $F_{n}(0)=1\ ,\quad
F_{n}(w)=(w-a_{0})F_{n-1}(w)-na_{n}-\sum\limits_{i=0}^{n-1}a_{n-i}F_{i}(w)\ .$
Furthermore, combining (12) and (11) we have that $\sum\limits_{n\geq
0}F_{n}(g(\zeta))z^{-n}=\frac{z}{z-\zeta}+\sum\limits_{m,n>0}mc_{nm}z^{-m}\zeta^{-n}=\sum\limits_{n>0}\left(\frac{w}{z}\right)^{n}+\sum\limits_{m,n>0}mc_{nm}z^{-m}\zeta^{-n}$
so that  $F_{n}(g(z))=z^{n}+\sum\limits_{m\geq 1}c_{nm}z^{-m}\ ,$ which is
(10) in our definition, up to the factor of $n$ which will be more convenient
for our succeeding discussions. . They have a remarkable connection with the
Bieberbach conjecture 262626 We recall the statement of the Bierberbach
Conjecture, proven by de Branges. For univalent holomorphic function with
Taylor series of the form $f(z)=z+\sum\limits_{n\geq 2}a_{n}z^{n}$ (such
functions are called Schlicht, or simple/plain), the coefficients have the
property that $|a_{n}|\leq n$ for all $n\geq 2$. . The Bieberbach conjecture
is a bound on the coefficients of functions univalent on the unit disk, and
there’s a very nice book [21], an AMS publication, on the solution to the
Bieberbach conjecture by de Branges, and these Grunsky coefficients played the
major role in the establishment of this conjecture in its early days272727
There is a very recent paper on the appearance and relevance of Bieberbach/de
Brange as well as Grunsky coefficients in scattering amplitudes in quantum
field theories [45]. .

In our definition (10), we have written $F_{n}(f)$ in this way with an $n$ in
front in order to take advantage of the symmetry of $h_{m,n}$ in $m$ and $n$.
In fact that is the definition of the Grunsky coefficients

$h_{m,n}(f)=T_{n}(f)|_{q^{m}}\ .$ (13)

There will be other introductions to the Faber polynomials and these Hecke
operators $T_{n}$ later 282828 In [62], a particularly nice characterization
of the Faber polynomial is as given. Consider, as always, a function
$f(q)=q^{-1}+\sum\limits_{n\geq 1}a_{n}q^{n}$ for nome $q=\exp(2\pi iz)$ with
${\rm Im}(z)>0$, as in (9). Then, for each $n\in\mathbb{Z}_{>0}$, there is a
unique monic polynomial $F_{n}$ such that  $F_{n}(f(q))=q^{-n}+{\cal O}(q)\
,\quad\mbox{ as }q\rightarrow 0\ .$ These are the Faber polynomials. Depending
on the Taylor series of $f(q)$, the first few are  $F_{0}(z)=1,\ F_{1}(z)=z,\
F_{2}(z)=z^{2}-2a_{1},\ F_{3}(z)=z^{3}-3a_{1}z-3a_{2}\ .$ More generally, we
have  $F_{n}(z)=\det(z\mathbb{I}-A_{n})\ ,\qquad
A_{n}:={\tiny\left(\begin{matrix}a_{0}&1&&&&\\\ 2a_{1}&a_{0}&1&&&\\\
\vdots&\vdots&\vdots&&\\\ (n-2)a_{n-3}&a_{n-4}&a_{n-5}&\ldots&1&\\\
(n-1)a_{n-2}&a_{n-3}&a_{n-4}&\ldots&a_{1}&1\\\
na_{n-1}&a_{n-2}&a_{n-3}&\ldots&a_{1}&a_{0}\end{matrix}\right)}$ .

It’s not obvious from the above expression that $h_{m,n}$ is symmetric in $m$
and $n$, but it is, and we can see it with a slightly different generating
function for the Grunsky coefficients.

##  3 Hecke operators and Faber Polynomials

I now turn to Part 2 of my lectures, having alluded to the Hecke operators.
There are Hecke operators, and often in the books they assume that the
function on which the Hecke operator is acting has a weight greater than zero,
whereas the functions we are interested in all have weight zero. You have a
group action acting linear fractionally on $z$, and on the modular function
$j$ this is given by, for all $n\geq 1$

$T_{n}(j(z))=\frac{1}{n}\sum_{\begin{subarray}{c}ad=n\\\ 0\leq
b<d\end{subarray}}j\left(\frac{az+b}{d}\right)=\frac{1}{n}F_{n}(j(z))\ .$ (14)

The effect of the Hecke action is to replace the pole of order 1 of the
$j$-function with a pole of order $n$ at infinity. Moreover, the action of the
Hecke operator preserves the space of modular functions. Hence, $T_{n}(j(z))$
is a rational function of $j(z)$, and so this going to be a polynomial in
$j(z)$. $T_{n}(j(z))$ can be expressed both as a $q$-series and as a
polynomial in $j(z)$. In fact, we can define the $j$-function by the fact that
there is an action of the Hecke operator defined in terms of sum over the
function valued on sublattices. Let’s have a quick look at this, which is
standard.

(a)  $\stackrel{{\scriptstyle T_{2}}}{{\longrightarrow}}$ (b)

Figure 2: (a) Fundamental region of the lattice
$\Lambda=\mathbb{Z}+z\mathbb{Z}$. (b) The 3 sublattices of index $n=2$, viz.,
$\mathbb{Z}+2z\mathbb{Z}$, $\mathbb{Z}+(1+2z)\mathbb{Z}$ and
$2\mathbb{Z}+z\mathbb{Z}$.

In part (a) of Figure 2, this is the fundamental region, up to orientation and
homothety. The Hecke operator $T_{n}$ maps the lattice
$\Lambda={\mathbb{Z}}+z.{\mathbb{Z}},\ z=\omega_{1}/\omega_{2}$ such that $\Im
z>0$, to sublattices $\\{\Lambda_{i}\\}$ of index $n$, so induces an action on
functions defined on these lattices 292929 The fundamental fact here is that
sublattices of index $n$ are in one-one correspondence with integer matrices
${\tiny\left(\begin{matrix}a&b\\\ 0&d\end{matrix}\right)}$ with $a>0$,
$b=0,1,\ldots,d-1$ and $ad=n$. For example, at $n=2$, we have 3 such matrices,
${\tiny\left(\begin{matrix}1&0\\\
0&2\end{matrix}\right)},{\tiny\left(\begin{matrix}1&1\\\
0&2\end{matrix}\right)}$ and ${\tiny\left(\begin{matrix}2&0\\\
0&1\end{matrix}\right)}$, corresponding to the lattices
$\mathbb{Z}+2z\mathbb{Z}$, $\mathbb{Z}+(1+2z)\mathbb{Z}$ and
$2\mathbb{Z}+z\mathbb{Z}$. . For example, for $n=2$, there are 3 lattices of
fundamental region twice that of $\Lambda$ and the Hecke action is given by

$T_{2}:\ f(z)\mapsto\
\frac{1}{2}\left(f(2z)+f\left(\frac{z}{2}\right)+f\left(\frac{z+1}{2}\right)\right)\
.$ (15)

This is drawn in part (b) of Figure 2. I’ve used $z$ here because $\tau$ is
used later.

For each of the functions we are interested in there will be a discrete
subgroup $G_{f}$ of ${\mbox{PSL}_{2}({\mathbb{R}})}$ with respect to which $f$
is modular, and in [14] there is a discussion on how much is being fixed. The
action of $G_{f}$ is linear fractional. For us, the transformation as an
element of $G_{f}$

$f\left(\frac{az+b}{cz+d}\right)=(cz+d)^{k}f(z)$ (16)

is not affected by the automorphy factor $(cz+d)^{k}$ because we are working
with functions of weight $k=0$, and this is rather important since the
behavior is different 303030 That is, the $j$-function is an absolute
invariant $j\left(\frac{az+b}{cz+d}\right)=j(z)$ for
${\tiny\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)}\in
PSL(2;\mathbb{Z})$. Indeed, for weight $k$ objects, the standard definition
[76] of the Hecke operator is
$T_{n}(f(z))=n^{k-1}\sum_{\begin{subarray}{c}ad=n\\\ 0\leq
b<d\end{subarray}}d^{-k}f\left(\frac{az+b}{d}\right)\ .$ for all
$n\in\mathbb{Z}_{\geq 1}$. . There is also the notion of the level , which is
the smallest $N$ such that ${\Gamma}(N)\subset G_{f}$ 313131 The congruence
groups are defined with some modulo $N$ relation. For example, the principal
congruence subgroup is $\Gamma(N):=\\{A\in PSL(2;\mathbb{Z})\,\left|\,A\equiv
I\bmod N\right.\\}$. .

Classically, for all $\gcd(n,N)=1$, $T_{n}(f)$ is a polynomial in $f$ ($f$ as
in (9)). This last statement is the critical one that enable us to generalize
the action of the Hecke operator, and the generality is that if one defines
the $j$-function with the normal Hecke operator acting as a polynomial then
what we do is we preserve the action of the (Faber) polynomial and redefine
the Hecke operator. Indeed, since for all $n\geq 1$ we have

$T_{n}(f(z))=\frac{1}{n}\sum_{\begin{subarray}{c}ad=n\\\ 0\leq
b<d\end{subarray}}f\left(\frac{az+b}{d}\right)=\frac{1}{n}F_{n}(f(z))\ ,$ (17)

the modular level is one, and therefore is a rational function of $j$. Hence,
using (9) where $f$ is normalized to have a simple pole at infinity and a zero
constant term, implies that $T_{n}(f(z))$ is a polynomial in $j(z)$, and thus
$f=j$. This enables us to define the $j$-function from the property that the
Hecke operator on $j$ averaging over sublattices of index $n$ is the Faber
polynomial associated with $j$.

I want to say quite a bit about this important polynomial. It’s easy to find
this polynomial from a computational and algorithmic points of view. Start off
with (9) and look at various powers of $f$ you’ll get various negative powers
of $q$ on the right hand side of (17), and by forming linear combinations of
these powers of $f$ on the left hand side of (17) we can eliminate all but the
largest negative power of $q$, and if there is a constant term we can put it
into the polynomial on the left hand side. That makes it clear that this
polynomial is very simply determined from (9) and is unique. What this
polynomial is doing is replacing the simple pole at infinity in (9) by an
order $n$ pole also at infinity in (17). And if we write, for $f$ as in (9),

$F_{n}(f)=q^{-n}+n\sum_{m\geq 1}h_{m,n}(f)q^{m}.$ (18)

the coefficients $h_{m,n}$ are the _Grunsky coefficients_. They have a lot of
interesting properties too 333333 Let us re-iterate this point. We saw in the
footnotes above that the Faber polynomials are the unique degree $n$ monic
polynomials bringing $q^{-1}+{\cal O}(q)$ to $q^{-n}+{\cal O}(q)$ as
$q\rightarrow 0$. Now, our Hecke operator on $j$ of weight 0 as in (17), thus
all $T_{n}(j(z))$ are invariant under $PSL(2;\mathbb{Z})$ since the sum gets
permutated by the action of the modular group. Whence they must be rational
functions in $j(z)$ since the $j$-invariant, being a Hauptmodul 323232 I want
to use the word, rather than call things Hauptmodul because Hauptmodul has a
nasty plural, and we would like to try to call them principal modulus, and we
have done that in some papers.  , generates the function field of invariants.
However, since it has no poles in the upper half plane, they must in fact be
polynomials. In fact, we find that $T(j(q))=q^{-n}+{\cal O}(q)$ as
$q\rightarrow 0$. By uniqueness then, $T_{n}(j)$ must be (up to overall
normalization) the Faber polynomials!  .

###  3.1 Replicable Functions: Norton’s Basis

We can define replicable functions $\\{f^{(m)}\\}$, say, by the same
$q$-expansion as in (9)

$f^{(m)}(z)=\frac{1}{q}+\sum_{k\geq 1}h_{k}^{(m)}q^{k}\ ,$ (19)

I am using the superscript $(m)$ here, and I was talking to John Conway in the
breakfast about this. I think the notation has to be changed. I’ve been using
small letters for coefficients and character, and I’ve used a little $h_{k}$
here instead of $a_{k}$. You can use $a_{k}$ if you wanted to. But the I use
the $h_{k}$ to remind you of the characters of the Monster in the special case
when you restrict to the Monster. We can define a collection of functions here
by, for all $n\geq 1$

$\widehat{T}_{n}(f)=\frac{1}{n}\sum_{\stackrel{{\scriptstyle ad=n}}{{0\leq
b<d}}}f^{(a)}\left(\frac{az+b}{d}\right)=\frac{1}{n}F_{n}(f)\ .$ (20)

What we’ve done is that we kept the Faber action in the above and replaced the
sum over representatives of the sublattices of the function $f$ by the
functions $f^{(a)}$. Those of you who want a glimpse of the future, I can give
you the relation between $f$ and $f^{(a)}$ with reference to the Monster.

To an element of the Monster $g$ there corresponds a function $f_{g}$,
corresponding for the moonshine for this function on the element $g$, then
raising $f$ to the $a-$th replicate power $f^{(a)}$ is the same as replacing
$g$ by $g^{a}$. You will get all this in a short time. There is no need to
make reference to the Monster in order to define this 343434 Historically, the
concept of replicability came about from Conway-Norton’s initial observation
[14] that the moonshine functions (McKay-Thompson series) obeyed certain
functional identities, which they called replication. This is the reason for
studying the type of recurrences in §3.5. The motivation in defining it in the
manner of the present section is to generalize the remarkable fact that action
of the $n$-th Hecke operator on $j$ is the $n$-th Faber polynomial in $j$.
Thus a function of the expansion type (9) is replicable if there exists a
family of function $\\{f^{(a)}\\}$, called replicable functions of $f$ such
that the generalized Hecke operator on these
$\frac{1}{n}\sum\limits_{\stackrel{{\scriptstyle ad=n}}{{0\leq
b<d}}}f^{(a)}\left(\frac{az+b}{d}\right)$ is the Faber polynomial in $f$,
i.e., $\frac{1}{n}F_{n}(f)$. .

There is an inductive definition, and there is only going to be one term,
which is $f^{(n)}$, in the sum in (20), and so you can take out the rest of
this sum and put it on the right hand side and that with will involve only
$f^{(a)}$ with $a<n$ and you have an inductive definition of these functions.
I’ll say more about these Hecke operators.

Now, Norton did all this unaware that it had all been done 70 years before and
before that. Perhaps the easy definition of the Grunsky coefficients $h_{m,n}$
is in terms of this generating function

$\ln\left(\frac{f(p)-f(q)}{\frac{1}{p}-\frac{1}{q}}\right)=-\sum_{m,n\geq
1}h_{m,n}p^{n}q^{m}$ (21)

with $p=e^{2\pi is},\ q=e^{2\pi iz}$. You take the logarithm of the difference
of the function evaluated at two different arguments $p$ and $q$. Remember
that $f(p)$ (resp. $f(q)$) will start with $1/p$ (resp. $1/q$), so dividing
through by $1/p-1/q$ gets rid of the singularities and you end up with a quite
nice series, namely the series on the right hand side of (21). That’s the
generating function for the $h_{m,n}$ and you can expand that to have

$Eq.~{}\eqref{eqn:h_m,n-Norton}=\ln\left(1-pq\sum_{k\geq
1}h_{k}\frac{p^{k}-q^{k}}{p-q}\right)$ (22)

and there are certain consequences of the expansion one of them is that the
$h_{m,n}$’s are polynomial in the $h_{k}$’s with $k\leq m+n$; the value $m+n$
provides you with the grading. I’m using the convention $h_{k}=a_{k}=h_{k,1}$.
When we compare coefficients in the last expansion with the $\ln$ term, we get
this recursive expression

$h_{r,s}=h_{r+s-1}+\frac{1}{r+s}\sum_{m=1}^{r-1}\sum_{n=1}^{s-1}h_{m+n-1}(r+s-m-n)h_{r-m,s-n}.$
(23)

Indeed, in (22), the terms $\displaystyle\frac{p^{k}-q^{k}}{p-q}$ are part of
$p^{s}q^{t}$ where $s+t=k-1$, but you are multiplying by $pq$ so you get a
term involving the expression involving the $h_{k}$’s and the $h_{m,n}$
corresponding to the appropriate exponent. If we call $r+s$ the grade then all
$h_{r-m,s-n}$ in (23) have a lower grade. The $h_{r,s}$ are not integers, but
the largest denominator is $(r,s)$. I believe that’s correct in a sense.
Suppose we want to compute a $q$-coefficient of the function, then that is
typically given by the term $h_{r+s-1}$. So we have a choice: if we want to
compute $h_{k}$ then we can choose $r+s=k+1$ so that $k=r+s-1$, and we can do
that what will give us the initial step, and the game is to try to find a pair
which has a reduced sum.

Simon Norton 353535 Following [62], we can proceed with this formal definition
of a replicable function. Consider a function the form (9), and write its
corresponding Faber polynomial, with Grunsky coefficients $h_{m,n}$ as in
(10). Then $f$ is replicable if $h_{m,n}=h_{r,s}$ whenever
$\gcd(m,n)=\gcd(r,s)$ and ${\rm lcm}(m,n)={\rm lcm}(r,s)$. Equivalently, we
can define replicable functions using the Hecke operators (for weight 0). The
function of our form (9) is replicable, if for each positive interger $n$ and
positive divisor $a|n$, there are functions $f^{(a)}$ of the form of (9) such
that $\widehat{T}_{n}(f):=F_{n}(f(q))=\sum_{\stackrel{{\scriptstyle
ad=n}}{{0\leq b<d}}}f^{(a)}\left(\frac{az+b}{d}\right)$. The functions
$f^{(a)}$ are called replication powers and have the property that
$f^{(k)}(q)=q^{-1}+\sum\limits_{i\geq
1}\left(k\sum\limits_{d|k}\mu(d)h_{dki,\frac{k}{d}}\right)q^{i}\ ,$ (24) where
$\mu$ is the standard Möbius $\mu$-function. has a definition of a replicable
function which is that a function is replicable if
$h_{m,n}=h_{m^{\prime},n^{\prime}}$ whenever $mn=m^{\prime}n^{\prime}$ and
$\gcd(r,s)=\gcd(m,n)$. This is an important definition and we can take
advantage of this in computing the coefficients of a replicable function from
(23).

The following picture shows how to compute the coefficient of $q^{k}$. With
$k$ fixed this gives us a choice of $r,\ s$, so we can draw the line
$x+y=r+s=k+1$, and then the game is to find a point $(r,s)$ on that line that
dominates, if we are lucky, some other point $(r^{\prime},s^{\prime})$ with
the same $\gcd$ and $\rm{lcm}$. We then have the hyperbola $xy=rs$. So if we
can find this $(r^{\prime},s^{\prime})$ on a lower line then we start with, we
can go to this line and proceed to do the same thing again, and each time we
do that there are two possibilities – there is or there is not a line below
it. If the point exists we carry on. If it does not exist we mark the
parameter for the line. So here there is a bunch of lines here, and it turns
out – this is Norton’s straight theorem – that there are 12 values for which
you can’t reduce them further. These are

$23,\ 19,\ 17,\ 11,\ 9,\ 8,\ 7,\ 5,\ 4,\ 3,\ 2,\ 1\ .$

These 12 values are the values of $k$ so that every coefficient of a
replicable function is a polynomial in these 12 values of $h_{k}$. This is
called Norton basis, and that’s a very fundamental result. Now, Conway was
talking yesterday about the work of Atkin, Fong and Smith [2], and by the way,
Borcherds. Well, if Atkin, Fong and Smith had got this theorem at the time
they did their computations they would have only needed at worst 24
coefficients to establish the result of the moonshine conjecture and these
modular functions. But they didn’t have it at the time and so that wasn’t
accessible to them. This is the general version, and this provides a basis for
all replicable functions.

I want to say a bit more about this, something special, in a minute. If the
function has odd level, that means for the moonshine functions and that means
that for the conjugacy class containing the element has odd order, you don’t
need more than these few at the bottom, in fact, $1,\ 2,\ 3,\ 5$ are
sufficient to do the thing. There is something [12] rather special when you
have odd level, but in general you need all the above 12 elements. I’ll show a
quite neat proof of the theorem in a minute. This is a restatement of the
condition of replicability by Norton saying that

$h_{m,n}=h_{\rm{lcm}(m,\ n),\gcd(m,n)}\ .$ (25)

Think of the Smith normal form of a $2\times 2$ matrix perhaps. You have a
matrix with $m$ and $n$ on the diagonal and it’s equivalent to a matrix with
$\gcd(m,n)$ and $\rm{lcm}(m,\ n)$ on the diagonal. If you check the
coefficients $h_{i}^{(k)}$, where $f^{(k)}(z)=\sum h_{i}^{(k)}q^{i}$, you can
see that

$h_{i}^{(k)}=k\sum_{d|k}\mu(d)h_{k/d,dki},\quad i>0,\ h_{-1}^{(k)}=1,\
h_{0}^{(k)}=0\ .$ (26)

From this, inverting the above, you deduce for all $r,s\in{\mathbb{N}}$

$h_{r,rs}=\sum_{d|r}\frac{1}{d}h_{r^{2}s/d^{2}}^{(d)}\ .$ (27)

This can be rewritten as

$h_{m,n}=\sum_{d|(m,n)}\frac{1}{d}h_{mn/d^{2}}^{(d)}\ .$ (28)

This is the final result.

Now those of you who have read the useful book _A course in Arithmetic_ by
Serre [76], the second half of the book is devoted to things of interest to us
(modular forms), and mentions the Leech lattice and various things to do with
theta functions, you will find a formula very like this without the
superscript $(d)$, and if you follow Serre’s proofs they go through pretty
well word for word in this more general situation.

I’d like to just show you the proof of the Norton basis theorem [66] done by
Cummins [15] because it’s very neat, and maybe other versions around are not
as neat as this 363636 Cf. [63] for more discussions on the algorithms. Of
course, one sees the beginning of the supersingular primes here. .

######  THEOREM 3.0 (Norton Basis Theorem).

The $q$-coefficients of a replicable function are polynomials in $h_{k},\ k\in
B=\\{1,\ 2,\ 3,\ 4,\ 5,\ 7,\ 8,\ 9,\ 11,\ 17,\ 19,\ 23\\}$.

So what we want to do is to prove that for everything that is not in the basis
we can actually reduce the sum $m+n$. So you need to prove that there exists
$m,\ n,\ m^{\prime},\ n^{\prime}\in{\mathbb{N}}$ such that

  * 1.

$m+n=N$, where $N$ is given;

  * 2.

$\mathop{{\rm lcm}}(m,n)=\mathop{{\rm lcm}}(m^{\prime},\ n^{\prime})$;

  * 3.

$\gcd(m,n)=\gcd(m^{\prime},n^{\prime})$;

  * 4.

$m^{\prime}+n^{\prime}<m+n$.

###### Proof.

The first remark is that if we find some result which is true for any number
$N$ then the result can be true for $kN$. That’s a useful thing to look at,
and that means that we don’t need to have common factors in the subscripts.
Here are the cases to go through one by one:

  * i.

$N=2^{k}$ not 2, 4 or 8; we can always look at what we believe to be the basis
and see what we need not worry about. For $N=16$, here is a pair $m=1,\ n=15,\
m^{\prime}=3,\ n^{\prime}=5$. All else is a power of $2=16t$.

  * ii.

For $N$ odd and $2^{a}+1,\ a\geq 4,\ N\geq 17$, we have $m=2^{a}-2,\ n=3,\
m^{\prime}=2^{a-1}-1,\ n^{\prime}=6$.

  * iii.

$N$ odd $N\neq 1+2^{k}$, you subtract 1 from it to get $m=N-1,\ n=1,\
m^{\prime}2^{-r}(N-1),\ n^{\prime}=2^{r}$ with $2^{r}|(N-1)$ is a whole
divisor of $N-1$. You can follow that through $N-1>2^{r}\ \Rightarrow\
N>2^{r}+1\ \Rightarrow\ N(2^{r}-1)>2^{2r}-1\ \Rightarrow\
N>2^{-r}(N-1)+2^{r}$.

  * iv.

Then $N$ even not a power of 2. In this situation $N$ is a going to be a
product of 2, 4 or 8 with 3, 5 or 9. And these cases we look at individually.
For 40 take $m=1,\ n=39,\ m^{\prime}=3,\ n^{\prime}=13$. For 36, $m=1,\ n=35,\
m^{\prime}=5,\ n^{\prime}=7$ also works for 72.

This reduces all the ones that can be reduced, and what you have left over is
in the Norton basis, and this proves what the Norton basis actually is. ∎

###  3.2 Elastica

Now something of interest that maybe someone throw some light on here. If you
write down the Norton basis in this way

|  1 |  2 |  3 |  4 |  5 |  |  7 |  8 |  9 |  |  11 |  |  |  |  |  |  17 |  |  19 |  |  |  |  23
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
23 |  |  |  |  19 |  |  17 |  |  |  |  |  |  11 |  |  9 |  8 |  7 |  |  5 |  4 |  3 |  2 |  1 |

we have a symmetry 373737 In that a number is accompanied by a gap, and vice
versa, except for positions 4 and 10.  in the Norton basis with the exception
of the boxes that are supposed to contain 4 and 10. This was noticed by
Matsutani who is in Yokohama, and he wondered whether this has to do with
Weierstrass gaps [48]. Some of you might know about Weierstrass gaps and the
implications of it if there is some connection but that I don’t know 383838 We
recall the statement of the Weierstraß gap theorem: For a compact genus $g>0$
Riemann surface $X\ni x$, there exists exactly $g$ numbers
$1=n_{1}<n_{2}<\ldots<n_{g}<2g$ such that there does not exist a holomorphic
function on $X\backslash x$ with a pole of order $n_{i}$ at $x$. . A remark to
make is that when you are working with odd level functions, these 1, 2, 3 and
5 are the relevant entries that you need for the basis. Now what about 4 and
10? Well, you can write down expressions for 4 and 10. Basically speaking you
can express $a_{10}$ in terms of $a_{4}$, and I think there might well be an
argument for putting 10 in the above tables and leaving things as they are
otherwise, as you can express $a_{4}$ in terms of $a_{10}$. There is a little
complication which I don’t want to talk about.

This is an interesting remark and this leads to some other work of Matsutani
[56] which suggests that there might be some connection with the variational
problem of Euler and these replicable functions. The variational problem of
Euler is what you get when you take an _elastica_ , and elastica is Euler’s
word. It’s about 1750 or so.

What is an elastica? Take a metal ruler and push it in from the ends it will
bend, and the question is what is the curve that you get when you bend it and
that is a variational problem on the integral on the square of the curvature
over the arc length. That was solved completely by Euler 393939 q.v. Leonhard
Euler, “Methodus inveniendi lineas curvas maximi minimive proprietate
gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti,
chapter Additamentum 1”, eulerarchive.org, E065, 1744. . But, if you
generalized it a bit into what Matsutani calls a quantized version [57] you
get some interesting objects, and genus 0 functions come up, rather than
replicable functions. But whether there is a connection or not, I don’t know
404040 The reader is referred to [16] for a discussion on the significance of
the genus zero property.  . So that’s a curiosity which might be worth
pursuing.

###  3.3 Faber Polynomials and Symmetric Functions

Let me come back to the Faber polynomials. They haven’t been much studied,
really. I’d like to mention some things here. I think the first of the
following identities is perhaps the most important one to remember because
it’s easy to remember

$1+\sum_{n\geq
1}h_{n}t^{n}=\prod_{i}(1-x_{i}t)^{-1}=\exp\left(\sum\frac{t^{n}}{n}\cdot\sum_{i}x^{n}_{i}\right)\
.$ (29)

The $h_{n}$’s are called the complete homogeneous symmetric functions 414141
It is also called the plethystic exponential and has been the key to a
programme of counting gauge-invariant operators in quantum field theories
[31]. . Symmetric functions can be expressed in terms of $x_{1},\
x_{2}\cdots$, and the $h_{n}$ is of degree $n$ and is a sum over the $x_{i}$’s
whether or not they are equal. So you are looking at sums of $x_{i}$’s where
$x_{i_{1}}\geq x_{i_{2}}\geq\cdots$. The functions on the right hand side of
(29) are generating functions for these homogeneous symmetric functions 424242
This has been interpreted as fugacity-inserted plethystic exponential of a
Hilbert series in the context of D-brane gauge theories [7] and as Witt
vectors in [32]. .

McDonald [58] on symmetric functions shows that there is an involution that
transforms the homogeneous symmetric functions to the elementary ones, and
thus involution changes the generating functions. So there is a relation
between

$1+\sum_{n\geq
1}b_{n}t^{n}=\prod_{i}(1+x_{i}t)=\exp\left(\sum\frac{-t^{n}}{n}\cdot\sum_{i}x^{n}_{i}\right)$
(30)

and (29).

Now, what are the Faber polynomial doing? Well, there is a completely
different notion from what I’ve been talking about and that is that the Faber
polynomials are related to a change of basis for a symmetric function. There
are six standard bases for symmetric functions, five of which are well known,
one of which is the Doubilet basis, and is called the forgotten symmetric
functions [Dou]. Anyhow, one could take one of the above functions and
multiply it by $1/t$ so that everything is shifted by 1. You take the
following matrix (you have to be careful about these Faber polynomials)

$A_{n}={\left(\begin{array}[]{ccccc}b_{1}&1&0&\cdots&0\\\
2b_{2}&b_{1}&1&\ddots&\vdots\\\ 3b_{3}&b_{2}&\ddots&\ddots&0\\\
\vdots&\vdots&&\ddots&1\\\ nb_{n}&b_{n-1}&&&b_{1}\\\ \end{array}\right)}\ .$
(31)

We have $F_{n}(b_{1},\cdots,b_{n})=\det(A_{n})$ 434343 That’s the matrix, and
I got into trouble when talking to Serre about this because we were using
different notations and he objected very strongly to this. Note that the
notation in footnote 28 is the unshifted version. . Write
$F_{n}(z)=F_{n}(z,b_{2},\cdots,b_{n})$ with $F_{0}(f)=1,\ F_{1}(f)=f,\
F_{2}(f)=f^{2}-2b_{2},\ F_{3}(f)=f^{3}-3b_{2}f-3b_{3}$. One can think of the
Faber polynomial as $F_{n}(z)$, and remember that I shifted everything by 1 in
(30) (by dividing by $q$); the function $f$ we are dealing with here is

$f(q)=\frac{1}{q}\left(1+\sum_{n\geq 1}b_{n}q^{n}\right)\ .$

For replicable functions $b_{1}=0$ and $b_{k}=h_{k-1}$. Moreover, the
$F_{n}(z)$ are isobaric, meaning homogeneous in the subscripts, in the sense
that if you replace $z$ by $b_{1}$ these are isobaric polynomials in the
$b_{i}$’s. It’s very easy to get signs wrong, for me anyhow, and if you get
things correct for the third one I think you are ok. These are Faber
polynomials, and they come about from solving the Newton relations which I
just described in terms of the roots of the polynomial $F_{n}(z)$ and its
coefficients 444444 That is, the recursion relations for the Faber polynomials
described in footnote 25. . So that’s what these Faber polynomials really are.

As I said yesterday, historically it’s quite interesting that they were
described by Faber in 1903 and mathematicians know them, and there are
certainly due to someone earlier than Faber, but it might be one of this
fairly folklorist things that goes back and that might be even predates
Newton. There isalso Girard 454545 Albert Girard (1595-1632), worked on
fundamental theorem of algebra, symmetric polynomials, Fibonacci numbers,
inter alia. , but I don’t know quite what role he had to play. Anyhow, Faà di
Bruno is the person who predates Faber, and I don’t know what date we are
talking about, probably $1857$; he’s well known for the $n$-th derivative of
the composition of two functions 464646 Francesco Faà di Bruno (1825-88), cf.
“Sullo sviluppo delle Funzioni”, Annali di Scienze Matematiche e Fisiche, 6:
479-480, 1855 and “Note sur une nouvelle formule de calcul differentiel”, The
Quarterly Journal of Pure and Applied Mathematics, 1: 359-360, 1857. .

###  3.4 Norton’s Conjecture on Replicable Functions

Now, I’d like to make a statement on this main outstanding conjecture about
replicable functions and nobody, as far as I know, has tried to solve it, but
it’s not quite as simple as one might wish. This is Norton’s main conjecture
and it is that:

######  CONJECTURE 1 (Norton’s Conjecture).

A function $f$ of the form (9) is replicable if either

  * 0.

$f(q)=1/q+cq$ with $c\in\\{0,\ -1,\ 1\\}$ as we are working with integer
coefficients. Respectively for these values of $c$ we get the $\exp,\ \sin$
and $\cos$ functions 474747Recall that the nome $q=\exp(2\pi i\tau)$., which
we call “Modular Fictions” and to be ignored henceforth.

Or, surprisingly using the modular polynomial you can get some results about
these things, which are consequences of the modular polynomial for them.
Things like the $\cos$ of twice the angle is a polynomial of the $\cos$
whereas the $\sin$ of twice the angle is not a polynomial in the $\sin$. This
was proved by C. Cummins again [15], that this is all there is 484848Indeed,
these are the Chebyshev polynomials; the reader is referred to [62] for
discussions on how these famous polynomials are the simplest replicable
functions..

  * 1.

There exists $N:\ {\Gamma}_{0}(N)\subset
G_{f}\subset\mbox{Nor}({\Gamma}_{0}(N))$, where Nor is the normalizer inside
${\mbox{PSL}_{2}({\mathbb{R}})}$, such that the compact Riemann surface
$\widehat{G_{f}/{\mathfrak{H}}}$ obtained by adding a finite set of
inequivalent cusps has genus 0, $f$ is a principal modulus of this Riemann
surface, and $G_{f}$ is commensurable 494949 Recall that $G_{f}$ was defined
in (16) as the modular subgroup for which $f$ is invariant (weight $k=0$).
with ${\mbox{PSL}_{2}({\mathbb{Z}})}$.

The maximal groups on the right side are called _Helling groups_ , and there
is a paper by Conway [13] called “Understanding groups like ${\Gamma}_{0}(N)$”
which was referred to by J. Duncan. It’s really a pretty piece of work in that
he also proves Helling’s theorem.

What is needed is a proof of this result. I think the proper way to do this is
to use a two-sided decomposition with respect to GL2 of the adèles, and in
that way one should be able to pick up the primes dividing the Monster’s
order, which are the primes you find inside the above levels $N$, namely the
15 supersingular primes.

What has been done to now, I might say something about computations if there
is time, it’s been done by some people; I had some visitors that did some
work, C. Cummins did some, but the bulk of this stuff was done by S. Norton
and I could describe it. Basically speaking, it was a very local picture that
was used to do the computations to find these replicable functions. There are
one or two functions that you know classically from Weber’s [78] and
Schläfli’s [73] work, and from there you can build up other functions which
are closely related to replicable functions. You just keep looking and making
sure that the genus is zero. But it is conceivable that something has been
left out, it’s unlikely but it’s possible.

The number 616 is the number of replicable functions there are (with integer
coefficients). That in itself is a quite interesting number, and there is
amusing reference to J. Conway’s remark. Remember that he was talking
about505050 The group is defined as follows. Consider a graph $G_{p,q,r}$ with
a single tri-vertex, say $a$ and 3 strands, each consisting of respectively
$p$, $q$ and $r$ nodes joined up; call these $b_{1,\ldots,p}$,
$c_{1,\ldots,q}$ and $d_{1,\ldots,r}$. Thus there are $p+q+r+1$ nodes in
total. The group $Y_{p,q,r}$ is a Coxeter-type group with one generator
associated to each of the nodes and presentation  $Y_{p,q,r}=\langle
a,b_{1,\ldots,p},c_{1,\ldots,q},d_{1,\ldots,r}|(gg^{\prime})^{O(g,g^{\prime})}=(ab_{1}b_{2}ac_{1}c_{2}ad_{1}d_{2})^{10}=1\rangle\
,$ where $g$ and $g^{\prime}$ are any of the $a,b,c,d$ generators and that
$O(g,g^{\prime})=3$ (respectively 2) if $g$ and $g^{\prime}$ are adjacent
(respectively, not adjacent). It was shown [46] that $Y_{5,5,5}\simeq
Y_{4,4,4}\simeq\mathbb{M}\wr C_{2}$, the wreath product of the monster with
the cyclic group of order 2.  a group $Y_{555}$ or perhaps $Y_{666}$. Well,
$666$ as you know occurs in the Bible, it’s sort of a bad number, and I was
giving this talk in Norway to the mathematical department and I mentioned this
$616$ as the number of replicable functions and a person in the back went out
and he came back to me with a transparency on which there is something called
the “Oxyrhynchus Papyrus” from few hundred A. D. 515151 Manuscripts discovered
in the C19th near Oxyrhynchus, Egypt, dating from 1st to 6th century AD.,
mostly housed in the Ashmolean in Oxford.  In the papyrus, which is written in
Greek– it’s very readable, it says that the real number shouldn’t be 666; it
should be 616. So that was very quite amusing. I don’t know why or how he came
across to this.

Here is another piece of numerology, entirely frivolous but nevertheless
grounded on some deep mathematics, which may amuse you. The Leech lattice
$\Lambda_{24}$, the famous even unimodular lattice in 24 dimensions, can be
constructed from the even Lorentzian unimodular lattice $II_{25,1}$ in 26
dimensions using a Weyl vector $w=(0,1,2,3,\ldots,23,24;70)$. Then
$\Lambda_{24}$ is realized as $w^{\perp}/w$. That $w$ is indeed an integral
vector in $II_{25,1}$ follows from the remarkable Diophantine condition 525252
Discussions on this equation and the emergence of 42 from $j(q)$ are presented
in [42], in a volume in honour of J. H. Conway. The number of pages of the
present notes is, of course, 42.

$1^{2}+2^{2}+3^{2}+\ldots+23^{2}+24^{2}=70^{2}\ .$ (32)

Now, consider the first 24 $q$-series coefficients of the (normalized) modular
$J$-function, viz, $196884,21493760,864299970,20245856256,\ldots$ (that is we
do not include the constant term 744 nor $1/q$ and start from ${\cal O}(q)$).
Sum the squares of these 24 numbers and compute it modulo 70, you will get 42,
which we all know to be the answer to the ultimate question of life, the
universe and everything. So, here again, you see how Moonshine encodes all
things.

###  3.5 Mahler’s Recurrence Relations

Now, there is another interesting paper [1] where we used recurrences to build
up these series. What is not very well known is the recurrence that we use
which people call Borcherds’ recurrence 535353 The sort of recurrences which
arise from Borcherds’ proof is the remarkable one such as
$j(p)-j(q)=\left(\frac{1}{p}-\frac{1}{q}\right)\prod\limits_{m,n=1}^{\infty}(1-p^{n}q^{m})^{c_{nm}}\
,$ where $c_{k}$ are the q-expansion coefficients of $j(q)$. We will see more
recurrences in §3.5. , which is fair enough, is really due to Mahler and
Mahler wrote his paper in 1974, published in 1976 in the journal of the
Australian Math. Soc. [52].

He was a very remarkable number theorist. He was a cripple and his father knew
Carl Ludwig Siegel. So he got to get a rather good education without going
through the usual processes of university and then he became a refugee in
Manchester for 30 years and then he went from an assistant professor to a
personal Chair, in Canberra where he lived and died not a long ago.

He wrote me a letter in 1982 or 1983 in which he said that he thought his
paper has something to do with moonshine, and if I would come to Canberra and
discuss it with him before he dies. I was in Montreal and he was in Australia.
I went out there and unfortunately I got ill and I came back. So we never
actually met, but Mahler recurrence is the recurrence that does the
calculations.

We observe that (q.v. also [43])

$\displaystyle 360+256=$ $\displaystyle 616$ $\displaystyle 120+2\times 248=$
$\displaystyle 616\ .$ (33)

Maybe you can explain this. I’ll give you the Mahler recurrence. I’m just
going back a bit. I already remarked upon

$f^{(n)}(nz)=\frac{1}{n}F_{n}(f(z))-\frac{1}{n}\sum_{\begin{subarray}{c}ad=n\\\
0\leq b<d\\\ a<n\end{subarray}}f^{(a)}\left(\frac{az+b}{d}\right)\ ,$

and that $f^{(n)}(nz)$ is invariant under $z\ \mapsto\ z+1/n$. Cummins and
Norton have proved the replicability of rational Hauptmoduln [17], and
remember that the replicability property is essentially combinatorial. You are
just saying that things work nicely with Hecke operators whereas if you want
to prove results about replicable being Hauptmodul you have to go from the
combinatorial side to the analytic and it’s a much more difficult thing to do.

There is some very remarkable work done by a student of Arne Meurman in
Sweden. His name is Dmitry Kozlov [51]. Masao Koike is the first to realize
what the generalized Hecke operator is. And if I have done it correctly. When
$n=p$ is prime, the classical Hecke operator can be expressed in terms of the
the Atkin $U_{-}$-operator and if you know it for $p$ you know it for all
integers because of the multiplicative property, and the Adams’
$V_{-}$-operator

$\displaystyle T_{p}$ $\displaystyle=\frac{1}{p}V_{p}+U_{p}$ $\displaystyle
V_{p}$ $\displaystyle:\ f(q)\ \mapsto\ f(q^{p})$ $\displaystyle U_{p}$
$\displaystyle:\ a_{n}q^{n}\ \mapsto\ a_{pn}q^{n}\ .$ (34)

Koike recognizes this for the twisted Hecke operator in the very early days

$\widehat{T}_{p}=\frac{1}{p}\Psi^{p}\circ{V}_{p}+U_{p}$ (35)

with $\widehat{V}_{p}=\Psi^{p}\circ V_{p}:\ f(q)\ \mapsto\ f^{(p)}(q^{p})$.
Now if you go back to the moonshine to remember what this does for you, this
means that the coefficients are traces which are the sums of the initial
$p-$th power function. That’s the context of moonshine but the rest is true
for replicable functions generally.

There are some characteristic classes associated with this, and this is called
the Bott Cannibalistic Class. So when I was in Harvard I asked Bott about it,
but he told me he had forgotten so I don’t think we’ll get any further there.
By the way there is a remarkable meeting in a week or two in Montreal, I think
in June, a week on Bott’s legacy with very good speakers including Witten and
Atiyah among others in the University of Montreal 545454 This is the
conference “A Celebration of the Mathematical Legacy of Raoul Bott”, CRM, June
9-13, 2008, Montreal. One can find the proceedings in CRM Proceedings &
Lecture Notes, Volume 50, AMS 2010. . I just make a passing remark is that
$f\equiv f^{(p)}\bmod\,p$ because in $f^{(p)}$ the coefficients are sums of
$p$-th powers of the coefficients of $f$. This is the work of Koike, and there
is a nice survey article by him in Japanese in Sugaku, and there is an English
version in number 160 of the AMS translations [49].

I think Mahler must be the first or the only mathematician, unless Conway has
done the same thing, to publish a calculator program in the Royal Society of
London Proceedings, but he did that in an subsequent paper where he does his
computations. What’s extremely interesting about Mahler is that for each prime
$p$ he has a recurrence relation to compute the coefficients of these
functions and he is really computing the $q$-coefficients of the $j$-function.
There are several people who have done this for $j$, where the actual setup
will work for replicable functions in general. This was true by Kozlov’s
thesis which was on the $j$-function, but in fact it applies to all replicable
functions. The same with Mahler; he has this recurrence relations for all
prime $p$ for the $j$-function and he tries to use them in a broader context
to other functions that have arisen in [34] and others and it doesn’t work and
he doesn’t see why it doesn’t work.

Today we know why it doesn’t work. It’s because the level of the function and
the level of the Hecke operator are not coprime, and that’s the modification
that you need to go from Borcherds’ formula to Mahler’s formula. So I’m just
going to give it to you here. For $p=2$ Mahler has used

$\displaystyle f\left(\frac{z}{2}\right)+f\left(\frac{z+1}{2}\right)+f(2z)$
$\displaystyle=f(z)^{2}-2a_{1}$ $\displaystyle
f\left(\frac{z}{2}\right)f(2z)+f\left(\frac{z+1}{2}\right)f(2z)+f\left(\frac{z}{2}\right)f\left(\frac{z+1}{2}\right)$
$\displaystyle=2a_{2}f-f+2(a_{4}-a_{1})\ ,$ (36)

which might be look at as the elementary symmetric functions of degree 1 and 2
respectively in the function values $f\left(\frac{z}{2}\right),\
f\left(\frac{z+1}{2}\right)$ and $f(2z)$. This is for the $j$-function
originally, and then you have to make modifications. The modifications are
simple to make; whenever you see a 2 in the $f$-value, say in $f(2z)$ you
change it to $f^{(2)}(2z)$. These are the recurrence relations from Mahler

$\displaystyle a_{4k}$
$\displaystyle=\sum_{j=1}^{k-1}a_{j}a_{2k-j}+\frac{1}{2}(a_{k}^{2}-a_{k}^{(2)})\
,$ $\displaystyle a_{4k+}$
$\displaystyle=a_{2k+3}\sum_{j=1}^{k}a_{j}a_{2k+2-j}+\frac{1}{2}(a_{k+1}^{2}-a_{k+1}^{(2)})+\frac{1}{2}(a_{2k}^{2}-{a_{2k}^{(2)}})-a_{2}a_{2k}+$
$\displaystyle\qquad\qquad+\sum_{j=1}^{k}{a_{j}^{(2)}}a_{4k-4j}+\sum_{j=1}^{2k-1}(-1)^{j}a_{j}a_{4k-j}\
,$ $\displaystyle a_{4k+2}$
$\displaystyle=a_{2k+2}+\sum_{j=1}^{k-1}a_{j}a_{2k+1-j}\ ,$ $\displaystyle
a_{4k+3}$
$\displaystyle=a_{2k+4}\sum_{j=1}^{k+1}a_{j}a_{2k+3-j}+\frac{1}{2}(a_{2k+1}^{2}-{a_{2k+1}})-a_{2}a_{2k+1}+$
(37)
$\displaystyle\qquad\qquad+\sum_{j=1}^{k}{a_{j}^{(2)}}a_{4k+2-4j}+\sum_{j=1}^{2k}(-1)^{j}a_{j}a_{4k+2-j}\
.$

which do come from

$\displaystyle
f\left(\frac{z}{2}\right)+f\left(\frac{z+1}{2}\right)+f^{(2)}(2z)$
$\displaystyle=f(z)^{2}-2a_{1}$ $\displaystyle
f\left(\frac{z}{2}\right)f^{(2)}(2z)+f\left(\frac{z+1}{2}\right)f^{(2)}(2z)+f\left(\frac{z}{2}\right)f\left(\frac{z+1}{2}\right)$
$\displaystyle=2a_{2}f-f+2(a_{4}-a_{1})\ ,$ (38)

and these are Borcherds’ relations. That’s the difference between Mahler’s and
Borcherds’.

Notice that the original ones are universal, and if you have any function in
this case of odd level these are the recurrence relations for its coefficients
and if you have a function of even level the you have to know what $f^{(2)}$
is in order to make use them; you have to use coefficients from $f^{(2)}$ in
order to build up these relations. They come in a group of four. They are
extremely good for computing several hundreds coefficients quite easily, but
if you speak to someone like Atkin few hundreds is nothing; he computes the
first 10,000.

Now, to go back to some remarks a bit earlier on. We find
$f^{(p)}=\sum_{n}h_{n}^{(p)}q^{n}$ and

$h_{n}^{(p)}=ph_{pn,p}-pa_{p^{2}n}\ .$ (39)

On $\mathbb{M}:\ f_{\langle g\rangle}\ \longrightarrow\ f_{\langle
g\rangle}^{(k)}=f_{\langle g^{k}\rangle}$, this is the replication formula I
mentioned to you 555555 Thus we come full circle back to the Monster
$\mathbb{M}$. Recall that for conjugacy classes, and in particular of cyclic
groups $\langle g\rangle$, one can write down McKay-Thompson series, these
obey replication identities. The question then is whether group structure in
$\mathbb{M}$, such as the power map $g\mapsto g^{k}$ are reflected in the
replicable functions $f\mapsto f^{(k)}$. . Since $|\mathbb{M}|<\infty$ there
exists $k=k_{0}$ such that $g^{k_{0}}=id$ and so $f_{\langle
g^{k_{0}}\rangle}=f_{\langle id\rangle}=j$. So in a sense you can think of the
functions we start with as replication roots of the $j$-function.

## Acknowledgements

JM is grateful to the organizers for the Kashiwa conference for their warm
hospitality and that of the NSERC of Canada. YHH would like to thank the
Science and Technology Facilities Council, UK, for grant ST/J00037X/1, and the
Chinese Ministry of Education, for a Chang-Jiang Chair Professorship at NanKai
University.

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