Open Access

A proof of the Thompson moonshine conjecture

Research in the Mathematical Sciences20163:36

DOI: 10.1186/s40687-016-0084-7

Received: 1 July 2016

Accepted: 1 September 2016

Published: 14 December 2016

Abstract

In this paper, we prove the existence of an infinite-dimensional graded supermodule for the finite sporadic Thompson group Th whose McKay–Thompson series are weakly holomorphic modular forms of weight \(\frac{1}{2}\) satisfying properties conjectured by Harvey and Rayhaun.

Mathematics Subject Classification

11F22 11F37

1 Introduction and statement of results

One of the greatest accomplishments of twentieth-century mathematics was certainly the classification of finite simple groups. The study of the representation theory of one of these groups, the Monster group \(\mathbb {M}\), the largest of the 26 sporadic simple groups, revealed an intriguing connection to modular forms: McKay and Thompson [41] were the first to observe that the dimensions of irreducible representations of the Monster group are closely related to Klein’s modular invariant
$$\begin{aligned} J(\tau )=q^{-1}+\sum _{n=1}j_nq^n=q^{-1}+196\,884q+21\,493\,760q^2+864\,299\,970q^3+O(q^4), \end{aligned}$$
the Hauptmodul for the full modular group. Here and throughout, \(\tau =u+iv,\,u,v\in \mathbb {R}\), is a variable living in the complex upper half-plane \(\mathfrak {H}\) and \(q:=e^{2\pi i\tau }\). More precisely, the first irreducible representations of \(\mathbb {M}\) have dimensions
$$\begin{aligned} \chi _1(1)=1,\quad \chi _2(1)=196\,883,\quad \chi _3(1)=21\,296\,876,\quad \chi _4(1)=842\,609\,326, \end{aligned}$$
so that one has (for example)
$$\begin{aligned} j_1&=\chi _1(1)+\chi _2(1),\\ j_2&=\chi _1(1)+\chi _2(1)+\chi _3(1),\\ j_3&=2\chi _1(1)+2\chi _2(1)+\chi _3(1)+\chi _4(1). \end{aligned}$$
Thompson further observed that a similar phenomenon occurs when one considers combinations of \(\chi _j(g)\) for other \(g\in \mathbb {M}.\) Based on theses observations, he conjectured in [40] that there should exist an infinite-dimensional graded \(\mathbb {M}\)-module that reflects these combinations. Conway and Norton [14] made this more precise (the so-called Monstrous Moonshine conjecture), conjecturing that for each conjugacy class of \(\mathbb {M}\) there is an explicit associated genus 0 subgroup of \({\text {SL}}_2(\mathbb {R})\) whose normalized Hauptmodul coincides with the so-called McKay–Thompson series (see Sect. 2.3 for a definition of this term) of the conjugacy class with respect to the module. An abstract proof of this conjecture (i.e., one whose construction of the module is done only implicitly in terms of the McKay–Thompson series) was announced by Atkin–Fong–Smith [20, 36]. Their proof was based on an idea of Thompson. Later, the full conjecture was proven by Borcherds [2] using a vertex operator algebra previously constructed by Frenkel et al. [22].

Conway and Norton also observed in [14] that Monstrous Moonshine would imply Moonshine phenomena for various subgroups of the Monster. Queen [32] computed the Hauptmoduln associated with conjugacy classes of several sporadic groups, among them the Thompson group. Note, however, that the Moonshine phenomenon we prove in this paper is not directly related to this generalized moonshine considered by Queen, but more reminiscent of the following. In 2011, Eguchi et al. [19] observed connections like the ones between the dimensions of irreducible representations of the Monster group and coefficients of the modular function J for the largest Mathieu group \(M_{24}\) and a certain weight \(\tfrac{1}{2}\) mock theta function. Cheng et al. [12, 13] generalized this to Moonshine for groups associated with the 23 Niemeier lattices, the non-isometric even unimodular root lattices in dimension 24, which has become known as the Umbral Moonshine Conjecture. Gannon proved the case of Mathieu moonshine in [23], and the full Umbral Moonshine Conjecture was then proved in [18] by Duncan, Ono, and the first author.

In [25], Harvey and Rayhaun conjecture Moonshine for the Thompson group Th, a sporadic simple group of order
$$\begin{aligned} 90\,745\,943\,887\,872\,000= 2^{15} \cdot 3^{10} \cdot 5^3 \cdot 7^2 \cdot 13 \cdot 19 \cdot 31 \end{aligned}$$
(see Sect. 2.1 for definitions and notation).

Conjecture 1.1

There exists a graded Th-supermodule
$$\begin{aligned} W=\bigoplus _{\begin{array}{c} m=-3 \\ m\equiv 0,1\ \left( {\text {mod}} 4\right) \end{array}}^\infty W_m, \end{aligned}$$
where for \(m\ge 0\) the graded component \(W_m\) has vanishing odd part if \(m\equiv 0\pmod 4\) and vanishing even part if \(m\equiv 1\pmod 4\), such that for all \(g\in Th\) the McKay–Thompson series
$$\begin{aligned} \mathcal {T}_{[g]}(\tau ):=\sum _{\begin{array}{c} m=-3 \\ m\equiv 0,1\ \left( {\text {mod}} 4\right) \end{array}}^\infty \mathrm{strace}_{W_m}(g)q^m \end{aligned}$$
is a specifically given weakly holomorphic modular form (see Sect. 2) of weight \(\tfrac{1}{2}\) in the Kohnen plus space.

Here, we prove this conjecture.

Theorem 1.2

Conjecture 1.1 is true. Moreover, if \(g\in Th\) does not lie in the class 12A or 12B (see Remark 3.3), the McKay–Thompson series \(\mathcal {T}_{[g]}(\tau )\) is the unique weakly holomorphic modular form in \(M_\frac{1}{2}^{!,+}(4|g|,\psi _{[g]})\) (see Sect. 2 for notation) satisfying the following conditions:
  • its Fourier expansion is of the form \(2q^{-3}+\chi _2(g)+(\chi _4(g)+\chi _5(g))q^4+O(q^5)\), where \(\chi _j\) is the \(j{\mathrm{th}}\) irreducible character of Th as given in Tables 1, 2, 3 and 4, and all its Fourier coefficients are integers.

  • if |g| is odd, then the only other pole of order \(\tfrac{3}{4}\) is at the cusp \(\frac{1}{2|g|}\); otherwise, there is only the pole at \(\infty \). It vanishes at all other cusps.

If \(|g|\ne 36\), it suffices to assume that the Fourier expansion is of the form \(2q^{-3}+\chi _2(g)+O(q^4)\).
The proof of Theorem 1.2, like the proofs of the Mathieu and Umbral Moonshine Conjectures, relies on the following idea. For each \(n\ge 0\), the function defined by
$$\begin{aligned} \omega _n{:}\ Th \rightarrow \mathbb {C},\quad g\mapsto \alpha _{[g]}(n), \end{aligned}$$
where we write
$$\begin{aligned} \mathcal {T}_{[g]}(\tau ):=\sum _{n=-3}^\infty \alpha _{[g]}(n)q^n \end{aligned}$$
for the tentative McKay–Thompson series conjectured by Harvey and Rayhaun, is a complex-valued class function on the Thompson group. Therefore, we need to show that \((-1)^n\omega _n\) is a character of Th for every n, which is equivalent to the assertion that
$$\begin{aligned} (-1)^n\omega _n(g)=\sum _{j=1}^{48} m_j(n)\chi _j(g), \end{aligned}$$
(1.1)
where \(m_1,\ldots ,m_{48}\) are nonnegative integers and \(\chi _1,\ldots ,\chi _{48}\) are the irreducible characters of Th as defined in Tables 1, 2, 3 and 4. Using a variant of Brauer’s characterization of generalized characters due to Thompson (see, e.g., [36, Theorem 1.1]), one can reduce this to a finite calculation.

The rest of the paper is organized as follows. In Sect. 2, we recall some relevant definitions on supermodules, harmonic Maaß forms, and the construction of the (tentative) McKay–Thompson series in [25]. In Sect. 3, we show that these series are in fact all weakly holomorphic modular forms (instead of harmonic weak Maaß forms) with integer Fourier coefficients and that all the multiplicities \(m_j\) in (1.1) are integers. Section 4 is concerned with the proof of the positivity of these multiplicities, which finishes the proof of Theorem 1.2. Finally, in Sect. 5, we give some interesting observations connecting the McKay–Thompson series to replicable functions.

2 Preliminaries and notation

2.1 Supermodules

We begin by introducing the necessary definitions and notations in Conjecture 1.1.

Definition 2.1

A vector space V is called a superspace, if it is equipped with a \(\mathbb {Z}/2\mathbb {Z}\)-grading \(V=V^{(0)}\oplus V^{(1)}\), where \(V^{(0)}\) is called the even and \(V^{(1)}\) is called the odd part of V. For an endomorphism \(\alpha \) of V respecting this grading, i.e., \(\alpha (V^{(i)})\subseteq V^{(i)}\), we define its supertrace to be
$$\begin{aligned} \mathrm{strace}(\alpha )={\text {trace}}\left( \alpha |_{V^{(0)}}\right) -{\text {trace}}\left( \alpha |_{V^{(1)}}\right) . \end{aligned}$$
Now let G be a finite group and \((V,\rho )\) a representation of G. If the G-module V admits a decomposition into an even and odd part as above which is compatible with the G-action, we call V a G-supermodule. For a G-subsupermodule W of V and \(g\in G\), we then write
$$\begin{aligned} \mathrm{strace}_W(g):=\mathrm{strace}\left( \rho |_W(g)\right) . \end{aligned}$$
Note that \(\mathrm{strace}(g)\) only depends on the conjugacy class of g, which we denote by [g].

2.2 Harmonic Maaß forms

Harmonic Maaß forms are an important generalization of classical, elliptic modular forms. In the weight 1 / 2 case, they are intimately related to the mock theta functions, a term coined by Ramanujan in his famous 1920 deathbed letter to Hardy. It took until the first decade of the twenty-first century before work by Zwegers [43], Bruinier and Funke [7] and Bringmann and Ono [5, 6] established the “right” framework for these enigmatic functions of Ramanujan’s, namely that of harmonic Maaß forms. Since then, there have been many applications of harmonic Maaß forms both in various fields of pure mathematics, see for instance [1, 4, 9, 16], among many others, and mathematical physics, especially in regard to quantum black holes and wall crossing [15] as well as Mathieu and Umbral Moonshine [12, 13, 18, 23]. For a general overview on the subject, we refer the reader to [31, 42].

Recall the definition of the congruence subgroup
$$\begin{aligned} \varGamma _0(N):=\left\{ \gamma =\begin{pmatrix} a &{}\quad b \\ c &{}\quad d \end{pmatrix} \in {\text {SL}}_2(\mathbb {Z}){:}\,N|c\right\} \end{aligned}$$
of the full modular group \({\text {SL}}_2(\mathbb {Z})\).

Definition 2.2

We call a smooth function \(f{:}\,\mathfrak {H}\rightarrow \mathbb {C}\) a harmonic (weak)1 Maaß form of weight \(k\in \tfrac{1}{2}\mathbb {Z}\) of level N with multiplier system \(\psi \), if the following conditions are satisfied:
  1. (1)
    We have \(f|_k\gamma (\tau )=\psi (\gamma )f(\tau )\) for all \(\gamma \in \varGamma _0(N)\) and \(\tau \in \mathfrak {H}\), where we define
    $$\begin{aligned} f|_k\gamma (\tau ):= {\left\{ \begin{array}{ll} (c\tau +d)^{-k}f\left( \frac{a\tau +b}{c\tau +d}\right) &{}\quad \text {if }k\in \mathbb {Z}, \\ \left( \left( \frac{c}{d}\right) \varepsilon _d\right) ^{2k}\left( \sqrt{c\tau +d}\right) ^{-2k}f\left( \frac{a\tau +b}{c\tau +d}\right) &{}\quad \text {if }k\in \frac{1}{2}+\mathbb {Z}. \end{array}\right. } \end{aligned}$$
    with
    $$\begin{aligned} \varepsilon _d:= {\left\{ \begin{array}{ll} 1 &{}\quad d\equiv 1\pmod {4}, \\ i &{}\quad d\equiv 3\pmod 4. \end{array}\right. } \end{aligned}$$
    and where we assume 4|N if \(k\notin \mathbb {Z}\).
     
  2. (2)
    The function f is annihilated by the weight k hyperbolic Laplacian,
    $$\begin{aligned} \varDelta _k f:=\left[ -v^2\left( \frac{\partial ^2}{\partial u^2}+\frac{\partial ^2}{\partial v^2}\right) +ikv\left( \frac{\partial }{\partial u}+i\frac{\partial }{\partial v}\right) \right] f\equiv 0. \end{aligned}$$
     
  3. (3)

    There is a polynomial \(P(q^{-1})\) such that \(f(\tau )-P(e^{-2\pi i\tau })=O(e^{-cv})\) for some \(c>0\) as \(v\rightarrow \infty \). Analogous conditions are required at all cusps of \(\varGamma _0(N)\).

     
We denote the space of harmonic Maaß forms of weight k, level N and multiplier \(\psi \) is denoted by \(H_k(N,\psi )\), where we omit the multiplier if it is trivial.

Remark 2.3

  1. (1)

    Obviously, the weight k hyperbolic Laplacian annihilates holomorphic functions, so that the space \(H_k(N,\psi )\) contains the spaces \(S_k(N,\psi )\) of cusp forms (holomorphic modular forms vanishing at all cusps), \(M_k(N,\psi )\) of holomorphic modular forms, and \(M_k^!(N,\psi )\) of weakly holomorphic modular forms (holomorphic functions on \(\mathfrak {H}\) transforming like modular forms with possible poles at cusps).

     
  2. (2)

    It should be pointed out that the definition of modular forms resp. harmonic Maaß forms with multiplier is slightly different in [25], where the multiplier is included into the definition of the slash operator \(f|_k\gamma \), so that multipliers here are always the inverse of the multipliers there.

     

It is not hard to see from the definition that harmonic Maaß forms naturally split into a holomorphic part and a non-holomorphic part (see, for example, equations (3.2a) and (3.2b) in [7]).

Lemma 2.4

Let \(f\in H_{k}(N,\psi )\) be a harmonic Maaß form of weight \(k\ne 1\) such that \(\psi \left( \left( {\begin{matrix} 1 &{}\quad 1 \\ 0 &{}\quad 1\end{matrix}}\right) \right) =1\). Then there is a canonical splitting
$$\begin{aligned} f(\tau )=f^+(\tau )+f^-(\tau ), \end{aligned}$$
(2.1)
where for some \(m_0\in \mathbb {Z}\) we have the Fourier expansions
$$\begin{aligned} f^+(\tau ):=\sum \limits _{n=m_0}^\infty c_f^+(n)q^n, \end{aligned}$$
and
$$\begin{aligned} f^-(\tau ):=\sum \limits _{\begin{array}{c} n=1 \end{array}}^\infty \overline{c_f^-(n)}n^{k-1}\varGamma (1-k;4\pi nv)q^{-n}, \end{aligned}$$
where \(\varGamma (\alpha ;x)\) denotes the usual incomplete Gamma-function.

In the theory of harmonic Maaß forms, there is a very important differential operator that associates a weakly holomorphic modular form to a harmonic Maaß form [7, Proposition 3.2 and Theorem 3.7], often referred to as its shadow.2

Proposition 2.5

The operator
$$\begin{aligned} \xi _{k}{:}\,H_{k}(N,\psi )\rightarrow S_{2-k}\left( N,\overline{\psi }\right) ,\,f\mapsto \xi _{k}f:=2iv^{k}\overline{\frac{\partial f}{\partial \overline{\tau }}} \end{aligned}$$
is well defined and surjective with kernel \(M_{k}^!(N,\nu )\). Moreover, we have that
$$\begin{aligned} (\xi _{k}f)(\tau )=-(4\pi )^{1-k}\sum \limits _{n=1}^\infty c_f^-(n)q^n \end{aligned}$$
and we call this cusp form the shadow of f.
The \(\xi \)-operator can also be used to define the Bruinier–Funke pairing
$$\begin{aligned} \{\cdot ,\cdot \}{:}\,S_{2-k}(N,\overline{\psi })\times H_k(N,\psi ),\quad (g,f)\mapsto \{g, f\}:=\langle g, \xi _k(f)\rangle , \end{aligned}$$
where \(\langle \cdot ,\cdot \rangle \) denotes the Petersson inner product on the space of cusp forms.

We will later use the following result [7, Proposition 3.5].

Proposition 2.6

For \(f=f^++f^-\) as in Lemma 2.4 and \(g=\sum _{n=1}^\infty b_nq^n\in S_{2-k}(N,\overline{\psi })\) such that f grows exponentially only at the cusp \(\infty \) and is bounded at all other cusps of \(\varGamma _0(N)\), we have that
$$\begin{aligned} \{g,f\}=\sum _{n<0} c_f^+(n)b(-n). \end{aligned}$$
If f has poles at other cusps, the pairing is given by summing the corresponding terms using the q-series expansions for f and g at each such cusp.

2.3 Rademacher sums and McKay–Thompson series

Here, we recall a few basic facts about Poincaré series, Rademacher sums, and Rademacher series. For further details, the reader is referred to [10, 11, 17] and the references therein.

An important way to construct modular forms of a given weight and multiplier is through Poincaré series. If one assumes absolute and locally uniform convergence, then the function
$$\begin{aligned} P^{[\mu ]}_{\psi ,k}(\tau ):=\sum _{\gamma \in \varGamma _\infty \setminus \varGamma _0(N)} \overline{\psi }(\gamma )q^\mu |_k\gamma , \end{aligned}$$
where \(\varGamma _\infty =\left\langle \pm T\right\rangle \) with \(T=\left( {\begin{matrix} 1 &{}\quad 1 \\ 0 &{}\quad 1 \end{matrix}}\right) \) denotes the stabilizer of the cusp \(\infty \) in \(\varGamma _0(N)\) and \(\mu \in \frac{\log (\psi (T))}{2\pi i}+\mathbb {Z}\), transforms like a modular form of weight k with multiplier \(\psi \) under the action of \(\varGamma _0(N)\) and is holomorphic on \(\mathfrak {H}\). In fact it is known that we have absolute and locally uniform convergence for weights \(k>2\), and in those cases, \(P^{[\mu ]}_{\psi ,k}\) is a weakly holomorphic modular form, which is holomorphic if \(\mu \ge 0\) and cuspidal if \(\mu >0\).
For certain groups and multiplier systems, one can obtain conditionally, locally uniformly convergent series, now called Rademacher sums, for weights \(k\ge 1\), by fixing the order of summation as follows. Let for a positive integer K
$$\begin{aligned} \varGamma _{K,K^2}(N):=\left\{ \left( {\begin{matrix} a &{}\quad b \\ c &{}\quad d \end{matrix}}\right) \in \varGamma _0(N)\, : \, |c|<K\text { and }|d|<K^2\right\} . \end{aligned}$$
One can then define the Rademacher sum
$$\begin{aligned} R^{[\mu ]}_{\psi ,k}(\tau )=\lim \limits _{K\rightarrow \infty } \sum _{\gamma \in \varGamma _\infty \setminus \varGamma _{K,K^2}(N)} \overline{\psi }(\gamma )q^\mu |_k\gamma . \end{aligned}$$
Rademacher observed [33] that one can in addition regularize the summands individually to obtain convergence for weights \(k<1\), see, e.g., equations (2.26) and (2.27) in [11]. He originally applied this to obtain an exact formula for the coefficients of the modular j-function.
In this paper, we especially need to look at Rademacher sums of weights \(\tfrac{1}{2}\) and \(\tfrac{3}{2}\) for \(\varGamma _0(4N)\) with multiplier
$$\begin{aligned} \psi _{N,v,h}\left( \gamma \right) :=\exp \left( -2\pi iv\frac{cd}{Nh}\right) , \end{aligned}$$
(2.2)
where \(\gamma =\left( {\begin{matrix} a &{}\quad b \\ c &{}\quad d \end{matrix}}\right) \in \varGamma _0(4N)\) and vh are integers with \(h|\gcd (4N,96)\). First, we establish convergence of these series.

Proposition 2.7

For a positive integer N and multiplier \(\psi =\psi _{N,v,h}\) as in (2.2), the Rademacher sums \(R^{[-3]}_{\psi ,\frac{1}{2}}(\tau )\) and \(R^{[3]}_{\overline{\psi },\frac{3}{2}}\) converge locally uniformly on \(\mathfrak {H}\) and therefore define holomorphic functions on \(\mathfrak {H}\).

Proof

By following the steps outlined in [10, Section 5] to establish the convergence of weight \(\tfrac{1}{2}\) Rademacher series with a slightly different multiplier system (related to that of the Dedekind eta function) mutatis mutandis, we find that the Rademacher sums we are interested in converge, assuming the convergence at \(s=\tfrac{3}{4}\) of the Kloosterman zeta function
$$\begin{aligned} \sum _{c=1}^\infty \frac{K_\psi (-3,n,4Nc)}{(4Nc)^{2s}} \end{aligned}$$
with
(2.3)
where the \({}^*\) at the sum indicates that it runs over primitive residue classes modulo \(c,\,\overline{d}\) denotes the multiplicative inverse of d modulo c, and \(e(\alpha ):=\exp (2\pi i\alpha )\) as usual. We omit the subscript if \(\psi =1\). In order to establish positivity of the multiplicities of irreducible characters in Sect. 4, we will show not only convergence of this series, but even explicit estimates for its value, which will complete the proof. \(\square \)
Since the Rademacher sum \(R^{[-3]}_{\psi ,\frac{1}{2}}(\tau )\) is 1-periodic by construction, it has a Fourier expansion, which can (at least formally) be established by standard methods. Projecting this function to the Kohnen plus space then yields the function
$$\begin{aligned} Z_{N,\psi }(\tau ):=q^{-3}+\sum _{\begin{array}{c} n=0 \\ n\equiv 0,1\ \left( {\text {mod}} 4\right) \end{array}}^\infty A_{N,\psi }(n)q^n, \end{aligned}$$
(2.4)
where \(A_{N,\psi }\) is given by
$$\begin{aligned} \begin{aligned} A_{N,\psi }(0)&:=\frac{\pi \sqrt{3}}{2 N^\frac{3}{2}}(1-i)\sum _{c=1}^\infty \left( 1+\delta _{\mathrm{odd}}(Nc)\right) \frac{K_\psi (-3,0,4Nc)}{(4Nc)^\frac{3}{2}},\\ A_{N,\psi }(n)&:=\frac{\pi \sqrt{2}}{4N}\left( \frac{3}{n}\right) ^\frac{1}{4}(1-i)\sum _{c=1}^\infty (1+\delta _{\mathrm{odd}}(Nc))\frac{K_\psi (-3,n,4Nc)}{4Nc}I_\frac{1}{2}\left( \frac{\pi \sqrt{3n}}{Nc}\right) . \end{aligned} \end{aligned}$$
(2.5)
Here, \(I_\frac{1}{2}\) denotes the usual modified Bessel function of the first kind of order \(\tfrac{1}{2}\) and
$$\begin{aligned} \delta _{\mathrm{odd}}(k):= {\left\{ \begin{array}{ll} 1 &{}\quad k\text { odd,} \\ 0 &{}\quad k\text { even.} \end{array}\right. } \end{aligned}$$
For each conjugacy class [g] of the Thompson group Th, we associate integers \(v_g\) and \(h_g\) (where \(h_g|96\)) as specified in Table 5 and the character \(\psi _{[g]}:=\psi _{|g|,v_g,h_g}\), where |g| denotes the order of g in Th, as well as a finite sequence of rational numbers \(\kappa _{m,g}\) which are also given in Table 5 and define the function
$$\begin{aligned} \mathcal {F}_{[g]}(\tau ):=2Z_{|g|,\psi _{[g]}}(\tau )+\sum _{\begin{array}{c} m>0 \\ m^2\mid h_g|g| \end{array}}\kappa _{m,g}\vartheta (m^2\tau ), \end{aligned}$$
(2.6)
with
$$\begin{aligned} \vartheta (\tau ):=\sum _{n\in \mathbb {Z}}q^{n^2}. \end{aligned}$$
This is going to be the explicitly given weakly holomorphic modular form (see Proposition 3.1) mentioned in Conjecture 1.1, meaning that we have
$$\begin{aligned} \mathcal {T}_{[g]}(\tau )=\mathcal {F}_{[g]}(\tau ) \end{aligned}$$
for all conjugacy classes [g] of Th.

We now prove and recall some important facts about Rademacher sums that we shall use later on. As in [10, Propositions 7.1 and 7.2], one sees the following.

Proposition 2.8

The Rademacher sum \(R^{[-3]}_{\psi ,\frac{1}{2}}(\tau )\) with \(\psi \) as in (2.2) is a mock modular form of weight \(\tfrac{1}{2}\) whose shadow is a cusp form with the conjugate multiplier \(\overline{\psi }\), which is a constant multiple of the Rademacher sum \(R^{[3]}_{\psi ,\frac{3}{2}}\).

Next, we establish the behavior of Rademacher sums at cusps. Here, we have to take into account that the sums we look at are projected into the Kohnen plus space which might affect the behavior at cusps. For a function \(f\in M_{k+\frac{1}{2}}^!(\varGamma _0(4N))\), where k is an integer and N is odd, the projection of f to the plus space is defined by
$$\begin{aligned} f|\text {pr}=(-1)^{\lfloor \frac{k+1}{2}\rfloor }\frac{1}{3\sqrt{2}}\sum _{v=-1}^2( f|B\cdot A_v)+\frac{1}{3} f, \end{aligned}$$
where
$$\begin{aligned} B=\begin{pmatrix} 4 &{}\quad 1 \\ 0 &{}\quad 4 \end{pmatrix}\quad \text {and}\quad A_v=\begin{pmatrix} 1 &{}\quad 0 \\ 4Nv &{}\quad 1 \end{pmatrix}. \end{aligned}$$
Using this projection operator, one sees that the following is true. The proof is similar in nature to that of Proposition 3 in [28] and is carried out in some detail (for the special case where \(N=p\) is an odd prime) in [24, Section 2].

Lemma 2.9

Let N be odd and \(f\in H_{k+\frac{1}{2}}(\varGamma _0(4N))\) for some \(k\in \mathbb {N}_0\), such that
$$\begin{aligned} f^+(\tau )=q^{-m}+\sum _{n=0}^\infty a_nq^n \end{aligned}$$
for some \(m>0\) with \(-m\equiv 0,(-1)^k\pmod 4\) has a non-vanishing principal part only at the cusp \(\infty \) and is bounded at the other cusps of \(\varGamma _0(4N)\). Then, the projection \(f|\text {pr}\) of f to the plus space has a pole of order m at \(\infty \) and has a pole of order \(\frac{m}{4}\) either at the cusp \(\frac{1}{N}\) if \(m\equiv 0\pmod 4\) or at the cusp \(\frac{1}{2N}\) if \(-m\equiv (-1)^k\pmod 4\) and is bounded at all other cusps.

Proof

In order to compute the expansion of \(f|\text {pr}\) at a given cusp \(\mathfrak {a}=\frac{a}{c}\), we compute
$$\begin{aligned} f|\text {pr}|\sigma _\mathfrak {a}, \end{aligned}$$
where \(\sigma _\mathfrak {a}=\left( {\begin{matrix} a &{}\quad b \\ c &{}\quad d \end{matrix}}\right) \in {\text {SL}}_2(\mathbb {Z})\). Multiplying out the matrices, we see that this is (up to a constant factor) equal to
$$\begin{aligned} (-1)^{\lfloor \frac{k+1}{2}\rfloor }\frac{1}{3\sqrt{2}}\sum _{v=-1}^2 f|\begin{pmatrix} (4+4Nv)a+c &{}\quad (4+4Nv)b+d \\ 16Nva+4c &{}\quad 16Nvb+4d \end{pmatrix}+\frac{1}{3}\, f|\sigma _\mathfrak {a}. \end{aligned}$$
(2.7)
By assumption, this function can only have a pole at \(\infty \) if the denominator of the fraction \(\frac{(4+4Nv)a+c}{16Nva+4c}\) in lowest terms (where we allow the denominator to be 0 which we interpret as \(\infty \)) is divisible by 4N, which is easily seen to imply that N|c. Since there are only three inequivalent cusps of \(\varGamma _0(4N)\) whose denominator is divisible by N, represented by \(\infty ,\,\frac{1}{N},\,\frac{1}{2N}\), we can restrict ourselves to
$$\begin{aligned} \sigma _{\frac{1}{N}}=\begin{pmatrix} 1 &{}\quad 0 \\ N &{}\quad 1 \end{pmatrix}\quad \text {and}\quad \sigma _{\frac{1}{2N}}=\begin{pmatrix} 1 &{}\quad 0 \\ 2N &{}\quad 1 \end{pmatrix}. \end{aligned}$$
Plugging first \(\sigma _\frac{1}{N}\) into (2.7), we find for each \(v=-1\ldots 2\) that
$$\begin{aligned} f|B\cdot A\cdot \sigma _\frac{1}{N}=f|\begin{pmatrix} 4+4Nv+N &{}\quad \beta _v \\ 16Nv+4N &{}\quad \delta _v \end{pmatrix}\cdot \begin{pmatrix} 1 &{}\quad \delta _v-4\beta _v \\ 0 &{}\quad 16 \end{pmatrix}=Cf\left( \frac{\tau +\delta _v-4\beta _v}{16}\right) , \end{aligned}$$
where \(\beta _v,\delta _v\in \mathbb {Z}\) such that
$$\begin{aligned} \det \begin{pmatrix} 4+4Nv+N &{}\quad \beta _v \\ 16Nv+4N &{}\quad \delta _v \end{pmatrix}=1 \end{aligned}$$
and \(C\in \mathbb {C}\) is a constant that a priori depends on v, but by working out the corresponding automorphy factors, one sees with easy, elementary methods that it does indeed not. Note that it does, however, depend on N. Furthermore, it is not hard to see that the difference \(\delta _v-4\beta _v\) runs through all residue classes modulo 16 that are congruent to N modulo 4. This implies that in
$$\begin{aligned} \sum _{v=-1}^2 f\left( \frac{\tau +\delta _v-4\beta _v}{16}\right) \end{aligned}$$
only powers of \(q^\frac{1}{16}\) survive whose exponent is divisible by 4. Since by assumption we have that \(f|\sigma _\frac{1}{N}=O(1)\) as \(\tau \rightarrow \infty \), we therefore see that \(f|\text {pr}\) has a pole of order \(\frac{m}{4}\) at the cusp \(\tfrac{1}{N}\) if and only if m is divisible by 4.
For the cusp \(\tfrac{1}{2N}\), the argumentation is analogous. One finds that
$$\begin{aligned} f|B\cdot A\cdot \sigma _\frac{1}{2N}=f|\begin{pmatrix} 2+2Nv+N &{}\quad \beta _v' \\ 8Nv+4N &{}\quad \delta _v' \end{pmatrix}\cdot \begin{pmatrix} 2 &{} \quad \delta _v-4\beta _v \\ 0 &{}\quad 8 \end{pmatrix}=C_v'f\left( \frac{2\tau +\delta _v-4\beta _v}{8}\right) , \end{aligned}$$
where \(\beta _v',\delta _v',C_v'\) have the analogous meaning as \(\beta _v,\delta _v,C\) above, with the only difference that \(C_v'\) actually does depend on v. The dependence on v is so that in the summation only powers of \(q^\frac{1}{4}\) with exponents \(\equiv (-1)^k\pmod 4\) survive, which implies our Lemma. \(\square \)

For even N, it turns out that the Rademacher series are automatically in the plus space. This follows immediately from the next lemma.

Lemma 2.10

Let \(m, n\in \mathbb {Z}\) such that \(m\not \equiv n\pmod 4\) and \(c\in \mathbb {N}\) be divisible by 8. Then, we have
$$\begin{aligned} K(m,n,c)=0 \end{aligned}$$
with K(mnc) as defined in (2.3).

Proof

We write \(c=2^\ell c'\) with \(\ell \ge 3\) and \(c'\) odd. By the Chinese Remainder Theorem, one easily sees the following multiplicative property of the Kloosterman sum,
$$\begin{aligned} K(m,n,c)=K\left( m\overline{c'},n\overline{c'},2^\ell \right) \cdot S\left( m\overline{2^\ell },n\overline{2^\ell },c'\right) , \end{aligned}$$
(2.8)
where
is a Salié sum and \(\overline{c'}\) denotes the inverse of \(c'\) modulo \(2^\ell \) and \(\overline{2^\ell }\) denotes the inverse of \(2^\ell \) modulo \(c'\).
Therefore, it suffices to show the lemma for \(c=2^\ell \) with \(\ell \ge 3\). The case where \(\ell =3\) can be checked directly, so assume \(\ell \ge 4\) from now on. In this case, it is straightforward to see that
$$\begin{aligned} \left( \frac{2^\ell }{d}\right) =\left( \frac{2^\ell }{d+2^{\ell -1}}\right) ,\quad \varepsilon _d=\varepsilon _{d+2^{\ell -1}},\quad \text {and}\quad \overline{d+2^{\ell -1}}=\overline{d}+2^{\ell -1}. \end{aligned}$$
This yields that for \(m\not \equiv n\pmod 2\) we have that
$$\begin{aligned} \left( \frac{2^\ell }{d+2^{\ell -1}}\right) \varepsilon _{d+2^{\ell -1}}\cdot e\left( \frac{m(\overline{d+2^{\ell -1}})+n(d+2^{\ell -1})}{2^\ell }\right) =-\left( \frac{2^\ell }{d}\right) \varepsilon _d e\left( \frac{m\overline{d}+nd}{2^\ell }\right) \end{aligned}$$
for all odd \(d\in \{1,\ldots ,2^{\ell -1}-1\}\), so that the summands in the Kloosterman sum pair up with opposite signs, making the sum 0 as claimed.
If m and n have the same parity, but are not congruent modulo 4, a similar pairing also works. In this case, we find through similar reasoning that for \(\ell \ge 5\) we have
$$\begin{aligned} \left( \frac{2^\ell }{d}\right) \varepsilon _d e\left( \frac{m\overline{d}+nd}{2^\ell }\right)&=\left( \frac{2^\ell }{d+2^{\ell -1}}\right) \varepsilon _{d+2^{\ell -1}}\cdot e\left( \frac{m(\overline{d+2^{\ell -1}})+n(d+2^{\ell -1})}{2^\ell }\right) \\&=-\left( \frac{2^\ell }{d+2^{\ell -2}}\right) \varepsilon _{d+2^{\ell -2}}\cdot e\left( \frac{m(\overline{d+2^{\ell -2}})+n(d+2^{\ell -2})}{2^\ell }\right) . \end{aligned}$$
Again, we can pair summands with opposite signs, proving the lemma. \(\square \)

From the preceding two lemmas, we immediately find that the following is true.

Proposition 2.11

For any \(g\in Th\), the function \(Z_{|g|,\psi _{[g]}}\) is a mock modular form which has a pole of order 3 at \(\infty \), a pole of order \(\tfrac{3}{4}\) at \(\frac{1}{2N}\) if N is odd, and vanishes at all other cusps.

Proof

As described in Appendix E of [10], we see that the Rademacher sums \(R^{[-3]}_{\psi _{[g]},\frac{1}{2}}(\tau )\) have only a pole of order 3 at \(\infty \) and grow at most polynomially at all other cusps. By Lemmas 2.9 and 2.10, we see that the poles are as described in the proposition. The vanishing at all remaining cusps follows as in [8, Theorem 3.3]. \(\square \)

3 Identifying the McKay–Thompson series as modular forms

In this section, we want to establish that the multiplicities of each irreducible character are integers. To this end, we first establish the exact modularity and integrality properties of the conjectured McKay–Thompson series \(\mathcal {F}_{[g]}(\tau )\), which are stated without proof in [25].

Proposition 3.1

For each element g of the Thompson group, the function \(Z_{|g|,\psi _{[g]}}(\tau )\) as defined in (2.4) lies in the space \(M_{\frac{1}{2}}^{+,!}(4|g|,\psi _{[g]})\le M_{\frac{1}{2}}^{+,!}(N_{[g]})\) with \(N_{[g]}\) as in Table 5.

Proof

As we know from Proposition 2.8, we have
$$\begin{aligned} \xi _{\frac{1}{2}}Z_{|g|,\psi _{[g]}}\in S^+_\frac{3}{2}(4|g|,\overline{\psi _{[g]}})\le S_\frac{3}{2}^+(N_{[g]}). \end{aligned}$$
The space \(S_\frac{3}{2}^+(N_{[g]})\) now turns out to be zero-dimensional for
$$\begin{aligned}&[g]\in \left\{ 1A, 2A, 3A, 3B, 3C, 4A, 4B, 5A, 6A, 6B, 6C, 7A, 8A, 9A, 9B, 10A, 12A, 12B, 12C, \right. \\&\qquad ~~~~~\left. 13A, 18A, 36A, 36B, 36C \right\} , \end{aligned}$$
which is directly verifiable using the built-in functions for spaces of modular forms in for example Magma [3]. Furthermore, we have the Bruinier–Funke pairing (see Proposition 2.6) combined with Proposition 2.11 which tell us that
$$\begin{aligned} \left\{ \xi _\frac{1}{2} Z_{|g|,\psi _{[g]}},Z_{|g|,\psi _{[g]}}\right\} \overset{\cdot }{=}c_{Z_{|g|,\psi _{[g]}}}^-(3), \end{aligned}$$
(3.1)
where the dot above the equal sign indicated an omitted multiplicative (nonzero) constant. More precisely, we can apply Proposition 2.6 to the Rademacher sum \(R_{\psi ,\frac{1}{2}}^{[-3]}\) and then use the same reasoning as in the proof of Lemma 2.9 to see that projection to the plus space only alters this value by a multiplicative nonzero constant, since the additional pole at the cusp \(\frac{1}{2|g|}\) (if |g| is odd) is directly forced by the plus space condition.
From (3.1), we can now deduce, because the Petersson inner product is positive definite on the space of cusp forms, that the shadow of \(Z_{|g|,\psi _{[g]}}\) must be 0 if every \(f\in S_\frac{3}{2}^+(N_{[g]})\) is \(O(q^4)\). Again, this can be checked using built-in features of Magma, therefore showing the claim for
$$\begin{aligned}{}[g]\in \left\{ 14A, 19A, 20A, 28A, 31A, 31B\right\} . \end{aligned}$$
For the remaining 18 conjugacy classes, one can use the same arguments as above, but with the refinement that instead of looking at the full space \(S_\frac{3}{2}^+(N_{[g]})\), one looks at the (usually) smaller space \(S_\frac{3}{2}^+(4|g|,\overline{\psi _{[g]}})\). Since computing bases for these spaces is not something that a standard computer algebra system can do without any further work, we describe how to go about doing this. Let \(f\in S_\frac{3}{2}^+(4|g|,\overline{\psi _{[g]}})\) for some conjugacy class [g]. Then \(f\cdot \vartheta \) is a modular form of weight 2 with the same multiplier and level (respectively, trivial multiplier and level \(N_{[g]}\)). Using programs3 written by Rouse and Webb [34], one can verify that the algebra of modular forms of level \(N_{[g]}\), so in particular the space \(M_2(N_{[g]})\), is generated by eta quotients. One can also compute a generating system consisting of eta quotients for all remaining \(N_{[g]}\) that still need to be considered. Since one can actually compute Fourier expansions of expressions like
$$\begin{aligned} (f|\gamma )(\tau ) \end{aligned}$$
for \(f(\tau )=\prod _{\delta |N}\eta (\delta \tau )^{r_\delta }\) an eta quotient and \(\gamma \in {\text {SL}}_2(\mathbb {Z})\) explicitly, see, e.g., [27, Proposition 2.1] it is straightforward linear algebra to compute a basis of the space \(M_2(4|g|,\overline{\psi _{[g]}})\) and from there go down to \(S_\frac{3}{2}^+(4|g|,\overline{\psi _{[g]}})\). A Magma script computing dimensions and bases of these spaces can be obtained from the second author’s homepage. Using this script, we find that \(\dim S^+_\frac{3}{2}(4|g|,\overline{\psi _{[g]}})=0\) for
$$\begin{aligned}{}[g]\in \left\{ 8B,9C,12D,15A,15B,24A,24B,24C,24D,27A,27B,27C\right\} \end{aligned}$$
and for all remaining conjugacy classes [g], we find that every \(f\in S^+_\frac{3}{2}(4|g|,\overline{\psi _{[g]}})\) is \(O(q^4)\). This completes the proof. \(\square \)

Proposition 3.2

For each \(g\in Th\), the functions \(\mathcal {F}_{[g]}(\tau )=\sum _{n=-3}^\infty c_{[g]}(n)q^n\) as defined in (2.6) are all weakly holomorphic modular forms of weight \(\tfrac{1}{2}\) for the group \(\varGamma _0(N_{[g]})\) in Kohnen’s plus space with integer Fourier coefficients at \(\infty \).

Proof

We have established in Proposition 3.1 that the Rademacher series \(Z_{|g|,\psi _{[g]}}(\tau )\) are all weakly holomorphic modular forms of weight \(\tfrac{1}{2}\) for \(\varGamma _0(N_{[g]})\) in the plus space. The given theta corrections are holomorphic modular forms of the same weight and level, hence so is their sum. Furthermore, theta functions do not have poles, so that all poles of \(\mathcal {F}_{[g]}\) come from the Rademacher series which has a pole of order 3 only at the cusps of \(\varGamma _0(N_{[g]})\) lying above the cusp \(\infty \) on the modular curve \(X_0(4|g|)\). Hence, the function
$$\begin{aligned} \mathcal {F}_{[g]}(\tau )\cdot G(\tau ), \end{aligned}$$
where \(G(\tau )=q^3+O(q^4)\in S^+_{2k-\frac{1}{2}}(\varGamma _0(4|g|)\) is a cusp form with integer Fourier coefficients, is a weight 2k holomorphic modular form with trivial multiplier under the group \(\varGamma _0(N_{[g]})\). Now by the choice of \(G(\tau )\), we have that \(\mathcal {F}_{[g]}(\tau )\) has integer Fourier coefficients if and only if \(\mathcal {F}_{[g]}(\tau )\cdot G(\tau )\) has integer Fourier coefficients, which is the case if and only if the Fourier coefficients of this modular form are integers up to the Sturm bound [37]
$$\begin{aligned} \frac{k}{6}[{\text {SL}}_2(\mathbb {Z}){:}\,\varGamma _0(N_{[g]})]. \end{aligned}$$
In order to compute the necessary Fourier coefficients exactly without relying on the rather slow convergence of the Fourier coefficients of the Rademacher series, one can construct linear combinations of weight \(\tfrac{1}{2}\) weakly holomorphic eta quotients again using the programs4 written by Rouse and Webb [34] which have the same principal part at \(\infty \) (and the related cusp \(\frac{1}{2N}\)) as the Rademacher series (see Proposition 2.11) and the same constant terms as the theta corrections if there are any, wherefore their difference must be a weight \(\tfrac{1}{2}\) holomorphic cusp form for \(\varGamma _0(N_{[g]})\) with trivial multiplier, which by the Serre–Stark basis theorem [35] is easily seen to be 0.5 The largest bound up to which coefficients need to be checked turns out to be 384 for \([g]=24 CD \). \(\square \)

Remark 3.3

As it turns out, the theta correction
$$\begin{aligned} f(\tau )=-\vartheta (4\tau )+3\vartheta (36\tau ) \end{aligned}$$
\([g]=12AB\) that Harvey and Rayhaun [25, Table 5] give transforms with a different multiplier than the Rademacher series \(Z_{12,\psi _{12AB}}(\tau )\): As one computes directly from the fact that
$$\begin{aligned} -\vartheta (\tau )+3\vartheta (9\tau ) \end{aligned}$$
transforms with the multiplier \(\psi _{3,1,3}\) under the group \(\varGamma _0(12),\,f(\tau )\) transforms under the group \(\varGamma _0(48)\) with the multiplier \(\psi _{12,1,3}\), while \(Z_{12,\psi _{12AB}}(\tau )\) transforms with the multiplier \(\psi _{12,7,12}\). But since both multipliers become trivial on the group \(\varGamma _0(144)\), the proposition remains valid.

We can now establish the uniqueness claim in Theorem 1.2 very easily.

Proposition 3.4

For \([g]\notin \{12A,12B,36A,36B,36C\}\), we have that the function \(\mathcal {F}_{[g]}(\tau )\in M_\frac{1}{2}^{!,+}(4|g|,\psi _{[g]})\) is the unique function satisfying the conditions given in Theorem 1.2:
  • its Fourier expansion is of the form \(2q^{-3}+\chi _2(g)+O(q^4)\) and all its Fourier coefficients are integers.

  • if |g| is odd, then the only other pole of order \(\tfrac{3}{4}\) is at the cusp \(\frac{1}{2|g|}\); otherwise, there is only the pole at \(\infty \). It vanishes at all other cusps.

For \(|g|=36,\,\mathcal {F}_{[g]}(\tau )\) is uniquely determined by additionally fixing the coefficient of \(q^4\) to be \(\chi _4(g)+\chi _5(g)\).

Proof

As we have used already, the function \(2Z_{|g|,\psi _{[g]}}(\tau )\) has the right behavior at the cusps so that \(\mathcal {F}_{[g]}(\tau )-2Z_{|g|,\psi _{[g]}}(\tau )\) is a holomorphic weight \(\frac{1}{2}\) modular form. As it turns out, in all cases but the one where \(|g|=36\), this space is at most two dimensional, which can be seen by the Serre–Stark basis theorem if \(\psi _{[g]}\) is trivial or through a computation similar to the one described in the proof of Proposition 3.1 if the multiplier is not trivial. Hence prescribing the constant and first term in the Fourier expansion determines the form uniquely. If \(|g|=36\), the space of weight \(\frac{1}{2}\) modular forms turns out to be 3 dimensional, so that fixing one further Fourier coefficient suffices to determine the form uniquely. \(\square \)

We ultimately want to study the multiplicities of the irreducible characters of Th. To this end, we now consider the functions
$$\begin{aligned} \mathcal {F}_{\chi _j}(\tau ):=\frac{1}{|Th|}\sum _{g\in Th} \overline{\chi _j(g)}\mathcal {F}_{[g]}=\sum _{n=-3}^\infty m_j(n)q^n \end{aligned}$$
with \(m_j(n)\) as in (1.1), the generating functions of the multiplicities. We want to show that all those numbers \(m_j(n)\) are integers. A natural approach for this would be to view \(\mathcal {F}_{\chi _j}\) as a weakly holomorphic modular form of weight \(\tfrac{1}{2}\) and level
$$\begin{aligned} N_{\chi _j}:=\text {lcm}\left\{ N_{[g]}{:}\quad \chi _j(g)\ne 0\right\} \end{aligned}$$
and then use a Sturm bound type argument as in the proof of Proposition 3.2. However, these levels turn out to be infeasibly large in most cases. For example, we have that
$$\begin{aligned} N_{\chi _1}=2\,778\,572\,160, \end{aligned}$$
so one would have to compute at least a few 100 million Fourier coefficients of \(\mathcal {F}_{\chi _1}\) to make such an argument work, which is entirely infeasible.

This bound can be reduced substantially, however, by breaking the problem into many smaller problems involving simpler congruences, each of which requires far fewer coefficients to prove.

We proceed by a linear algebra argument. Let \(\mathbf {C}\) be the coefficient matrix containing the coefficients of the alleged McKay–Thompson series for each conjugacy class. In theory, we have that \(\mathbf {C}\) is a \(48\times \infty \) matrix. In practice, we take \(\mathbf {C}\) to be a \(48\times B\) matrix with B large. Let \(\mathbf {X}\) be the \(48\times 48\) matrix with columns indexed by conjugacy classes of Th and rows indexed by irreducible characters, whose \((\chi _i,[g])\)th entry is
$$\begin{aligned} \mathbf {X}_{(\chi _i,[g])}=\chi _i(g)\cdot \frac{|[g]|}{|Th|}. \end{aligned}$$
Using the first Schur orthogonality relation for characters,
$$\begin{aligned} \sum _{g\in Th}\chi _i(g)\chi _j(g)={\left\{ \begin{array}{ll}|Th| &{}\quad \text { if } i=j,\\ 0&{}\quad \text { otherwise,}\end{array}\right. } \end{aligned}$$
we have that the rows of the matrix \(\mathbf {m}:=\mathbf {X} \mathbf {C}\) which are indexed by the characters \(\chi _i\) are exactly the multiplicity values under consideration for the given character.
The matrix \(\mathbf {C}\) does not have full rank. Besides the duplicated series (such as \(\mathcal {T}_{[12A]}=\mathcal {T}_{[12B]}\)), we have additional linear relations given in Appendix B.1. As some of these relations involve the theta functions used as correction terms in the construction, let us define \(\mathbf {C}^+\) to be the matrix extending \(\mathbf {C}\) to include the coefficients of the theta functions \(\vartheta (n^2\tau )\), for \(n=1,2,3,6,9.\) Then there are matrices \(\mathbf {N},\mathbf {N}^*\) of dimensions \(48\times 35\) and \(35\times 53\), respectively, so that \(\mathbf {N}^* \mathbf {C}^+\) has full rank and
$$\begin{aligned} \mathbf {m}=\mathbf {X} \mathbf {C}=\mathbf {X} \mathbf {N} \mathbf {N}^*\mathbf {C}^+. \end{aligned}$$
We construct the matrix \(\mathbf {N}^*\) by taking a \(53\times 53\) identity matrix indexed by conjugacy classes and removing the rows corresponding to one of each duplicate series and also the highest level conjugacy class (as ordered for instance by Table 5) appearing in each of the linear relations. The matrix N may be constructed starting with a \(35\times 35\) identity matrix, adding in columns to reconstitute the removed conjugacy classes and removing the columns corresponding to the theta series.
The rows of \(\mathbf {N}^*\mathbf {C}^+\) exhibit additional congruence as listed in Appendix B.2. For each prime p dividing the order of the Thompson group, we construct a matrix \(\mathbf {M}_p\) to reduce by these congruences. We start as before with a \(35\times 35\) identity matrix. Then for each congruence listed, we replace the row of the matrix with index given by the highest weight conjugacy class appearing in the congruence with a new row constructed to reduce by that congruence. Given the congruence
$$\begin{aligned} \sum _{g} a_g\mathcal {T}_{[g]}\equiv 0\pmod {p^s}, \end{aligned}$$
where the \(a_g\in \mathbb {Z},\) the new row will be given by
$$\begin{aligned} \sum _g p^{-s} a_g\mathfrak {e}_{[g]}, \end{aligned}$$
where \(\mathfrak {e}_{[g]}\) is the elementary basis element.
For instance, in the case \(p=7,\) we have the congruence
$$\begin{aligned} \mathcal {T}_{[1A]}- \mathcal {T}_{[7A]} \equiv 0\pmod {7^2}. \end{aligned}$$
This tells us that the [7A]-th row of \(M_7\) should be
$$\begin{aligned} 7^{-2} \left( \mathfrak {e}_{[7A]}- \mathfrak {e}_{[1A]} \right) . \end{aligned}$$
Assuming for the moment the validity of these congruences, we have that \(\mathbf {M}_p\mathbf {N}^*\mathbf {C}^+\) is an integer matrix. Moreover, by construction \(\mathbf {M}_p\) is invertible (depending on ordering, we have that \(\mathbf {M}_p\) is lower triangular with non-vanishing main diagonal).
In each case, we have computationally verified that the matrix
$$\begin{aligned} \mathbf {X} \mathbf {N} \mathbf {M}_p^{-1} \end{aligned}$$
is rational with p-integral entries. Since
$$\begin{aligned} \mathbf {m}=\left( \mathbf {X} \mathbf {N} \mathbf {M}_p^{-1}\right) \cdot \left( \mathbf {M}_p\mathbf {N}^*\mathbf {C}^+\right) . \end{aligned}$$
is the product of two p-integral matrices, we have that every multiplicity must also be p-integral.

The congruences listed in Appendix B.2 were found computationally by reducing the matrix \(\left( \mathbf {N}^*\mathbf {C}^+\right) \pmod p\) and computing the left kernel. After multiplying by a matrix constructed similar to \(\mathbf {M}_p\) above so as to reduce by the congruences found, the process was repeated. The list of congruences given represents a complete list, in the sense that the matrix \(\left( \mathbf {M}_p\mathbf {N}^*\mathbf {C}^+\right) \) both is integral and has full rank modulo p.

Many of the congruences can be easily proven using standard trace arguments for spaces of modular forms of level pN to level N. For uniformity, we will instead rely on Sturm’s theorem following the argument described above. The worst case falls with any congruence involving the conjugacy class 24CD. These occur for both primes \(p=2\) and 3. The nature of the congruences, however, does not require us to increase the level beyond the corresponding level \(N_{24CD}=1152\). There is a unique normalized cusp form of weight 19 / 2 and level 4 in the plus space. This form vanishes to order 3 at the cusp \(\infty \) and to order 3 / 4 at the cusp 1 / 2. This is sufficient so that multiplying by this cusp form moves these potential congruences into spaces of holomorphic modular forms of weight 10,  level 1152. The Sturm bound for this space falls just shy of 2000 coefficients. This bound could certainly be reduced by more careful analysis, but this is sufficient for our needs. The congruences were observed up to 10,000 coefficients. These computations were completed using Sage mathematical software [39].

Remark 3.5

A similar process can be used in the case of Monstrous Moonshine to prove the integrality of the Monster character multiplicities. This gives an (probably6) alternate proof the theorem of Atkin–Fong–Smith [20, 36]. As in the case of Thompson moonshine, we have calculated a list of congruences for each prime dividing the order of the Monster, proven by means of Sturm’s theorem. This is list complete in the sense that once we have reduced by the congruences for a given prime, the resulting forms have full rank modulo that prime. The Monster congruences may be of independent interest and are available upon request to the authors.

4 Positivity of the multiplicities

To establish the positivity of the multiplicities of the irreducible representations, we follow Gannon’s work mutatis mutandis. First, we notice that by the first Schur orthogonality relation for characters and the triangle inequality, for each irreducible representation \(\rho \) with corresponding character \(\chi \) of Th we have the estimate
$$\begin{aligned} \text {mult}_{\rho _j}(W_k)&=\sum _{[g]\subseteq Th} \frac{1}{|C(g)|}\mathrm{strace}_{W_k}(g)\overline{\chi (g)}\\&\ge \frac{|\mathrm{strace}_{W_k}(1)|}{|G|}\chi (1)-\sum _{[g]\ne [1]}\frac{|\mathrm{strace}_{W_k}(g)|}{|C(g)|}|\chi (g)|. \end{aligned}$$
Here, C(g) denotes the centralizer of g in Th and the summation runs over all conjugacy classes of Th. Thus to prove positivity, we show that \(|\mathrm{strace}_{W_k}(1)|\) always dominates all the others. To this end, we use the description of the Fourier coefficients of \(\mathcal {F}_{[g]}(\tau )\) in terms of Maaß–Poincaré series, see (2.4) and (2.6). We use the following elementary (and rather crude) estimates,
$$\begin{aligned} \begin{array}{ll} \left| I_{\frac{1}{2}}(x)-\sqrt{\frac{2x}{\pi }}\right| \le \frac{1}{5}\sqrt{\frac{2x^5}{\pi }}&{}\quad \text {for }0<x<1,\\ 0<I_{\frac{1}{2}}(x)\le \frac{e^x}{\sqrt{2\pi x}}&{} \quad \text {for }x>0,\\ |K_\psi (m,n,4c)|\le c/2&{} \quad \text {for all }c\in \mathbb {N}, \end{array} \end{aligned}$$
and set \(\delta _c=1+\delta _{\mathrm{odd}}(Nc)\) which has the obvious bounds \(1\le \delta _c\le 2\).
The convergence and bounds of the coefficients \(A_{N,\psi }\) rely on the convergence of the modified Selberg–Kloosterman zeta function
$$\begin{aligned} Z^*_\psi (m,n;s):=\sum _{c=1}^\infty (1-i)(1+\delta _{\mathrm{odd}}(Nc))\frac{K_\psi (m,n,4Nc)}{(4Nc)^{2s}}. \end{aligned}$$
(4.1)
The zeta function only converges conditionally at 3 / 4. The bounds we obtain are crude and very large, but they do not grow with n.
If we set
$$\begin{aligned} C_{[g]}(n):=\frac{4N}{\pi \sqrt{2}}\left( \frac{n}{3}\right) ^{1/4}A_{N,\psi }=\sum _{c=1}^{\infty }(1-i)(1+\delta _{\mathrm{odd}}(Nc))\frac{K_{\psi }(-3,n,4Nc)}{4Nc}I_{\frac{1}{2}}\left( \frac{\pi \sqrt{3n}}{Nc}\right) , \end{aligned}$$
then using the triangle inequality we find
$$\begin{aligned}&\left| D\right| - \left| R\right| -\left| \sqrt{8}(3n)^{1/4}Z^*_\psi (m,n;3/4)\right| \\&\quad \le |C(n)| \le \left| D\right| +\left| R\right| + \left| \sqrt{8}(3n)^{1/4}Z^*_\psi (m,n;3/4)\right| . \end{aligned}$$
Here, R is the absolutely convergent sum
$$\begin{aligned} \sum _{c=2}^{\infty }(1-i)(\delta _c)\frac{K_{\psi }(-3,n,4Nc)}{4Nc}\left( I_{\frac{1}{2}}\left( \frac{\pi \sqrt{3n}}{Nc}\right) -\sqrt{\frac{2\sqrt{3n}}{Nc}}\right) , \end{aligned}$$
and D is the dominant term coming from the \(c=1\) term in the expression for \(A_{N,\psi }.\) We may estimate |D| by
$$\begin{aligned} |D|&=\left| (1-i)(\delta _1)\frac{K_{\psi }(-3,n,4N)}{4N}\left( I_{\frac{1}{2}}\left( \frac{\pi \sqrt{3n}}{N}\right) -\sqrt{\frac{2\sqrt{3n}}{N}}\right) \right| \\&\le \sqrt{2}\frac{\delta _1}{2}\frac{\sqrt{N}}{\pi \sqrt{2}(3n)^\frac{1}{4}}e^{\frac{\pi \sqrt{3n}}{N}}\\&\le \frac{2\sqrt{N}}{\pi (3n)^\frac{1}{4}}e^{\frac{\pi \sqrt{3n}}{N}}. \end{aligned}$$
Here, we have used the second estimate for the Bessel function. We will only be interested in a lower bound for |D| when \([g]=[1A].\) In this case, we will just use the exact expression
$$\begin{aligned} D_{[1A]}=(-1)^{n}\left( I_{\frac{1}{2}}\left( \pi \sqrt{3n}\right) -\sqrt{2}(3n)^{1/4}\right) . \end{aligned}$$
If we set \(L:=\tfrac{\pi }{N}\sqrt{3n}\), the sum for \(2\le c\le L\) in R can be estimated as follows:
$$\begin{aligned}&\left| (1-i)\sum _{2\le c\le L}\delta _c\frac{K_{\psi }(-3,n,4Nc)}{4Nc}\left( I_{\frac{1}{2}}\left( \frac{\pi \sqrt{3n}}{Nc}\right) -\sqrt{\frac{2\sqrt{3n}}{Nc}}\right) \right| \\&\quad \le \sqrt{2}\sum _{2\le c\le L}\frac{\delta _c}{2}\frac{\sqrt{Nc}}{\pi \sqrt{2}(3n)^\frac{1}{4}}e^{\frac{\pi \sqrt{3n}}{Nc}} \le \frac{\sqrt{N}}{\pi (3n)^\frac{1}{4}} L^\frac{3}{2} e^{\frac{\pi \sqrt{3n}}{2N}}\\&\quad = \frac{\sqrt{3\pi n}}{N}e^{\frac{\pi \sqrt{3n}}{2N}}. \end{aligned}$$
For the terms of R with \(c\ge L\), we can use the first estimate on the Bessel function.
$$\begin{aligned}&\left| (1-i)\sum _{c> L}^{\infty }\delta _c\frac{K_{\psi }(-3,n,4Nc)}{4Nc}\left( I_{\frac{1}{2}}\left( \frac{\pi \sqrt{3n}}{Nc}\right) -\sqrt{\frac{2\sqrt{3n}}{Nc}}\right) \right| \\&\quad \le \sqrt{2}\left| \sum _{c> L}^{\infty }\frac{\delta }{2}\frac{\sqrt{2}\pi ^2 (3n)^\frac{5}{4}}{5(Nc)^\frac{5}{4}} \right| \\&\quad \le \frac{2 \pi ^2(3n)^\frac{5}{4}}{5N^\frac{5}{2}}\zeta \left( \frac{5}{2}\right) , \end{aligned}$$
where \(\zeta (s)\) denotes the Riemann zeta function. We now need only estimate \(Z^*_\psi (m,n;\tfrac{3}{4})\). It turns out that we can use Gannon’s estimates almost directly once we write the modified zeta function in a form sufficiently similar to the expressions he uses in his estimates. However, we will need to slightly modify some of Harvey and Rayhaun’s notation for v and h. Let \(\hat{h} = \frac{h}{(h,4)}\) and \(\hat{v}=\frac{4\nu }{(h,4)}\pmod {\hat{h}}\) so that \(\hat{h}\mid (N,24)\) and
$$\begin{aligned} \psi (4Nc,d)=\exp \left( -2\pi i \hat{v} \frac{cd}{\hat{h}}\right) . \end{aligned}$$
We note that in every case given we have that \(\hat{v}\equiv \pm 1\pmod {\hat{h}}.\)

With this notation, we have that

Proposition 4.1

Let \(n\ge 40\) with \(D=mn\) a negative discriminant. The Selberg–Kloosterman zeta function defined in (4) converges at \(s=3/4,\) with the following bounds.

If \(N\ne 2\), then
$$\begin{aligned} |Z^*_\psi (m,n;3/4)|&\le \frac{1}{4}\left( \prod _{p\mid 4N\hat{h}}\left( 1+\frac{1}{p}\right) \right) (1 + 2.13|D|^{1/8} \log |D|) \\&\quad \times \left( (6.124N^{35/6}\hat{h}^{47/6} - 3.09N^{23/4}\hat{h}^{31/4} + 64.32N^{29/6}{\hat{h}}^7 \right. \\&\quad -\, 23N^{19/4}\hat{h}^7)|D| + \left( .146N^{47/6}\hat{h}^{65/6} - .114N^{31/4}\hat{h}^{43/4} \right. \\&\quad \left. \left. +\, 2.51N^{35/6}\hat{h}^{10}- .74N^{23/4}\hat{h}^{10}\right) |D|^{3/2}\right) . \end{aligned}$$
If \(N=2\), then
$$\begin{aligned} |Z_\psi (m,n;3/4)|\le&\frac{1}{4}\left( \prod _{p\mid 4N\hat{h}}\left( 1+\frac{1}{p}\right) \right) (1 + 2.13|D|^{1/8} \log |D|)\times \left( 3872|D| + 213|D|^{3/2}\right) . \end{aligned}$$

Proof

Combining Eqs. (2.3) and (2.2), we can write the Kloosterman sum \(K_\psi (m,n,4Nc)\) as
(4.2)
Using a result by Kohnen [28, Proposition 5], we can write our Kloosterman sum as a sum over a more sparse set. Kohnen shows that
$$\begin{aligned}&\frac{1}{\sqrt{4Nc}}(1-i)\left( 1+\delta _{\mathrm{odd}}(Nc)\right) K(m,n,4Nc) \nonumber \\&\quad =\sum _{\begin{array}{c} \beta \pmod {2Nc}\\ \beta ^2\equiv mn\pmod {4Nc} \end{array}}\chi _{m}\left( \left[ Nc,\beta ,\frac{\beta ^2-mn}{4Nc} \right] \right) \exp \left[ 2\pi i \left( \frac{\beta }{2Nc}\right) \right] . \end{aligned}$$
(4.3)
Here, \([\alpha ,\beta ,\gamma ]\) is a positive definite integral binary quadratic form, in this case with discriminant mn, and \(\chi _\varDelta \) is the genus character defined as follows on integral binary quadratic forms with discriminant divisible by \(\varDelta \) by
$$\begin{aligned} \chi _\varDelta \left( Q\right) =\chi _\varDelta \left( [\alpha ,\beta ,\gamma ]\right) := {\left\{ \begin{array}{ll} \left( \frac{\varDelta }{R}\right) &{}\quad \text { if} {\text {gcd}}(\varDelta , \alpha ,\beta ,\gamma )=1 \text { where } Q \text { represents } R \\ 0 &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

We can write Eq. (4.2) in this form if we replace n with \(\tilde{n}=n-4\hat{v} c^2\cdot \frac{N}{\hat{h}}.\) Unfortunately, this makes the sum over a set of quadratic forms with discriminant \(m\tilde{n}\) which depends on c. This is not ideal for approximating the zeta function. To fix this, notice that if the quadratic form \(Q=[Nc,\beta ,\gamma ]\) has discriminant \(m\tilde{n},\) then the form \(Q'=[Nc,\beta ,\gamma '/\hat{h}]\) with \(\gamma '=\frac{\beta ^2-mn}{4Nc/\hat{h}}\) is a positive definite binary quadratic form, with discriminant mn and \(\gamma '\equiv mv c\pmod {\hat{h}}.\) This relation defines a bijection between such forms.

Let \(\mathcal {Q}_{N;\hat{h},m \hat{v} }(D)\) denote the set of quadratic forms \(Q=[Nc,\beta ,\gamma /\hat{h}]\) of discriminant D with \( c,\beta ,\gamma \in \mathbb {Z}\) and \(\gamma \equiv m\hat{v} c\pmod {\hat{h}},\) and let \(\mathcal {Q}_{N}(d)\) denote the set of quadratic forms \(Q=[Nc,\beta ,\gamma ]\) of discriminant d. Then, we have the bijection
$$\begin{aligned} \varphi _{\hat{h}, m \hat{v}}{:}\,\mathcal {Q}_{N;\hat{h},m \hat{v} }(D)\rightarrow \mathcal {Q}_{N;}\left( D-4m \hat{v} \alpha ^2N/\hat{h}\right) \end{aligned}$$
defined by
$$\begin{aligned} \varphi _{\hat{h}, m \hat{v}} \left[ N\alpha ,\beta ,\gamma /\hat{h}\right] =\left[ N\alpha ,\beta ,(\gamma -m\hat{v} \alpha )/\hat{h}\right] . \end{aligned}$$
We may drop the subscript of \(\varphi \) as it will generally be clear from context.
The set \(\mathcal {Q}_{N;\hat{h},m \hat{v}}(D)\) is acted upon by a certain matrix group which we denote by \(\varGamma _0(N;h,m\hat{v})\). This group consists of matrices \(\begin{pmatrix}a&{}\quad b/h\\ Nc&{}\quad d\end{pmatrix}\) of determinant 1 where each letter is an integer satisfying the relations
$$\begin{aligned} a\equiv \ell d \pmod {\hat{h}} \quad \text { and } \quad b\equiv \ell m\hat{v} c \pmod {\hat{h}}. \end{aligned}$$
Here, \(\ell \) is some number coprime to \(\hat{h}.\) This generalizes the groups \(\varGamma _0(N;h)=\varGamma _0(N;h,1)\) used by Gannon.

Proposition 4.2

Assume the notation above, and let \(Q_1,Q_2\in \mathcal {Q}_{N;\hat{h},\varDelta \hat{v} }(D)\) where \(\varDelta ,D\) are discriminants, with \(\varDelta \) fundamental and D divisible by \(\varDelta \). If \(Q_1\) and \(Q_2\) are related by the action of some \(M\in \varGamma _0(N;h,m\hat{v}),\) then
$$\begin{aligned} \chi _\varDelta \left( \varphi Q_1\right) =\chi _\varDelta \left( \varphi Q_2\right) \end{aligned}$$

Proof

Since \(\varDelta \) is a discriminant, the definition of \(\chi _\varDelta (Q)\) as a Kronecker symbol allows us to reduce the coefficients of the quadratic form Q modulo \(\varDelta .\) It is also multiplicative. If \(\varDelta =\varDelta _1\varDelta _2\) is a factorization into discriminants, then
$$\begin{aligned} \chi _\varDelta (Q)=\chi _{\varDelta _1}(Q)\chi _{\varDelta _2}(Q). \end{aligned}$$
We will want a factorization of \(\varDelta \) into discriminants \(\varDelta =\varDelta '\varDelta _h\) where \((\varDelta ',\hat{h})=1\) and \(\varDelta _h\) is divisible only by primes dividing \(\hat{h}.\) Since \(\varDelta \) is a fundamental discriminant and \(\hat{h}\) divides 24, this means that \(|\varDelta _h|\) also divides 24.
By construction, \(\varphi ~Q_1\equiv Q_1\pmod {\varDelta '}.\) As a determinant 1 matrix will not alter the integers represented by a quadratic form, we have that
$$\begin{aligned} \chi _{\varDelta '}(\varphi ~Q_1)=\chi _{\varDelta '}(Q_1)=\chi _{\varDelta '}(Q_2)=\chi _{\varDelta '}(\varphi ~Q_2) \end{aligned}$$
Suppose \(Q_1=[N\alpha ,\beta , \gamma /\hat{h}]\) and \(Q_2=Q_1|M\) where M is the matrix \(\begin{pmatrix}a &{}\quad b/\hat{h}\\ Nc&{}\quad d\end{pmatrix}\) with \(a\equiv \ell d \pmod {\hat{h}}.\) If we set \(\gamma '= (\gamma -m\hat{v} \alpha )/\hat{h},\) then a short calculation shows that
$$\begin{aligned} \varphi Q_2\equiv \left[ N(\alpha a^2 +\beta ac), \beta ad, d^2\gamma '\right] \pmod {\varDelta }. \end{aligned}$$
Since M has determinant 1, we have that a and d are coprime to \(\varDelta \). Since \(\varDelta _h\mid 24\), we have that \(a^2\equiv d^2\equiv 1\pmod {\varDelta _h}\). Moreover, by considering the discriminant, we see that \(\beta \) is even if \(\varDelta _h\) is, and \(4\mid \beta \) if \(8\mid \varDelta _h\). In either case, we find that \(\varDelta _h \) divides \(\beta N\). Therefore, we may further reduce to
$$\begin{aligned} \varphi Q_2\equiv [N\alpha , \beta \ell , \gamma '] \pmod {\varDelta _h}. \end{aligned}$$
The \(\ell \) does not change the possible numbers represented, and so we have that
$$\begin{aligned} \chi _{\varDelta _h}(\varphi ~Q_1)=\chi _{\varDelta _h}(\varphi ~Q_2), \end{aligned}$$
concluding the proof of Proposition 4.2. \(\square \)
Once again following Gannon, if we combine Eqs. (4.1), (4.3) and Proposition 4.2, we can write the modified Selberg–Kloosterman zeta function at \(s=3/4\) as
$$\begin{aligned} Z^*_\psi (m,n;3/4)&=\lim _{X\rightarrow \infty }\sum _{[Q]\in \mathcal {Q}_{N;\hat{h},\hat{v}}(mn)/\varGamma _0(N;\hat{h},m\hat{v})}\frac{\chi _m(\varphi _{\hat{h}, m \hat{v}} Q)}{\omega _Q}\\&\quad \,\sum _{\begin{array}{c} r,Ns\in Z,\\ (r,Ns/h)=1\\ 0<c(Q,r,Ns)<X \end{array}} \frac{1}{4Nc(Q,r,Ns)} \exp \left( 2\pi i \frac{\beta }{2Nc(Q,r,Ns)}\right) . \end{aligned}$$
Here, \(Q=[N\alpha ,\beta ,\gamma /\hat{h}],\) \(\omega _Q\) is the order of the stabilizer of Q in \(\varGamma _0(N;\hat{h},m\hat{v}),\) and \(c(Q,r,Ns)=\frac{Q(r,Ns)}{4N}\).

This equation is analogous to Equation (4.26) of [23], but differs in four main points: First, we have normalized the zeta function slightly differently. Second, the bijection \(\varphi _{\hat{h}, m\hat{v}}\) and Proposition 4.2 give a more general version of Gannon’s Lemma 5(b) allowing us to sum over \(\mathcal {Q}_{N;\hat{h},\hat{v}}(mn){/}\varGamma _0(N;\hat{h},m\hat{v})\) rather than \(\mathcal {Q}_{N;\hat{h}}(mn){/}\varGamma _0(N;\hat{h})\). Third, Gannon’s case was restricted to discriminants where the stabilizer could only be \(\{\pm I\},\) and so he replaces the \(\omega _Q\) term with a 2 in his equation. We will use this as a lower bound for \(\omega _Q.\) Fourth, his sum contains a power of \(-1\) while ours contains a genus character. In either case, the sign is constant for a given representative quadratic form Q.

Gannon estimates the inner sums in absolute value and the outer sum by bounding the number of classes of quadratic forms. His bounds for the size of \(\mathcal {Q}_{N;\hat{h}}(mn){/}\varGamma _0(N;\hat{h})\) are crude enough to also hold for the number of classes of \(\mathcal {Q}_{N;\hat{h},\hat{v}}(mn){/}\varGamma _0(N;\hat{h},m\hat{v})\). Proposition 4.1 follows from using Gannon’s bounds modified only to account for our differences in normalization. \(\square \)

Combining these estimates as described above, we find that each multiplicity of the irreducible components of \(W_n\) must alway be positive for \(n\ge 375.\) Explicit calculations up to \(n=375\) show that these multiplicities are always positive. The worst cases for the estimates with \(n\le 375\) arise from the trivial character or from estimating for Selberg–Kloosterman zeta function for the 24CD conjugacy class. These calculations were performed using Sage mathematical software [39].

5 Replicability

One important property of the Hauptmoduln occurring in Monstrous Moonshine is that they are replicable.

Definition 5.1

Let \(f(\tau )=q^{-1}+\sum _{n=0}^\infty H_nq^n\) be a (formal) power series with integer coefficients and consider the function
$$\begin{aligned} F(\tau _1,\tau _2)=\log \left( f(\tau _1)-f(\tau _2)\right) =\log \left( q_1^{-1}-q_2^{-1}\right) -\sum _{m,n=1}^\infty H_{m,n}q_1^mq_2^n, \end{aligned}$$
where \(\tau _1,\tau _2\in \mathfrak {H}\) are two independent variables and \(q_j=e^{2\pi i\tau _j},\,j=1,2\). We call f replicable, if we have that \(H_{a,b}=H_{c,d}\) whenever \(ab=cd\) and \(\gcd (a,b)=\gcd (c,d)\).

This property of the Hauptmoduln involved in Monstrous Moonshine in a sense reflects the algebra structure of the Monstrous Moonshine module, see [14, 30].

An important, but not immediately obvious fact is that any replicable function is determined by its first 23 Fourier coefficients, [21, 30].

Theorem 5.2

Let \(f(\tau )=q^{-1}+\sum _{n=1}^\infty a_nq^n\) be a replicable function. Then one can compute the coefficient \(a_n\) for any \(n\in \mathbb {N}\) constructively out of the coefficients
$$\begin{aligned} \{a_1,a_2,a_3,a_4,a_5,a_7,a_8,a_9,a_{11},a_{17},a_{19},a_{23}\}. \end{aligned}$$

A Maple procedure to perform this computation is printed at the end of [21].

In [18], there is also an analogous notion of replicability in the mock modular sense, which requires that the Fourier coefficients satisfy a certain type of recurrence. This is a special phenomenon occurring for mock theta functions, i.e., mock modular forms whose shadow is a unary theta function, satisfying certain growth conditions at cusps, see [26, 29].

It is now natural to ask about replicability properties of the McKay–Thompson series in the case of Thompson Moonshine. Let g be any element of the Thompson group and \(\mathcal {T}_{[g]}(\tau )=\mathcal {F}_{[g]}(\tau )\) be the corresponding McKay–Thompson series as in (2.6). As we have seen, this is a weekly holomorphic modular form of weight \(\tfrac{1}{2}\) living in the Kohnen plus space. To relate these to the Hauptmoduln and other replicable functions discussed in [21], we split into an even and an odd part
$$\begin{aligned} \mathcal {T}_{[g]}^{(0)}(\tau )=\sum _{m=0}^\infty \mathrm{strace}_{W_{4m}}q^{4m}\quad \text {and}\quad \mathcal {T}_{[g]}^{(0)}(\tau )=\sum _{m=0}^\infty \mathrm{strace}_{W_{4m-3}}q^{4m-3} \end{aligned}$$
in the notation of Conjecture 1.1. Letting
$$\begin{aligned} \vartheta ^{(0)}(\tau )=\vartheta (\tau )=\frac{\eta (2\tau )^5}{\eta (\tau )^2\eta (4\tau )^2}=\sum _{n\in \mathbb {Z}}q^{n^2} \end{aligned}$$
and
$$\begin{aligned} \vartheta ^{(1)}(\tau )=2\frac{\eta (4\tau )^2}{\eta (2\tau )}=\sum _{n\in \mathbb {Z}}q^{\left( n+\frac{1}{2}\right) ^2} \end{aligned}$$
we define the weight 0 modular functions
$$\begin{aligned} \mathbf {t}^{(j)}_{[g]}(\tau )=\frac{\mathcal {T}_{[g]}^{(j)}\left( \frac{\tau }{4}\right) }{\vartheta ^{(j)}(\tau )},\quad (j=0,1) \end{aligned}$$
and we set \(\mathbf {t}_{[g]}(\tau )=\mathbf {t}^{(0)}_{[g]}(\tau )+\mathbf {t}^{(1)}_{[g]}(\tau )\). Note that these functions do not have poles in \(\mathfrak {H}\).

As it turns out through direct inspection, these weight 0 functions are often replicable functions or univariate rational functions therein. We used the list of replicable functions given in [21] as a reference and found the identities given in Tables 6, which are all identities of the given form using the aforementioned table of replicable functions at the end of [21] and allowing the degree of the denominator of the rational function to be as large as 40.

Acknowledgements

The authors would like to thank Ken Ono for instigating this project and John Duncan for many invaluable conversations on the subject and helpful comments on an earlier version of this article. The first author was supported by the National Science Foundation grant DMS-1502390.

Footnotes
1

We usually omit the word “weak” from now on.

 
2

In the literature, the shadow is often rather associated with the holomorphic part \(f^+\) of a harmonic Maaß form f rather than to f itself.

 
5

A list of the eta quotients and linear combinations are available from the second author’s homepage.

 
6

We say probably because the proof of Atkin–Fong–Smith relies on results in Margaret Ashworth’s (later Millington) Ph.D. thesis (Oxford University, 1964, advised by A. O. L. Atkin), of which the authors were unable to obtain a copy.

 

Declarations

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Princeton University
(2)
Department of Mathematics and Computer Science, Emory University

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