A proof of the Thompson moonshine conjecture
 Michael J. Griffin^{1} and
 Michael H. Mertens^{2}Email author
https://doi.org/10.1186/s4068701600847
© The Author(s) 2016
Received: 1 July 2016
Accepted: 1 September 2016
Published: 14 December 2016
Abstract
In this paper, we prove the existence of an infinitedimensional 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 11F371 Introduction and statement of results
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 nonisometric 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.
Conjecture 1.1
Here, we prove this conjecture.
Theorem 1.2

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 \(\frac{1}{2g}\); otherwise, there is only the pole at \(\infty \). It vanishes at all other cusps.
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
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 twentyfirst 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].
Definition 2.2
 (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 definewith$$\begin{aligned} f_k\gamma (\tau ):= {\left\{ \begin{array}{ll} (c\tau +d)^{k}f\left( \frac{a\tau +b}{c\tau +d}\right) &{}\quad \text {if }k\in \mathbb {Z}, \\ \left( \left( \frac{c}{d}\right) \varepsilon _d\right) ^{2k}\left( \sqrt{c\tau +d}\right) ^{2k}f\left( \frac{a\tau +b}{c\tau +d}\right) &{}\quad \text {if }k\in \frac{1}{2}+\mathbb {Z}. \end{array}\right. } \end{aligned}$$and where we assume 4N if \(k\notin \mathbb {Z}\).$$\begin{aligned} \varepsilon _d:= {\left\{ \begin{array}{ll} 1 &{}\quad d\equiv 1\pmod {4}, \\ i &{}\quad d\equiv 3\pmod 4. \end{array}\right. } \end{aligned}$$
 (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)
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 \(\varGamma _0(N)\).
Remark 2.3
 (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)
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 nonholomorphic part (see, for example, equations (3.2a) and (3.2b) in [7]).
Lemma 2.4
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
We will later use the following result [7, Proposition 3.5].
Proposition 2.6
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.
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
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 \(R^{[3]}_{\psi ,\frac{3}{2}}\).
Lemma 2.9
Proof
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
Proof
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 \(M_{\frac{1}{2}}^{+,!}(4g,\psi _{[g]})\le M_{\frac{1}{2}}^{+,!}(N_{[g]})\) with \(N_{[g]}\) as in Table 5.
Proof
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
Remark 3.3
We can now establish the uniqueness claim in Theorem 1.2 very easily.
Proposition 3.4

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 \(\frac{1}{2g}\); otherwise, there is only the pole at \(\infty \). It vanishes at all other cusps.
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. \(\square \)
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.
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 (probably^{6}) 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
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.
Proof
We can write Eq. (4.2) in this form if we replace n with \(\tilde{n}=n4\hat{v} c^2\cdot \frac{N}{\hat{h}}.\) 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 \(\gamma '=\frac{\beta ^2mn}{4Nc/\hat{h}}\) 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.
Proposition 4.2
Proof
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
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
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].
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 DMS1502390.
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.
A list of the eta quotients and linear combinations are available from the second author’s homepage.
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
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