Coefficients of McKay-Thompson series and distributions of the moonshine module
aa r X i v : . [ m a t h . N T ] J a n COEFFICIENTS OF MCKAY-THOMPSON SERIES ANDDISTRIBUTIONS OF THE MOONSHINE MODULE
HANNAH LARSON
Abstract.
In a recent paper, Duncan, Griffin and Ono provide exact formulas for thecoefficients of McKay-Thompson series and use them to find asymptotic expressions for thedistribution of irreducible representations in the moonshine module V ♮ = L n V ♮n . Theirresults show that as n tends to infinity, V ♮n is dominated by direct sums of copies of theregular representation. That is, if we view V ♮n as a module over the group ring Z [ M ], thefree-part dominates. A natural problem, posed at the end of the aforementioned paper, isto characterize the distribution of irreducible representations in the non-free part. Here,we study asymptotic formulas for the coefficients of McKay-Thompson series to answer thisquestion. We arrive at an ordering of the series by the magnitude of their coefficients,which corresponds to various contributions to the distribution. In particular, we show howthe asymptotic distribution of the non-free part is dictated by the column for conjugacyclass 2A in the monster’s character table. We find analogous results for the other monstermodules V ( − m ) and W ♮ studied by Duncan, Griffin, and Ono. Introduction
Monstrous moonshine refers to the mysterious connection between the representation the-ory of the largest sporadic simple group M , known as the monster group , and the theoryfunctions which are invariant under the action of certain subgroups of GL ( Q ) + , known as modular functions . The first hints of monstrous moonshine came in the famous observationsof McKay and Thompson,(1.1) 1 = 1 , , , among others. The numbers on the left are coefficients in the Fourier expansion of the normalized j -function , the modular function defined by J ( τ ) := j ( τ ) −
744 = E ( τ )∆( τ ) − ∞ X n = − c ( n ) q n = q − + 196884 q + 21493760 q + . . . , where E ( τ ) is the Eisenstein series of weight 4, ∆( τ ) is the modular discriminant, and q := e πiτ . Meanwhile, the numbers in the summands on the right-hand side are dimensionsof irreducible representations of the monster group.The striking coincidences in (1.1), along with many similar observations, led Thompsonto conjecture [13] the existence of a naturally-defined infinite-dimensional monster module, V ♮ = L ∞ n = − V ♮n , satisfying dim( V ♮n ) = c ( n ) . Thompson then considered the other graded trace-functions that would arise from V ♮ ,(1.3) T g ( τ ) := q − + ∞ X n =1 c g ( n ) := ∞ X n = − tr( g | V ♮n ) q n , for g ∈ M , known as McKay-Thompson series , and found that their coefficients were alsoexpressible as simple sums of entries in the monster’s character table.The distinguishing feature of J ( τ ) is that it generates the field of SL ( Z )-invariant func-tions on the upper-half plane with at most exponential growth as I ( τ ) → ∞ , and is theunique such function with Fourier expansion beginning q − + O ( q ). Given any subgroupΓ ⊂ GL ( Q ) + commensurable with SL ( Z ), if the field of Γ-invariant functions with atmost exponential growth at the cusps is generated by a single function, then we call theunique generator with Fourier expansion beginning q − + O ( q ) the Hauptmodul for Γ. Thefamous monstrous moonshine conjecture of Conway and Norton states that for each g ∈ M ,the McKay-Thompson series T g ( τ ) is the Hauptmodul for a particular discrete subgroupΓ g ⊂ GL ( Q ) + , which they describe explicitly [6]. In highly celebrated works, Frenkel, Lep-owsky and Murman constructed the moonshine module V ♮ in [9, 10], and Borcherds provedthe monstrous moonshine conjecture in [2].More recent work shows that the monster also plays a role in quantum gravity, in particularin theories of chiral three-dimensional gravity [7, 8]. These theories take interest in a towerof monster modules V ( − m ) = ∞ M n = − m V ( − m ) n with V ( − = V ♮ . In [8], the authors consider the order m McKay-Thompson series arisingfrom these modules,(1.4) T ( − m ) g ( τ ) := q − m + ∞ X n =1 c g ( − m, n ) q n := q − m + ∞ X n =1 tr( g | V ( − m ) n ) q n , for g ∈ M , and show that they coincide with particular Rademacher sums of order m for thegroup Γ g (see Theorem 7.1 of [8]).It turns out that the multiplicities of irreducible representations in V ( − m ) n are expressibleas sums of entries in the monster’s character table weighted by the McKay-Thompson co-efficients c g ( − m, n ). We first give exact formulas for the coefficients c g ( − m, n ), similar tothose in Theorem 8.12 of [8], which are better suited to our purposes. Theorem 1.1.
Let g ∈ M , and let W g be the set corresponding to the Atkin-Lehner invo-lutions of Γ g given in the appendix. For any positive integers m and n , the coefficients of T ( − m ) g ( τ ) are exactly c g ( − m, n ) = X e ∈W g π r emn X c> K c ( g, e, − m, n ) c I (cid:18) πc √ emn (cid:19) , where K c ( g, e, − m, n ) is the Kloosterman sum defined in (2.1) and I ( x ) is the Bessel functionof the first kind. CKAY-THOMPSON COEFFICIENTS AND DISTRIBUTIONS OF THE MOONSHINE MODULE 3
From well-known asymptotics of the Bessel function and explicit computations of Kloost-erman sums, we arrive at the following asymptotic formula for these coefficients.
Theorem 1.2.
Let g ∈ M and let Γ g = N || h + W g as listed in the appendix. If ε = max( W g ) ,then as n → ∞ , we have c g ( − m, n ) ∼ ( mε ) / √ N n / · K N ( g, ε, − m, n ) · exp (cid:18) π √ εmnN (cid:19) , with the following exceptions: (1) If g is in conjugacy class or , m is odd, and n ≡ − m (mod 4) , then c g ( − m, n ) ∼ m / √ n / · K ( g, , − m, n ) · exp (cid:18) π √ mn (cid:19) . (2) If g is in conjugacy class , m ≡ , and n ≡ − m (mod 8) , then c g ( − m, n ) ∼ m / n / · K ( g, , − m, n ) · exp (cid:18) π √ mn (cid:19) (3) If g is in conjugacy class , m , and n ≡ − m (mod 3) , then c g ( − m, n ) ∼ m / n / · K ( g, , − m, n ) · exp (cid:18) π √ mn (cid:19) . (4) If g is in conjugacy class , m is odd, and n ≡ m (mod 4) , then c g ( − m, n ) ∼ m / √ n / · K ( g, , − m, n ) · exp (cid:18) π √ mn (cid:19) . Hence, the list of conjugacy classes in the appendix beginning , , , { , } , , { , } , , (4B , , (3C , , , . . . orders the order m McKay-Thompson series by the asymptotic magnitude of their coeffi-cients, where we remove a conjugacy class if K N ( g, ε, − m, n ) , which is periodic in m and n modulo N/ε , vanishes.
The ordering of conjugacy classes in Theorem 1.2 can be interpreted as an order forcontributions of the corresponding columns in the character table to the multiplicities ofirreducible representations in V ( − m ) n as n → ∞ . Let M i , for 1 ≤ i ≤ M i is the function denoted χ i in [5]. In addition, let m i ( − m, n ) be the multiplicity of M i in V ( − m ) n so that V ( − m ) n ≃ M i =1 M ⊕ m i ( − m,n ) i . In [8], the authors prove that as n → ∞ , m i ( − m, n ) ∼ dim( χ i ) m / √ n / | M | · e π √ mn . (1.5) HANNAH LARSON
In particular, the limit δ ( m i ( − m )) := lim n →∞ m i ( − m, n ) P i =1 m i ( − m, n )exists and is given by(1.6) δ ( m i ( − m )) = dim( χ i ) P j =1 dim( χ j ) = dim( χ i )5844076785304502808013602136 . Another way to phrase this result is that, as n tends to infinity, V ( − m ) n tends to directsums of copies of the regular representation. Viewing V ( − m ) as a module over the group ring Z [ M ], it is thus natural to decompose V ( − m ) n into a free part and a non-free part and askwhat can be said about the distribution of irreducible representations in the non-free part(see Problem 10.9 in [8]). Define nf i ( − m, n ) to be the multiplicity of M i in the non-free partof V ( − m ) n so that V ( − m ) n ≃ Z [ M ] ⊕ f ( − m,n ) ⊕ M i =1 M ⊕ nf i ( − m,n ) i where f ( − m, n ) is maximal. Remark.
The results of [8] show that as n → ∞ , we have f ( − m, n ) ∼ m / √ n / | M | · e π √ mn . Here, we use the order in Theorem 1.2 to find that the column of the monster’s charactertable corresponding to conjugacy class 2A dictates the asymptotic non-free distributions of V ( − m ) n . Writing χ i (2A) for the value of the character of M i on an element in conjugacy class2A and | | for the size of this conjugacy class, our precise result can be stated as follows. Theorem 1.3.
For any ≤ i ≤ , as n → ∞ , we have nf i ( − m, n ) ∼ (cid:18) dim( χ i )3301375 + χ i (2A) (cid:19) · | | m / | M | (2 n ) / · e π √ mn . Thus the following limit is well-defined δ ( nf i ( − m )) := lim n →∞ nf i ( − m, n ) P i =1 nf i ( − m, n ) . Corollary 1.4.
In particular, as n → ∞ , we have that δ ( nf i ( − m )) = dim( χ i )3301375 + χ i (2A) P j =1 dim( χ j )3301375 + χ j (2A) = dim( χ i ) + 3301375 χ i (2A)5845054856224474627181019136 . Remark.
It is natural to compare Theorem 1.3 and Corollary 1.4 to (1.5) and (1.6). Theappearance of two character values in our expression for the non-free distribution comes fromaccounting for negative character values, which are not considered in [8].
CKAY-THOMPSON COEFFICIENTS AND DISTRIBUTIONS OF THE MOONSHINE MODULE 5
Remark.
These methods can also be applied to the monster module W ♮ = L n W ♮n studiedby Duncan, Griffin, and Ono in their answer to a problem of Witten on the distribution ofblack holes (see Question 6.1 and § e nf i ( n ) be the multiplicity of M i in thenon-free part of W ♮n , then as n → ∞ , we have e nf i ( n ) ∼ (cid:18) dim( χ i )330175 + χ i (2A) (cid:19) | |√ | M |√ n + 1 · e π √ n +1) , and so δ ( e nf i ) = dim( χ i ) + 3301375 χ i (2A)5845054856224474627181019136 , providing a refinement to these distribution results.The following table, which should be compared with Table 1 in [8], illustrates the asymp-totics of Corollary 1.4 for χ , χ , χ and χ when m = 1. Let δ ( nf i ( − , n )) be the propor-tion of irreducible representations corresponding to χ i in the non-free part of V ( − n = V ♮n . n δ ( nf ( − , n )) δ ( nf ( − , n )) δ ( nf ( − , n )) δ ( nf ( − , n ))-1 1 0 0 00 0 0 0 01 1/2 1/2 0 02 1/3 1/3 0 0... ... ... ... ...40 4 . . . . × − . . . . × − . . . .
80 4 . . . . × − . . . . × − . . . . × − . . . .
120 1 . . . . × − . . . . × − . . . . × − . . . .
160 1 . . . . × − . . . . × − . . . . × − . . . .
200 7 . . . . × − . . . . × − . . . . × − . . . .
240 5 . . . . × − . . . . × − . . . . × − . . . .
280 5 . . . . × − . . . . × − . . . . × − . . . .
320 5 . . . . × − . . . . × − . . . . × − . . . .
360 5 . . . . × − . . . . × − . . . . × − . . . .
400 5 . . . . × − . . . . × − . . . . × − . . . . ... ... ... ... ... ∞ The exact values in the bottom row have the following decimal approximations: δ ( nf ( − ≈ . . . . × − δ ( nf ( − ≈ . × − δ ( nf ( − δ ( nf ( − ≈ . . . . . HANNAH LARSON
This paper is organized as follows. In Section 2, we recall the description of the moon-shine groups and construct Poincar´e series for these groups which are equal to the McKay-Thompson series. We then find exact expressions for their coefficients following the methodsin [3, 4]. Next, in Section 3, we study the Kloosterman sums appearing in these exact ex-pressions to arrive at the asymptotics in Theorem 1.2. In Section 4, we apply the asymptoticformulas to describe the non-free distributions.
Acknowledgements.
This research was started at the 2015 REU at Emory University.The author would like to thank Ken Ono for advising this project, Michael Griffin and LeaBeneish for helpful conversations and suggestions, and the NSF for its support.2.
McKay-Thompson coefficients
Recall that GL ( Q ) + acts on the upper half-plane H by fractional linear transformation:for γ = ( a bc d ) ∈ GL ( Q ) + and τ ∈ H , we set γτ := aτ + bcτ + d . Let Γ ⊂ GL ( Q ) + be a subgroupcommensurable with SL ( Z ). A modular function for Γ is a meromorphic function f : H → C which is invariant under the action of Γ, i.e. satisfies f ( γτ ) = f ( τ ) for all γ ∈ Γ.2.1.
The moonshine groups.
We now describe the groups Γ g having Hauptmodules whichare the McKay-Thopson series T g ( τ ). Recall that for a positive integer N , the congruencesubgroup Γ ( N ) ⊆ SL ( Z ) is defined as those matrices which are upper triangular mod N ,Γ ( N ) := (cid:26)(cid:18) a bc d (cid:19) ∈ SL ( Z ) : c ≡ N ) (cid:27) . For each exact divisor e | N with ( e, N/e ) = 1, the associated Atkin-Lenher involutions forΓ ( N ) are those matrices of the form W e = (cid:18) ae bcN de (cid:19) with determinant e . For each g ∈ M , we associate a group E g , denoted by a symbolΓ ( N | h ) + e, f, . . . (or just N | h + e, f . . . ), where h divides ( N,
24) and each e, f, . . . exactlydivides
N/h . This symbol stands for the group of matricesΓ ( N | h ) + e, f, . . . := (cid:18) /h
00 1 (cid:19) h Γ ( N/h ) , W e , W f , . . . i (cid:18) h
00 1 (cid:19) , where W e , W f , . . . are representative of Atkin-Lehner involutions for Γ ( N/h ). We write W g := { , e, f, . . . } for the set corresponding to Atkin-Lehner involutions in E g . The groups E g are eigengroups for T g ( τ ), meaning that T g ( γτ ) = σ g ( γ ) T g ( τ ) for γ ∈ E g , where σ g is agroup homomorphism from E g to the h th roots of unity. Conway and Norton provide thefollowing explicit description of σ g on generators of E g . Lemma 2.1 (Conway-Norton) . With the above notation, we have (1) σ g ( γ ) = 1 if γ ∈ Γ ( N h )(2) σ g ( γ ) = 1 if γ is an Atkin-Lehner involution of Γ ( N h ) inside E g (3) σ g ( γ ) = e − πih if γ = (cid:18) /h (cid:19) CKAY-THOMPSON COEFFICIENTS AND DISTRIBUTIONS OF THE MOONSHINE MODULE 7 (4) σ g ( γ ) = e − λ g πih if γ = (cid:18) N (cid:19) ,where λ g is − if N/h ∈ W g and otherwise. The group Γ g is the kernel of σ g inside E g , which is denoted by Γ ( N || h ) + e, f, . . . (or just N || h + e, f, . . . ). A complete list of the groups Γ g is in the appendix.2.2. Poincar´e series and exact expressions.
In Section 8.3 of [8], the authors build anice basis of Maass-Poincar´e series for Γ ( N ) and take certain combinations of them to obtainthe McKay-Thompson series. Here, we carry out the alternative construction suggested atthe beginning of that section, constructing Poincar´e series for the groups Γ g directly.Keeping with the notation of [3, 4, 8], let M s ( w ) := M ,s − ( | w | ) , where M ν,µ ( z ) is the M -Whittaker function. In addition, let τ = x + iy with x, y ∈ R , y > φ s ( τ ) := M s (4 πy ) e πix . Given a subgroup Γ ⊂ GL ( Q ) + commensurable with SL ( Z ), let Γ ∞ denote the stabilizerof ∞ in Γ. For each positive integer m , the Poincar´e series P s ( m, Γ) := X M ∈ Γ ∞ \ Γ φ s ( − m · M τ )converges for s with Re( s ) >
1. In general, these Poincar´e series are harmonic Maass forms,but when Γ has genus zero they are weakly holomorphic. In this case, taking the limit as s →
1+ along the real axis, we obtain a modular function P ( m, Γ) for Γ having a Fourierexpansion q − m + O ( q ) at infinity and no other poles. Since there is a unique such function,we must have T ( − m ) g ( τ ) = P ( m, Γ g ) . Thus, we can use the method in [3, 4] for determining coefficients of these Poincar´e series toprove Theorem 1.1. For a positive integer c , any g ∈ M , and integers m and n , we definethe Kloosterman sum (2.1) K c ( g, e, m, n ) := X ( a bc d ) ∈F g,e ( c ) exp (cid:18) πi ( ma + nd ) c (cid:19) , where the sum ranges over matrices in F g,e ( c ) := (cid:26)(cid:18) a bc d (cid:19) ∈ Γ g : 0 ≤ a, d < c, ad − bc = e (cid:27) . We can now prove our exact expression for the McKay-Thompson coefficients.
Proof of Theorem 1.1.
We argue exactly as in Section 1.3 of [4], but using the group Γ g and aslight modification to account for matrices with different determinants. Since the imaginarypart of τ transforms under ( a bc d ) by y ( ad − bc ) y | cτ + d | , we need to include an extra factor of e in front of y for each Atkin-Lehner involution W e in the sum over Γ ∞ \ Γ g / Γ ∞ in the lastequation on page 31 of [4]. Following through, this results in introducing a sum over e ∈ W g , HANNAH LARSON using the modified Kloosterman sum in (2.1), and changing the quantity B on page 33 to | m | c ey . Plugging this into the equation in terms of A and B on page 33 and carrying out thesimplification leading to Proposition 1.10, we obtain the expression in our Theorem 1.1. (cid:3) Asymptotic formulas for coefficients
First recall the asymptotics of the I -Bessel function,(3.1) I ( x ) ∼ e x √ πx (cid:18) − x + . . . (cid:19) . Suppose Γ g = N || h + W g for some g ∈ M . The Kloosterman sum K c ( g, e, − m, n ) vanishesfor all c N ) since F g,e ( c ) is empty in this case. Therefore, when it does not vanish,the term dominating the expression for c g ( − m, n ) in Theorem 1.1 is2 π r mεn · K N ( g, ε, − m, n ) N · I (cid:18) πN √ εmn (cid:19) ∼ ( mε ) / √ N n / · K N ( g, ε, − m, n ) · exp (cid:18) π √ εmnN (cid:19) , where ε = max( W g ). Thus, to prove Theorem 1.2, it suffices to show that if K N ( g, ε, − m, n )vanishes, then c g ( − m, n ) also vanishes or we are in one of the listed exceptions.The functions K N ( g, ε, − m, n ) are not hard to calculate: using the procedure on page 35of [8] one may evaluate σ g ( γ ) for all γ = ( a bc d ) ∈ E g with 0 ≤ a, d < c and det( γ ) = e to find F g,e ( c ) and then sum over this finite set. Remark.
We note that there is a typo on page 35 of [8]. The last b in the second bullet pointshould be an a .For convenience, we provide a collection of simplified expressions for the Kloostermansums which allows us to compute them all explicitly and determine when they vanish. Let ζ ℓ := e πi/ℓ . Lemma 3.1.
Suppose Γ g = N || h + W g . The following are true: (1) If N/h ∈ W g , then K N ( g, N/h, − m, n ) = ( h if n ≡ − m (mod h )0 otherwise. (2) If h = 1 , and ℓ = N/e for some e ∈ W g , then K N ( g, e, − m, n ) = X a ∈ ( Z /ℓ Z ) × ζ − am +( ae ) − nℓ . In particular, K N ( g, ε, − m, n ) = 0 if and only if one of the following holds: • n ≡ − m (mod ℓ ) for g and ℓ in the table belowg
4C 9B 12E 12I 18D 28C 36B ℓ • g is in and n ≡ m (mod 4) • g is in and n ≡ − m (mod 4) if m is odd or n ≡ − m (mod 8) if m is even • g is in and n ≡ m (mod 3) . CKAY-THOMPSON COEFFICIENTS AND DISTRIBUTIONS OF THE MOONSHINE MODULE 9 (3) If N/ h ∈ W g then K N ( g, N/ h, − m, n ) = ( ( − ( m + n ) /h · h if n ≡ − m (mod h )0 otherwise.The leading Kloosterman sums that do not fit into one of the above cases are
8D : K ( g, , − m, n ) = ζ m + n + ζ m +3 n + ζ m +5 n + ζ m +7 n , which is non-zero if and only if n ≡ m (mod 4) .
12G : K ( g, , − m, n ) = ζ m + n + ζ m +5 n + ζ m +2 n + ζ m +4 n , which is non-zero if and only if n ≡ − m (mod 2) .
15D : K ( g, , − m, n ) = ζ m + n + ζ m +2 n + ζ m +3 n + ζ m +4 n + ζ m +6 n + ζ m +7 n + ζ m +9 n + ζ m +11 n + ζ m +12 n + ζ m +13 n + ζ m +14 n , which is non-zero if and only if n ≡ − m (mod 3) .
24D : K ( g, , − m, n ) = ζ m + n + ζ m +3 n + ζ m +5 n + ζ m +7 n , which is non-zero if and only if n ≡ − m (mod 4) .
24G : K ( g, , − m, n ) = ζ m + n + ζ m +2 n + ζ m +4 n + ζ m +5 n + ζ m +7 n + ζ m +8 n + ζ m +10 n + ζ m +11 n , which is non-zero if and only if n ≡ − m (mod 4) . In order to prove Theorem 1.2, we need to know that the McKay-Thompson coefficientsvanish on certain arithmetic progressions corresponding to the vanishing of the Kloostermansums. To this end, we define a sieving operator which acts on functions f : H → C withFourier expansions f ( τ ) = P n a ( n ) q n by( f | S ℓ,r )( τ ) := X n ≡ r (mod ℓ ) a ( n ) q n = 1 ℓ ℓ − X k =0 ζ ℓ − krℓ f (cid:18) τ + kℓ (cid:19) . In addition, we define the projection operator as the sum of this operator over a square class( f | e S ℓ,r )( τ ) := X s ∈S r ( f | S ℓ,s )( τ ) , where S r := { ra (mod ℓ ) : ( a, ℓ ) = 1 } . It is well-known that sieving on Fourier coefficientsin an arithmetic progression preserves modularity, although it may change the group onwhich the function is modular. More precisely, if f ( τ ) is modular on Γ ( N ), then ( f | e S ℓ,r )( τ )is modular on Γ (lcm( N, ℓ )).We now prove several lemmas that determine which arithmetic progressions certain McKay-Thompson series are supported on. Lemma 3.2.
For g in one of the conjugacy classes listed below, T ( − m ) g ( τ ) has coefficients c g ( − m, n ) supported on n ≡ − m (mod ℓ ) where ℓ is specified in the table. g
4C 8E 9B 12E 12I 16B 18D 28C 36B ℓ Proof.
Note that for these ℓ we have e S ℓ,r = S ℓ,r . We observe that for the given ℓ , the function( T − mg | S ℓ, − m )( τ ) is again modular on Γ g . Then we need only check that the principal part ofthe Fourier expansion at each cusp is unaffected. The principal part at infinity is alreadysupported on this arithmetic progression, and the projection operator cannot introduce polesat the other cusps if there are no others to begin with. Hence, ( T − mg | S ℓ, − m )( τ ) = T − mg ( τ ) sothe coefficients are supported on the claimed arithmetic progression. (cid:3) We need additional results for g in conjugacy class 8D, 8E, 16B, 18A, or 24D when m iseven. For ease of notation, let T ( − m )8D ( τ ) , T ( − m )8E ( τ ), etc. denote T ( − m ) g ( τ ) for g in the specifiedconjugacy class. Lemma 3.3.
The following identities relating order m McKay-Thompson series are true: T ( − ( τ ) = T ( − ( τ ) = T (2 τ ) T ( − ( τ ) = T (2 τ ) T ( − ( τ ) = T ( − ( τ ) = T (4 τ ) T ( − ( τ ) = T (4 τ ) T ( − ( τ ) = T (2 τ ) T ( − ( τ ) = T (8 τ ) T ( − ( τ ) = T (4 τ ) T ( − ( τ ) = T (3 τ ) Proof.
In each case, we note that both sides are modular on Γ g for g in the conjugacy classin the subscript on the left. In addition, both have Fourier expansion q − m + O ( q ) at infinityand no other poles. Since there is a unique such function, they must be equal. (cid:3) Remark.
The identities involving 8E and 16B are special cases of Lemma 2.11 of [1] whichprovides an expression for T ( − m ) g ( τ ) as a combination of Hauptmoduln on lower levels hitwith Hecke operators for all g with Γ g = Γ ( N ). The other identities above suggest that thisresult generalizes to other moonshine groups. Lemma 3.4.
Suppose m is even and g is in conjugacy class , , , or . Then c g ( − m, n ) = 0 for all n
6≡ ± m (mod ℓ ) for ℓ = 4 , , , or respectively. When g is in ,this holds for all m .Proof. When m = ℓ , the claim follows from Lemma 3.3. Suppose that the claim holds when m = iℓ for all 0 < i ≤ k . We can write T ( − ( k +1) ℓ ) g ( τ ) as T ( − ℓ ) g ( τ ) · T ( − kℓ ) g ( τ ) plus a linearcombination of T ( − iℓ ) g ( τ ) for i ≤ k . Each term has coefficients supported on the arithmeticprogression n ≡ ℓ ), so the claim holds for all m ≡ ℓ ) by induction. For g not in 18A, if m = ℓ/
2, Lemma 3.3 together with Lemma 3.2 shows that T ( − m ) g ( τ ) hascoefficients c g ( − m, n ) supported on the progression n ≡ ℓ/ ℓ ). As before, supposethe claim holds when m = ℓ/ iℓ for all 0 ≤ i ≤ k . We can write T ( − ℓ/ − ( k +1) ℓ ) g ( τ ) as T ( − ℓ/ g ( τ ) · T ( − ( k +1) ℓ ) g ( τ ) plus a linear combination of T ( − ℓ/ − iℓ ) g ( τ ) for i ≤ k . The claim nowfollows by induction and the m ≡ ℓ ) case.If g is in 16B, we need the two base cases m = 2 and m = 6. These follow from Lemma3.3 and Lemma 3.2 and the fact that T ( − ( τ ) is a linear combination of T ( − ( τ ) T ( − ( τ ) CKAY-THOMPSON COEFFICIENTS AND DISTRIBUTIONS OF THE MOONSHINE MODULE 11 and T ( − ( τ ). Finally, for g in 18A, we use the identity T ( τ ) = T ( τ ) − T ( τ ) , together with Lemma 3.2, to establish the m = 1 and m = 2 base cases. Arguing byinduction as before completes the proof. (cid:3) Lemma 3.5. If Γ g = N || h + W g then c g ( − m, n ) = 0 for all n
6≡ − m (mod h ) .Proof. If we let γ = (cid:0) /h (cid:1) , then by Lemma 2.1 (3), we have T g ( τ ) = 1 h h − X j =0 e πij/h T g ( γ j τ ) = ( T g | S h, − )( τ ) , showing that T g ( τ ) is supported on terms q n with n ≡ − h ). Since T ( − m ) g ( τ ) is apolynomial in T g ( τ ), only terms T g ( τ ) k with k ≡ m (mod h ) may appear. Hence, T ( − m ) g ( τ )is supported on terms q n with n ≡ − k ≡ − m (mod h ). (cid:3) We can now prove Theorem 1.2.
Proof of Theorem 1.2.
Using Lemmas 3.2, 3.4, and 3.5 and the explicit computations ofKloosterman sums, we see that whenever K N ( g, ε, − m, n ) vanishes, so does c g ( − m, n ), exceptfor the special cases listed in the theorem. In these cases, it is clear that the listed termdominates and it is not hard to check that the Kloosterman sum does not vanish. Theordering listed in the appendix follows from comparing the arguments of the exponentials. (cid:3) Non-free distributions
At the beginning of Section 8.6 of [8], the authors use the orthogonality of characters todeduce that the functions defined by T ( − m ) χ i ( τ ) := 1 | M | X g ∈ M χ i ( g ) T ( − m ) g ( τ )are generating functions for the multiplicities m i ( − m, n ). In particular,(4.1) m i ( − m, n ) = 1 | M | X g ∈ M χ i ( g ) c g ( − m, n ) . By Theorem 1.2, we have that c ( − m, n ) exponentially dominates c g ( − m, n ) for all other g ∈ M . Thus we can see, as concluded in [8], that as n → ∞ , m i ( − m, n ) ∼ c ( − m, n ) | M | dim( χ i ) . The first order correction to this phenomenon comes from the terms weighted by the nextlargest McKay-Thompson coefficients in (4.1). With this idea, we can prove Theorem 1.3.
Proof of Theorem 1.3.
By Theorem 1.2, the coefficients c g ( − m, n ) for g in conjugacy class2A dominate all others with g not in 1 or 2A. The Kloosterman sum K ( g, , − m, n ) = 1 forall m and n so we can write c g ( − m, n ) ∼ m / (2 n ) / · e π √ mn . The number of regular representations we can pull out of V ♮n is limited by the character withthe smallest ratio of χ i (2A) to dim( χ i ). This minimum is achieved with the characters wehave labeled χ and χ , which have χ (2A)dim( χ ) = χ (2A)dim( χ ) = − . Therefore, when we add up the contributions to the non-free part from (4.1) for g in conjugacyclasses 1 and 2A, we find that as n → ∞ , nf i ( − m, n ) ∼ | || M | (cid:18) c g ( − m, n )3301375 dim( χ i ) + c g ( − m, n ) χ i (2A) (cid:19) ∼ (cid:18) dim( χ i )3301375 + χ i (2A) (cid:19) | | m / | M | (2 n ) / · e π √ mn , as desired. (cid:3) Appendix
Moonshine groups.
Here we list the symbols Γ g = N || h + e, f, . . . for each conjugacy classof the monster. As in [6], if h = 1, we omit the “ || W g = { } then we just write N || h , and if W g contains every exact divisor of N/h then we write N || h +. The labeling ofconjugacy classes is as in [5].1A 12A 2+2B 23A 3+3B 33C 3 ||
34A 4+4B 4 || ||
25A 5+5B 5 6A 6+6B 6 + 66C 6 + 36D 6 + 26E 66F 6 ||
37A 7+7B 78A 8+8B 8 || || || ||
49A 9+9B 910A 10+10B 10 + 510C 10 + 210D 10 + 1010E 1011A 11+12A 12+12B 12 + 4
CKAY-THOMPSON COEFFICIENTS AND DISTRIBUTIONS OF THE MOONSHINE MODULE 13
12C 12 || || || || || || || || || || || || || || || || || || || || , , , , , , || , , || || || || || || , , || , ,
42 44AB 44+45A 45+46AB 46 + 2346CD 46+47AB 47+48A 48 || || || || || || , , , , || , , || , , || , , , , || || , , || || || || Ordering by asymptotic magnitude of McKay-Thompson coefficients.
The asymp-totic formulas in Theorem 1.2 give rise to the following natural order of the conjugacyclasses corresponding to the asymptotic magnitude of their McKay-Thompson coefficients c g ( − m, n ). A class is understood to be removed if c g ( − m, n ) vanishes, which if it does,occurs periodically with n . We list classes X and Y in brackets { X , Y , . . . } whenlim n →∞ | c g ( − m, n ) || c g ′ ( − m, n ) | = 1for g in X and g ′ in Y. We group classes as (X , Y , . . . ) if the above limit takes on anotherconstant value, which may depend on congruence properties of m and n . The order withinround brackets may change depending on these congruence conditions. Finally, for classeswith exceptions listed in Theorem 1.2, we write [X] to indicate the placement of a classoutside of the exception and [X] ( ∗ ) to refer to the placement when m and n satisfy thehypotheses of exception ( ∗ ) in Theorem 1.2.1A , , , { , } , , { , } , , (4B , , (3C , , , { , } , , { , , } , , { , } , { , } , ( { , , } , , , (6D , { , } ) , , { , , } , { , } , , , ( { , } , { , } ) , (5B , , { , } , { , } , { , } , , { , , } , , (8C , , , , , { , } , ( { , } , , , { , } ) , , { , } , { , } , , { , } , { , } , (15B , , , , { , , } , , (10C , , , , , ( { , } , , , { , , , , } , , , ( { [8D] , [8E] , } , , { , } , , , , (12G , , , , (20D , { , } ) , , , , , { , } , , , (24F , , , (10E , , { , } , , , , , , { , } , , , ( { , } , , , , , [18A] , , , , , , [24D] , , , { , } , ([16B] , { [8D] (1) , [8E] (1) } ) , , , { [18A] (3) , , } , , , , (24J , [24D] (4) ) , [16B] (2) References [1] L. Beneish and H. Larson,
Traces of singular values of Hauptmoduln , Int. J. Numb. Th. , 1027 (2015).[2] R. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster , Proc. Nat. Acad. Sci. U.S.A. (1986), no. 10, 3068–3071.[3] K. Bringmann and K. Ono, Coefficients of harmonic mass forms , Partitions, q-Series and ModularForms, Developments in Mathematics, vol. 23, Springer New York, 2012, 23–38.[4] J. H. Brunier,
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Rademacher sums, moonshine and gravity , Commun. Number TheoryPhys. (2011), no. 4, 1–128.[8] J. F. R. Duncan, M. Griffin, and K. Ono, Moonshine , Research in the Mathematical Sciences, (2015),A11.[9] I. Frenkel, J. Lepowsky, and A. Murman, A natural representation of the Fischer-Griess Monster withthe modular function J as character , Proc. Nat. Acad. Sci. U.S.A. (1984), no. 10, Phys. Sci., 3256–3260.[10] I. Frenkel, J. Lepowsky, and A. Murman, A moonshine module for the moster , Math. Sci. Res. Inst. Publ.,vol. 3, Springer, New York, 1985, 231–273.[11] K. Harada and M. L. Lang,
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Some numerology between the Fischer-Griess Monster and the elliptic modular func-tion , Bull. London Math. Soc. (1979), no. 3, 352–353. Department of Mathematics, Harvard University, Cambridge, MA 02138
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