$p$-adic Properties of Hauptmoduln with Applications to Moonshine

The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the $j$-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the $j$-function satisfy congruences modulo $p^n$ for $p \in \{2, 3, 5, 7, 11\}$, which led to the theory of $p$-adic modular forms. We combine these two aspects of the $j$-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the Monster. We also find a distinguished class of subgroups of the Monster with graded characters satisfying such congruences.


Introduction and Statements of Results
The theory of monstrous moonshine arose from the remarkable observation of McKay and Thompson [31]  Here, the left-hand sides of the equations are the coefficients of the normalized modular j-function J(τ ) = j(τ ) − 744 = q −1 + 196884q + 21493760q 2 + · · · where q = e 2πiτ , and the right-hand sides are simple sums involving the dimensions of the irreducible representations of the monster group M: 1,196883,21296876,842609326,19360062527, . . . .
Thompson conjectured [30] that these identities could be explained by the existence of an infinite-dimensional graded monster module such that the graded dimension is given by J. More generally, the graded-trace functions Tr(g|V ♮ n )q n for the action of M on V ♮ , known as the McKay-Thompson series, were conjectured to be the normalized Hauptmoduln of certain genus zero groups Γ g ≤ GL + 2 (R) given in Conway-Norton [5]. That is, each T g was conjectured to be the unique generator T Γg of the function field of the genus zero curve Γ g \H * having q-expansion of the form q −1 + O(q) at ∞. Since all of the Hautpmoduln appearing in this paper will be normalized (meaning that they are bounded away from ∞ and have q-expansion q −1 + O(q) at ∞), we will henceforth omit the word "normalized" and refer to such functions simply as Hautpmoduln.
Frenkel-Lepowsky-Meurman [13,14] constructed V ♮ with the correct graded dimensions, and Borcherds [3] proved that the McKay-Thompson series were Hauptmoduln for the Γ g given by Conway-Norton. After being shown for the Monster, different incarnations of moonshine were later shown for other finite groups, such as the largest Mathieu group M 24 [15], and later the other 22 groups appearing in umbral moonshine [10].
Given the deep connections between J and the Monster, one might wonder whether these p-adic properties of J are special cases of a more general p-adic phenomenon taking place among the Hauptmoduln appearing in monstrous moonshine. To make this more precise, given a prime p and a modular function f , we say that f is p-adically annihilated if the q-series f |U n p uniformly converges to 0 in the p-adic limit as n → ∞. Given that J is padically annihilated for p ∈ {2, 3, 5, 7, 11}, we can then ask if other Hauptmoduln appearing in monstrous moonshine are as well. Our first main result is that p-adic annihilation is actually quite common among the Hauptmoduln of monstrous moonshine. In fact, out of the 171 Hauptmoduln in monstrous moonshine, we will show that 97 have p-adic annihilation for some prime p.
We further conjecture that Table 4.1 gives all the Hauptmoduln appearing in monstrous moonshine that are p-adically annihilated for any prime p (see Conjecture 4.2).
Once Theorem 1.1 has established a class of Hauptmoduln coming from monstrous moonshine with p-adic annihilation, we may next ask whether the structure of the Monster group informs p-adic properties of the Hauptmoduln. Specifically, we are interested in relating the power maps g → g m of the Monster to p-adic annihilation of Hauptmoduln. Theorem 1.2. Let T g be the Hauptmodul of a group appearing in Table 4.1, so that T g is p-adically annihilated by Theorem 1.1. Outside of the exceptions discussed in Section 4.4, we also have that T g m is p-adically annihilated.
Although Theorem 1.2 follows from Theorem 1.1, we will prove the two theorems in tandem, relying on the structure provided by Theorem 1.2 to make Theorem 1.1 easier to prove. As an illustration of Theorem 1.2, see Figure 1.1, which shows conjugacy classes with Hauptmoduln that are 2-adically annihilated and the power maps between them. For a full explanation of the notation used in this figure, and the corresponding figures for p = 3, 5, 7, 11, see Appendix B.
Finally, we consider which finite groups have infinite-dimensional representations with similar p-adic properties. We define a moonshine module for a finite group G to be a graded G-module V = ∞ n=−1 V n such that for each g ∈ G the graded trace T g = Tr(g|V n )q n associated to the action of g on V is the Hauptmodul of an order ord(g) conjugacy class of the Monster. We also require that the power maps of G interact with the Hauptmoduln in a way that mimics what occurs in monstrous moonshine; see Section 5 for the precise condition. For an irreducible character χ of G, we write m χ (n) for the multiplicity of χ appearing in the character of G acting on V n , and define the multiplicity generating function  We say that a moonshine module V for G is a p-adic moonshine module if M χ is padically annihilated for each irreducible character χ. We may then ask various questions about finite groups with p-adic moonshine modules, such as the number of such groups and which primes may divide their orders. In Section 5 we address these questions and give examples of groups with p-adic moonshine modules. In particular, we find that the groups in Table 1.2 have p-adic moonshine for the listed p in a slightly more general sense explained in Section 5.3. These groups arise as the centralizers of certain commuting pairs of elements of the Monster in the conjugacy class pA. p 2 3 5 7 11 C(pA 2 ) 2 2 · 2 E 6 (2) 3 2 × O + 8 (3) 5 2 × U 3 (5) 7 2 × L 2 (7) 11 2 #C(pA 2 ) 2 38 ·3 9 ·5 2 ·7 2 ·11·13·17·19 2 12 ·3 14 ·5 2 ·7·13 2 4 ·3 2 ·5 5 ·7 2 3 ·3·7 3 11 2 Table 1.2: Subgroups of the Monster with weakly p-adic moonshine Before proceeding, we outline the structure of this paper. We begin in Section 2 by proving technical lemmas that will be useful later in the paper. In Section 3, we extend Serre's theory of p-adic modular forms such that it becomes applicable to the groups appearing in monstrous moonshine, and we begin to see p-adic properties of Hauptmoduln. In Section 4, we prove Theorems 1.1 and Theorem 1.2 using both the theory of Section 3 and the interplay between power maps and p-adic properties. We conclude in Section 5 by considering finite groups with p-adic moonshine modules, and showing that only finitely many such groups exist. We also discuss examples of groups with p-adic moonshine, including those in Table 1

Preliminaries
In this section, we collect technical details and definitions that will be used later. We first describe the types of groups Γ whose Hauptmoduln will be studied. We then discuss various properties of operators on spaces of modular forms, most importantly the U p operator and the Atkin-Lehner involutions. We also give descriptions of which cusps a Hauptmodul may have poles at once U p is applied to it, and we give a modular form g with zeros at all such cusps. Finally, we discuss the trace of a modular form, which transforms modular forms on some Γ into modular forms on some Γ ′ ≥ Γ. These facts will ultimately be used to interpret Hauptmoduln as p-adic modular forms in Section 3.

n|h-Type Groups
Monstrous moonshine associates to each g ∈ M a Hauptmodul T g for some genus zero group Γ g ≤ GL 2 (R) + . This means that Γ g \H * is a genus zero curve and that T g is a generator for the function field such that T g is bounded away from the cusp ∞; moreover the q-expansion of T g at infinity begins T g = q −1 + O(q). Conway and Norton described the groups Γ g in [5], all of which take on a particular form which we reproduce here.
First we describe the normalizer of Γ 0 (N ) in PSL 2 (R). Let h be the largest integer such that h 2 |N and h|24, and set n = N/h. The normalizer of Γ 0 (N ) is given by e n/h w e where w e is the set of all matrices A = ae b/h cn de such that a, b, c, d ∈ Z and det A = e.
Here the notation x y means that x exactly divides y, i.e. that x|y and gcd(x, y/x) = 1.
Given integers e 1 , e 2 we set e 1 * e 2 = e 1 e 2 gcd(e 1 ,e 2 ) 2 , and under * the set of exact divisors of any integer N forms the abelian group (Z/2Z) n where n is the number of primes dividing N .
More generally, a class of subgroups called n|h-type groups is defined as follows. Let n be any positive integer and let h| gcd(n, 24). Set N = nh and w e as above, for e n/h. We define the group Γ 0 (n|h) = w 1 . We will often abuse notation and write w e for any element of w e , and we see that w e 1 w e 2 = w e 1 * e 2 . Since h|24 we have that m 2 ≡ 1 (mod h) for all m coprime to h, see [12,Lemma 1.1]. For a subgroup {1, e 1 , . . . , e n } of the group of exact divisors of n/h, we then define Γ 0 (n|h)+e 1 , e 2 , . . . , e n = Γ 0 (n|h), w e 1 , w e 2 , . . . , w en = w 1 ∪ w e 1 ∪ w e 2 ∪ · · · ∪ w en .
A group of this form is called an n|h-type group.
Setting N = nh, Γ 0 (n|h)+e 1 , . . . , e n normalizes both Γ 0 (n|h) and Γ 0 (N ), and the w e i are cosets of Γ 0 (n|h). When h = 1 we have Γ 0 (n|1) = Γ 0 (N ) and we denote the matrix by W e . The matrices W e for e N are called Atkin-Lehner involutions. Given an Atkin-Lehner involution W E on Γ 0 (N ), we can interpret this as an element of Γ 0 (n|h) via where h E is the largest divisor of h with h 2 E |E and e = E/h 2 E . In fact, setting AL(Γ) = {e nh : W e ∈ Γ and every prime dividing e also divides n/h} we have that this association gives a bijection For example, letting Γ = Γ 0 (8|2)+4 we have When dealing with n|h-type groups, it is standard to simplify notation in the following ways. When h = 1, we simply omit the |h, so that Γ 0 (n|1) = Γ 0 (n), and when all e n/h are included in a group, we simply write Γ 0 (n|h)+ so that Γ 0 (8|2)+ = Γ 0 (8|2)+4. We will also sometimes use the symbol n|h+e, f, . . . to represent the group Γ 0 (n|h)+e, f, . . . in order to save space, particularly in tables, so we might write 8|2+ instead of Γ 0 (8|2)+.
The n|h-type groups appear in monstrous moonshine as eigengroups of the Hauptmoduln. That is, the Hauptmodul T has an associated group Γ 0 (n|h)+e, . . . such that for all A in this group, T (Aτ ) = µT (τ ) for some h th root of unity µ. Conway-Norton [5] gave a rule for computing the eigenvalue λ corresponding to an element of Γ 0 (n|h)+e, . . . : (i) λ = 1 for any element of Γ 0 (N ), (ii) λ = 1 for all W e with e ∈ AL(Γ), Since the cosests x and y generate Γ 0 (n|h), we have Γ = x, y, W e : e ∈ AL(Γ) for any n|h-type group Γ. Hence Conway-Norton's rule uniquely determines a map λ : Γ → µ h where µ h denotes the group of h th roots of unity. We always use λ to denote this map. More generally, let Γ be an n|h-type group. An eigenvalue map is a homomorphism η : Γ → µ 2h such that Γ 0 (nh) ⊂ ker η. Then we we define When λ is the map given by Conway-Norton's rule, we have that Γ λ is an index h subgroup of Γ called the fixing group of Γ. However, Conway-Norton's rule does not always give a well-defined map, so Γ λ does not exist for every n|h-type group Γ. Ferenbaugh [12,Theorem 2.8] classified for which Γ Conway-Norton's rule is consistent, and we call such Γ admissible. There are 213 admissible n|h-type groups giving genus zero groups, including all 171 groups appearing in monstrous moonshine. Ferenbaugh also determined the structure of the quotient Γ/Γ 0 (nh), and therefore also the structure of Γ λ /Γ 0 (nh). In 174 cases, including the groups of monstrous moonshine, the latter quotient group has exponent 2; the remaining 3 groups are known as the "ghosts." For further discussion of which n|h-type groups appear in monstrous moonshine, see [7].
In the next section we study modular forms on n|h-type groups with given eigenvalue maps, and the action of the U p operator on such spaces of modular forms.

Action of U p on Hauptmoduln
Given an n|h-type group Γ with eigenvalue map η : Γ → µ 2h , we say a weight k weakly holomorphic modular form on Γ 0 (nh) is on Γ with eigenvalue map η or on (Γ, η) if where the weight k slash operator is defined by (2.1) below. By a weakly holomorphic modular form we mean a meromorphic modular form whose poles are supported on the cusps; on the other hand a modular form is assumed to be holomorphic everywhere. We write M k (Γ, η) for the space of weight k modular forms on Γ with eigenvalue map η. Similarly, we denote the space of weight k modular forms on Γ 0 (nh) invariant under all γ ∈ Γ η by M k (Γ η ).
Fix a prime p. In this section, we study U p applied to Hautpmoduln T , and more generally weakly holomorphic modular forms on Γ η or on Γ with eigenvalue map η. For our results to extend to n|h-type groups, the results of this section will be stated in the necessary general language. However the reader looking to use these results for modular forms on Γ 0 (N )+e, . . . should remember that this corresponds to taking h = 1 and ignoring eigenvalue maps in the following results.
Recall that the weight k slash operator | k γ for γ ∈ GL + 2 (R) is defined by If f is on SL 2 (Z) then for N ∈ N, we have f (N τ ) on Γ 0 (N ), and for d|N and e N , In terms of the slash operator, U p is defined on weight k modular forms by where S µ = 1 µ 0 p . This operator is independent of k and acts on Fourier expansions by a(n)q n |U p = a(pn)q n .
We first recall some basic facts about the U p operator (see [2, Section 2]).
The following lemma extends these facts from Γ 0 (N ) to n|h-type groups.
Lemma 2.2. Let p ∤ nh be prime, and let f be a weight k meromorphic modular form on a p α n|h-type group Γ with eigenvalue map η.
(a) Let x = 1 1/h 0 1 and y = 1 0 p α n 1 be the matrices of Section 2.1. Then (b) Let Γ ′ be the p β n|h-type group such that β = max{1, α − 1} and e ∈ AL(Γ ′ ) if and only if e ∈ AL(Γ) and p ∤ e. Let η ′ be an be an eigenvalue map on Γ ′ such that Then f |U p is on Γ ′ with eigenvalue map η ′ .
Proof. For 0 ≤ µ ≤ p − 1, let S µ = 1 µ 0 p denote the matrix appearing in the definition (2.3) of U p . The first identity of part (a) follows from the equation S µ x p = xS µ . The second identity follows from the equation where each * is an integer such that the matrix has determinant 1. Since p ∤ h, we have that p 2 ≡ 1 (mod h) (since h is a divisor of 24), so the matrix appearing in (2.4) is in Γ 0 (p α nh) and therefore fixes f . For part (b), note that since p ∤ h and β ≤ α + 1, the matrices x p and y p generate Γ 0 (p β n|h). Thus part (b) follows from part (a) and Lemma 2.1.
Let σ a : µ → µ a be an endomorphism of µ 2h . We set η σa = σ a •η. The preceding lemma says that if Γ is an pn|h-type group with p ∤ nh then U p is a map M k (Γ, η) → M k (Γ, η σp ). We record here a basic fact about decomposing the space M k (Γ η ).
Let Γ be a p α n|h-type group with eigenvalue map η such that Γ η is genus zero and p ∤ nh. Lemma 2.2 tells us under which group T |U p is modular. We know that T |U p is weakly holomorphic, and next characterize the cusps at which it may be unbounded. We first recall that a set of representatives for the cusps of Γ 0 (N ) is given by Moreover, if two cusps a b and c d (not necessarily representatives of the form above) are equivalent under Γ 0 (N ) and gcd(a, b) = gcd(c, d) = 1, then gcd(b, N ) = gcd(d, N ). Both of these facts follow from [9,Proposition 3.8.3].
In what follows, if Γ ≤ GL + 2 (R) is commensurate with SL 2 (Z) and s, s ′ ∈ P 1 (Q) are cusps, then we write s Γ ∼ s ′ to mean that s and s ′ are equivalent under Γ. If Γ = Γ 0 (N ) then we simply write s N ∼ s ′ .
Lemma 2.4. Let p ∤ nh be prime, and let Γ be a p α n|h-type group with eigenvalue map η. Let Γ ′ and η ′ be the p β n|h-type group and eigenvalue map defined in Lemma 2.2(b). Suppose f is a weakly holomorphic modular form on Γ with eigenvalue map η, so that f |U p is on Γ ′ with eigenvalue map η ′ by Lemma 2.2.
(a) If f is bounded away from ∞ then the poles of f |U p are supported on the cusps where A ′ is obtained from A by replacing each x with x p and each y with y p . Hence s ∼ ∞, as desired. The case α ≥ 2 is dealt with similarly, completing the proof of (a).
A similar argument gives part (b), where we must now use the fact that p ∤ e for e ∈ AL(Γ) to show that p ∤ e for all W e appearing in A. For part (c), one finds that f |U p may only have a pole at the cusp ∞; however, since it does not have a pole here by hypothesis, f |U p is bounded everywhere and hence constant.
Remark 2.5. Lemma 2.4(c) delivers a class of Hauptmoduln T for which T |U p = 0. For the Hauptmoduln from monstrous moonshine to which this applies, this property also follows from [22,Lemma 3.2]. Furthermore, if Γ is a n|h-type group and η is an eigenvalue map with η(x) = e 2πim/h , then any meromorphic modular form f = ∞ n=m a(n)q n on Γ with eigenvalue map η has a(n) = 0 if n ≡ m (mod h), since x sends q → e 2πim/h q. In particular, if h ≡ 0 (mod p) and T is the Hautpmodul on Γ λ , then f |U p = 0. Inspection of Table A.1 shows that the only monster Hauptmoduln with T |U p = 0 are those with h ≡ 0 (mod p) and those coming from Lemma 2.4(c).
In Section 3 we will need a modular form g on an n|h-type group Γ such that the zeros of g can cancel the poles of T |U p , whose locations were just determined. We will also need g to have certain properties modulo p.
To construct g, we will use the modular discriminant ∆(τ ) = η(τ ) 24 where η(τ ) is the Dedekind eta function. If a modular form f = ∞ n=m a(n)q n has rational coefficients, we set v p (f ) = inf n v p (a n ) where v p (a n ) = sup{r ∈ Z : p r | a n }.
Lemma 2.6. Let Γ be a pn|h-type group where p ∤ nh is prime and p ∤ e for all e ∈ AL(Γ). Let m = # AL(Γ) and set a = 12m(p − 1). Then there is modular form g on Γ of weight a with rational coefficients such that Specifically, g can be chosen to be Proof. First, let and a p = 12(p − 1). Note that g p (τ ) is invariant under Γ 0 (pn|h). For any e ∈ AL(Γ) On the other hand, we see that Since ∆ is nonzero on H, (2.5) shows that g p | ap W e is a modular form on Γ 0 (pnh) with rational coefficients. Moreover, (2.6) shows that each g p | ap W pe for e ∈ AL(Γ) vanishes to order (h * e)(p 2 − 1) at ∞, so that each g p | ap W e vanishes to order ≥ p 2 − 1 at the cusps s Moreover v p (g p | ap W p ) = 6(p + 1). Thus, we may set which clearly satisfies the conditions given.
Remark 2.7. The function in Lemma 2.6 is chosen for its large order zeroes, in contrast with symmetrizations of the function g = E a − p a/2 E a | a W p of [26,Lemma 8]. This will be computationally useful in Section 4.
We conclude with a few tools for working with q-expansions of mod p modular forms.
We will apply Sturm's bound to weakly holomorphic modular forms after multiplying by a power of the function from Lemma 2.6. We thus bound the pole orders of T |U p .
Lemma 2.9. Let f be a weakly holomorphic function on X 0 (p α N ), where p ∤ N , and let β = max{α, 1}. If r is the maximum order of a pole of f on X 0 (p α N ), then the poles of f |U p as a function on X 0 (p β nh) have order at most rp 2 when α = 0, and order at most rp otherwise.
Proof. The ramification index of each cusp of X 0 (pnh) over X 0 (nh) is a divisor of p and at most p. Thus, for the case α = 0, the maximum order of a pole of f pulled back to X 0 (pnh) is rp. The U p operator may be defined via the correspondence The projections have degree p, so T pulls back to a function on γΓ 0 (p β nh)γ −1 ∩ Γ 0 (p β nh) \H * with poles at most degree rp 2 when α = 0 and rp else. The other maps of the correspondence, which are pullback by the isomorphism and trace down to X 0 (p β nh), do not increase the maximum pole order, so the claim follows.

Trace Formulas
Following Serre's idea [26], we will apply the trace to view classical modular forms as p-adic modular forms of lower level. In this section we discuss a few properties of trace maps. Suppose Γ and Γ ′ are subgroups of GL + 2 (R) with Γ a finite index subgroup of Γ ′ . We define the trace Tr Γ\Γ ′ from Γ to Γ ′ to be the operation where {γ 1 , . . . , γ m } is a system of right coset representatives for Γ\Γ ′ . If f is modular on Γ then Tr Γ\Γ ′ is modular on the larger group Γ ′ . When Γ = Γ 0 (N ) and Γ ′ = Γ 0 (N ′ ), we simply write Tr N \N ′ for Tr Γ\Γ ′ . First consider Γ = Γ 0 (pN ) and Γ ′ = Γ 0 (N ). The following generalizes [26,Lemma 7].
(a) A set of representatives for Γ 0 (pN )\Γ 0 (N ) is given by then where W p is the corresponding Atkin-Lehner involution on Γ 0 (pN ).
Proof. Since [Γ 0 (N ) : Γ 0 (pN )] = p + 1, part (a) follows upon checking that the representatives are inequivalent modulo Γ 0 (pN ). One can also check that for any The remainder of this section will extend Lemma 2.10 to the more general context we need, for example tracing from Γ 0 (pN )+e, . . . to Γ 0 (N )+e, . . . for p ∤ N . More precisely, for a prime p ∤ nh, suppose that Γ is a pn|h-type group with eigenvalue map η with p ∤ e for all e ∈ AL(Γ). Let Γ ′ be the n|h-type group such that AL(Γ ′ ) = AL(Γ). We have Γ ⊂ Γ ′ , and can take η ′ to be the eigenvalue map on Γ ′ with η ′ |Γ = η. Since Γ ′ is generated by is an isomorphism. We set H = im(ι|Γ η ) ≤ Γ ′ , and consider the restricted isomorphism We have nearly proved the following lemma. Lemma 2.11. Let Γ be a pn|h-type group with eigenvalue map η such that p ∤ e for all e ∈ AL(Γ). Let Γ ′ be an n|h-type group with AL(Γ ′ ) = AL(Γ) with eigenvalue map η ′ such that η ′ |Γ = η. Then for any weight k modular form on Γ η we have The formula then follows from Lemma 2.10.
Remark 2.12. Lemma 2.11 only assumes that f is on Γ η . Under the stronger assumption that f is on (Γ, η), we obtain the finer result that Tr pnh\nh f is on (Γ ′ , η ′ ). To see this, let x and Y = y p be the generators of Γ 0 (pn|h)/Γ 0 (pnh), and choose appropriate representatives of W e which normalize Γ 0 (nh). Then, apply x, Y , and W e to both sides of (2.7).

p-adic Modular Forms
In this section, we extend of Serre's theory of p-adic modular forms from [26] to Hauptmoduln and n|h-type groups. In particular, we study the interaction between eigenvalue maps and the mod p weight filtration. These p-adic properties could be studied by extending the theory of Katz and others, but we choose to generalize Serre's original treatment in order to perform explicit calculations for our applications. Take p ≥ 5, and let Γ be an n|h-type group with eigenvalue map η. We first study M k (Γ, η) and its p-adic completion. In Section 3.1 we prove that Hauptmoduln T become p-adic modular forms on some (Γ ′ , η ′ ) under applications of U p . In Section 3.2 we extract structural results concerning ordinary spaces and the action of U p on these p-adic modular forms. Again, readers interested only in modular groups of the form Γ 0 (N )+e, . . . can take h = 1 and let eigenvalue maps be identically 1.
For an n|h-type group Γ with eigenvalue map η : Γ → µ 2h , we first define the spaces: , the Q-vector space of modular forms with rational q-expansion; , the Z (p) -module of modular forms with p-integral q-expansion; and If f reduces mod p to a form inM k (Γ, η), we abuse notation and write f ∈M k (Γ, η), and similarly for Γ η . We focus on M k (Γ, η), and the corresponding results for M k (Γ η ) follow from Lemma 2.3.
Following [26], we define a p-adic modular form on (Γ, η) to be a q- Similarly, a p-adic modular form on Γ is a p-adic modular form on (Γ, 1). [20,Corollary 4.4.2]). It follows that the weight of a p-adic modular form on Γ 0 (N ), defined as the limit of the k m in the space exists and does not depend on the choice of sequence (f m ). The same is true of p-adic modular forms on (Γ, η), since M k (Γ, η) ⊆ M k (Γ 0 (nh)). In particular, forms in M k (Γ, η) have trivial nebentypus, as required in [20]. The correct extension of the mod p weight filtration to modular forms for n|h-type groups will feature a quadratic twist. To this end, we first twist our eigenvalue maps. Definition 3.1. Let Γ be an n|h type group and p ∤ nh a prime. If η is an eigenvalue map on Γ, then the twist of η is the map η t : Γ → µ 2h defined by where · p denotes the Legendre symbol. Remark 3.2. If 1 denotes the trivial eigenvalue map 1(γ) = 1 for all γ, then η t = η1 t for all η. Also, if e p = 1 for all e ∈ AL(Γ) then η = η t . Below, we collect some useful facts and begin to see the relationship between eigenvalue map twists and the weight mod 2(p − 1). Let E k (τ ) denote the weight k Eisenstein series with constant term 1. Proposition 3.3. Let Γ be an n|h-type group with eigenvalue map η, and let p ≥ 5 be prime with p ∤ nh.

Producing p-adic Modular Forms from Hautpmoduln
Let Γ be a p α n|h-type group with eigenvalue map η where p ∤ nh is prime. Suppose Γ η is genus zero, and its Hautpmodul T is on Γ with eigenvalue map η. We show that for some β, T |U β p is a p-adic modular form on (Γ ′ , η ′ ) for some specified n|h-type group Γ ′ and eigenvalue map η ′ . We can take β = 1 whenever α ≤ 3, Lemma 3.4. Let Γ be a p α n|h-type group for p ∤ nh prime, and η be an eigenvalue map on Γ. Let f be a weight 0 weakly holomorphic modular form on (Γ, η) that is holomorphic away from ∞. Let Γ ′ be the n|h-type group with e ∈ AL(Γ ′ ) if and only if e ∈ AL(Γ) and p ∤ e. Let β = max{1, α − 1}, and let η ′ be the eigenvalue map on Γ ′ such that where x, y, y ′ are the generators of Γ 0 (p α n|h) and Γ 0 (n|h) given by Proof. By Lemma 2.2, f |U β p is a weakly holomorphic modular form on (Γ, ν) where Γ is the pn|h-type group Γ where e ∈ AL(G) if and only if e ∈ AL(Γ) and p ∤ e, and ν satisfies The remainder of the proof follows [26,Theorem 10]. To show that f |U β p is a p-adic modular form on (Γ ′ , η ′ ), we set for m ≥ 0 where g is the modular form on Γ given by Lemma 2.6. Since we know f m is a weakly holomorphic modular form on (Γ ′ , η ′ ) by Lemma 2.11. Lemma 2.4 shows that f |U β p has poles only at the cusps equivalent to 0 on Γ 0 (p), and since g has zeros at all such cusps, we know f |U β p g m is holomorphic for m sufficiently large. If a is the weight of g, the weight of f m is ap m , which p-adically converges to 0. Hence it suffices to show that f m → f |U β p in the p-adic limit. We compute that and since applying U p does not decrease the power of p dividing a q-expansion, we have Remark 3.5. If Γ is a p α n|h-type group with eigenvalue map η such that Γ η is genus zero, and the Hauptmodul T on Γ η is on (Γ, η), then Lemma 3.4 applies. Moreover, since T r is on (Γ, η σr ), the lemma also applies to powers of the Hauptmodul. In particular, polynomials in T are p-adic modular forms on Γ ′ η ′ after enough applications of U p .

Ordinary Spaces
Having produced p-adic modular forms from Hauptmoduln on certain n|h-type groups, we now study the action of U p . The key idea, developed by Serre on level 1 in [26], is that U p contracts mod p modular forms onto a finite-dimensional space. These structural results will allow us to verify p-adic annihilation of certain Hauptmoduln in Section 4.2.
We will take p ≥ 5 prime with p ∤ nh. For the Hecke operator T p , we have Since T p acts on M k (Γ 0 (nh)), we know U p acts oñ M k (Γ 0 (nh)). Furthermore, let Γ be an n|h-type group and η be an eigenvalue map. Since Ax = x p A, Ay p = yA, AW e = W e A for p ∤ e, Lemma 2.2 implies that T p : M k (Γ, η) → M k (Γ, η σp ). Hence U p :M k (Γ, η) →M k (Γ, η σp ), so we consider the spaceM which is stabilized by U p . This sum is direct if and only if η = η σp . We thus set and remind the reader thatM (Γ, η) α already encodes spaces with twisted eigenvalue map. We next show how U p contractsM (Γ, [η] p ) α onto a finite-dimensional space called the ordinary space. Ordinary spaces of p-adic modular forms were extensively studied by Hida [18]. We describe our ordinary spaces in the language of Serre's p-adic modular forms.
Proposition 3.6 (Ordinary decomposition). Let Γ be an n|h-type group with eigenvalue map η.
(a) We can writeM so that U p is bijective on the ordinary space S(Γ, [η] p ) α and locally nilpotent on N(Γ, [η] p ) α ; that is, for any N(Γ, [η] p ) α , we have f |U n p = 0 for n sufficiently large. Otherwise, Proposition 3.6 can be interpreted to mean that repeated application of U p reduces the weight of a modular form mod p to either 0 or p − 1. To accomplish this, we need to incorporate the twisted eigenvalue maps. More precisely, the filtration of f ∈M (Γ, [η] p ) α with respect to (Γ, η) is When (Γ, η) is clear from context, we will simply write w for w Γ,η . Similarly, the filtration w Γ of a modular form mod p with respect to Γ is the filtration with respect to (Γ, 1).
To prove Proposition 3.6 we generalize the following fact from [19].
Lemma 3.7. Suppose Γ = Γ 0 (N ). Then for modular forms f mod p on Γ we have We give a suitable modification of Lemma 3.7 for (Γ, η).
Lemma 3.8. Let Γ be an n|h-type group with eigenvalue map η.
Let t be the order of T in Γ ′′ /Γ 0 (nh), and let π m be the projection onto the e 2πim/t eigenspace of T . Since p ≥ 5, we know p ∤ t. Thus, if ϕ, ϕ ′ have the same weight and ϕ ≡ ϕ ′ (mod p), then π m (ϕ) ≡ π m (ϕ ′ ) (mod p). By Proposition 3.3, multiplying by some power ofF gives Let m be such that η(T ) = e 2πim/t and set π = π m if π pm = π m π m ⊕ π pm else which is projection onto the span of the eigenspaces of T specified by [η] p . Thus, we have π(g) ≡ π(f ) = f (mod p). Then π(g) ∈ M (p) Extending by Atkin-Lehner involutions W e is a similar computation. Define the projections as before, replacing T with the order t = 2 action W e . The functionF is not necessarily fixed by W e , but rather lies in the e p eigenspace of W e . Thus where ε = 1 2 1 − e p , and we have π(g) ∈ M (p) Proof of Proposition 3.6. Let k be as in the statement of Proposition 3.6. Set Lemma 3.8 shows that these spaces satisfy the conditions of the proposition. Proposition 3.6 will be fundamental for proving p-adic annihilation in Section 4.

p-adic Annihilation
In this section, we restrict our focus to the n|h-type groups appearing in monstrous moonshine, and we prove the following theorem, first mentioned in the introduction, which gives a class of Hauptmoduln from monstrous moonshine that are p-adically annihilated for small primes p.

Remark 4.3.
Given an admissible n|h-type group Γ, we remark here on a method for showing T Γ is not p-adically annihilated for a given p. By Lemma 3.4, after applying U p enough times we may assume that p ∤ nh. Then after further applying U p , we have that T |U α p ∈M p−1 (nh) for some α. SinceM p−1 (nh) is finite-dimensional, it is then straightforward to verify that T is not p-adically annihilated. For small levels and primes, this method is easy to apply; for example one finds that J is not 13-adically annihilated since J|U 2 13 ≡ J|U 13 (mod 13). However, this method quickly starts requiring many coefficients of the Hauptmoduln and basis elements of M p−1 (nh), particularly when p is large.
Empirically, it appears that for p = 2, 3 one may use a similar approach with the spacẽ M 4 (nh).

Remark 4.4.
It is known that J|U p ≡ 0 (mod p) for any p ≥ 13 [25], and see [11] for further study of such congruences.
In Section 4.1, generalizing formulas of Conway and Norton [5], we write down compression formulas which relate Hauptmoduln on different groups via-à-vis power maps. These relations will reduce the verification of Theorem 4.1 to a smaller set of groups, which will be easier to verify computationally. In Section 4.2, we utilize the theory developed in Section 3 to prove annihilation for the entries in the table of Theorem 4.1 with p ≥ 5. In Section 4.3, we use separate techniques due to Lehner and verify the remaining entries, corresponding to p = 2, 3. These techniques are sufficiently explicit to give rates of p-adic annihilation in certain cases. Finally, we discuss in Section 4.4 a connection between p-adic annihilation of Hauptmoduln and the group structure of the monster group.

Compression Formulas
Throughout this section, we use the following notation. If Γ is the the group Γ 0 (n|h) + e 1 , e 2 , . . . , then we write Γ d for the group Γ 0 (n ′ |h ′ ) + e ′ 1 , e ′ 2 , . . . where n ′ = n/ gcd(n, d), h ′ = h/ gcd(h, d), and e ′ 1 , e ′ 2 , . . . are the divisors of n ′ /h ′ among e 1 , e 2 , . . . . This notation comes from [5], where it is explained that for any element g of the Monster, if T g is the Hauptmodul for the group Γ λ corresponding to g from moonshine, then T g d is the Hauptmodul for Γ d λ . Additionally, if Γ is the group Γ 0 (n|h) + e 1 , e 2 , . . . , then we write Γ, w e for the group Γ 0 (n|h) + e 1 , e 2 , . . . , e 1 * e, e 2 * e, . . . generated by Γ and w e , if it exists. In this section, we will also adopt the notation T Γ for the Hautpmodul on Γ λ .
The formula relating the Hauptmodul for g p with that of g for g ∈ M appears in [5, §8]. The following relations are of a similar form, and they allow us to connect the p-adic properties of Hauptmoduln on closely related groups. if r = 1 and p ∤ e for w e ∈ Γ, Remark 4.6. If r = 1, p ∤ e for w e ∈ Γ, and Γ, w p appears in monstrous moonshine, then part (b) follows from part (a) and (4.1). However, part (b) holds even when Γ, w p is not admissible. For example 4T 6|3 |U 2 2 = T 6|3 − T 3|3 even though 6|3+ is not admissible. Similarly, if r = 2, w p 2 ∈ Γ, and Γ p , w p appears in monstrous moonshine, then part (d) follows from parts (a) and (c), but part (d) holds even when Γ p , w p is not admissible.
In each case, there are only finitely many Hauptmoduln satisfying the hypotheses, and for each, we may apply Lemma 2.9, use Sturm's bound, and check that sufficiently many of the coefficients are 0. These relations reduce the number of Hauptmodul one needs to check to show that the Hauptmoduln in Theorem 4.1 are indeed annihilated. For example, (a) implies that if Γ is an pn|h-type group with p ∤ n and T Γ is p-adically annihilated, then so is T Γ,wp . Note that many of these formulas allow us to prove that T Γ p is p-adically annihilated from the fact that T Γ is, which explains much of the structure in the figures in Appendix B. In some sense, Proposition 4.5 suggests that p-adic properties of Hauptmoduln must be closely related to moonshine modules, since they tend to be preserved under power maps in the underlying group. Since we have already proved that T is p-adically annihilated in the cases where T |U p = 0 (see Remark 2.5), we have altogether reduced the verification of Theorem 4.1 to the following much smaller check.   One has a significant amount of freedom in choosing these representatives -we have chosen those most conducive to performing computations when p = 2, 3. When p ≥ 5 we will prove Theorem 4.1 directly in Section 4.2 rather than using the reduction given here.

Annihilation via Ordinary Spaces
In this section, we will prove Theorem 4.1 for p ≥ 5 using the theory of p-adic modular forms developed in Section 3. The key observation relating the theory of p-adic modular forms to p-adic annihilation is the following easy consequence of Proposition 3.6. Using the mfslashexpansion and mfatkin functions in Pari [28], one can compute the actions of x, y, and all Atkin-Lehner involutions on M p−1 (Γ, λ t ). Then using elementary linear algebra, it is easy to find a basis for S(Γ, [λ] p ) 0 ⊆M p−1 (Γ, λ t ). We performed this computation for various n|h groups Γ appearing in monstrous moonshine with small values of n with p ∤ n (specifically, n ≤ 24 for p = 5, n ≤ 11 for p = 7, and n ≤ 7 for p = 11). The Γ for which S(Γ, [λ] p ) 0 ⊆ F p are given in Table 4.3. Applying Lemma 3.4 to the Hauptmoduln of Table 4.1 then proves Theorem 4.1 for p ≥ 5. We note that every group from Table 4.3 corresponds to at least one group from Theorem 4.1. It is worth noting that these methods also apply to Hauptmoduln not appearing in monstrous moonshine. For example, if Γ = Γ 0 (2|2) and T is the Hauptmodul on Γ λ then Lemma 3.4 gives that T |U 5 is a 5-adic modular form on (Γ, λ). Using the method above, one can compute that S(Γ, [λ] p ) 0 = 0, so that T is 5-adically annihilated.
Although the methods developed here do not directly give rates of annihilation, the following observation held in all cases from Table 4.3, when checked with 10, 000 coefficients.
bounding the rate of annihilation from below by 1/(m + 1). Moreover this choice of m appears to be tight. We pose the question of whether these observations continue to hold to in general.

Additional p-adic Annihilation
The ordinary spaces of Section 3.2 and the annihilation verified in Section 4.2 were restricted to p ≥ 5. In this section, we use different techniques to explicitly verify p-adic annihilation for the six groups appearing in Corollary 4.7 for the primes p = 2, 3.   Proof. Throughout, we use T to denote the normalized Hauptmodul T plus some constant. The constant will be specified by writing T as an eta quotient given in [5], and the relevant ones are recorded in Table 4.5. We follow the proofs given in [23,24] for the j-function, We first compute expansions of T at each cusp, and write T|U p as a rational function in a Hauptmodul by subtracting off the principal part. Below, we write out the calculation for p = 3 and the group 6+2, and record only the formulas for the rest. These formulas can also be verified by checking that the first finitely many coefficients match. Let For W 2 a suitable Atkin-Lehner matrix on Γ 0 (6), we have and upon substituting τ → 3τ we obtain the cusp expansion We next subtract off this principal part with −T(τ ). Sending τ → −1/6τ and applying the η functional equation to the quotients in Table 4.5, we find where e is as in Table 4 where f is a polynomial with f (T(τ )) = T(pτ ) + O(q) (different for each group). Write where the coefficients b j are listed in Table 4 This equation has roots W (τ + λ) for λ ∈ {0, . . . , p − 1}. If S ℓ denotes the sum of the ℓ-th power of these roots, then we have p −eℓ−1 S ℓ = Z ℓ |U p . We show that p α(ℓ−1) (Z ℓ |U p ) ∈ p α Z · Z[p α Z, p α ] for all ℓ ≥ 1. Equivalently, we check that S ℓ ∈ p (e−α)ℓ+1+α p α Z · Z[p α Z, p α ]. Lehner uses Newton sums to set up an induction, relating S ℓ to S j for j < ℓ. A similar computation works for our cases. The coefficients in Table 4.7 imply the base case ℓ = 1. By Newton sums we have where b j , p h = 0 for j, h ≥ p + 1 and S 0 = ℓ. For ℓ ≤ p we rewrite the Newton sum as By construction we have that p j ∈ p e+1+α+vp(b j )−α p α Z · Z[p α Z, p α ] and applying the inductive hypothesis gives We have the inequality e + 1 + α + v p (b j ) − α ≥ (e − α)j, which one checks explicitly using the values of b j . In particular, p j S ℓ−j ∈ p (e−α)ℓ+1+α p α Z · Z[p α Z, p α ]. We also check that and we check that p j S ℓ−j ∈ p (e−α)ℓ+1+α p α Z · Z[p α Z, p α ] as above, which completes the induction. Thus, we have shown that Repeatedly applying the result of the induction thus gives the rates of annihilation claimed in Proposition 4.10.
Remark 4.11. These explicit techniques also apply when p ≥ 5, as long as T |U p is modular on a genus zero group. For example, if T is the Hauptmodul for 2+, then T |U 5 is modular on 10+2, which is genus zero. There are many examples for p ≥ 5 where this does not hold, e.g. if T the Hauptmodul for 2+ then T |U 7 is modular on 14+2, which is genus one. For this example, the arguments of this section can be modified to obtain explicit lower bounds on the annihilation rate, e.g. by working with bivariate polynomials in appropriate meromorphic modular forms instead of single-variate polynomials in the Hauptmodul, since the latter do not exist for the genus one curve corresponding to 14+2. In general, however, we rely on the theory of Section 3 to prove p-adic annihilation as in Section 4.2.

Preservation of Annihilation under Power Maps
We discuss the relationship between Hauptmoduln p-adic annihilation and power maps in the monster. We have already seen hints of this in Section 4.1. Unlike previous sections, this section will not be used elsewhere, except as motivation for Section 5.
Definition 4.12. For a fixed Hauptmodul T g and an integer d, we say that p-adic annihilation is preserved under the d-th power map if T g p-adically annihilatd implies T g d is p-adically annihilated.
These relationships are depicted in Appendix B for primes p ≤ 11, which we expect are the only primes with annihilation. In general, d will be clear and we just say p-adic annihilation is preserved. We first give conceptual explanations for this preservation of p-adic annihilation, when it holds. We then characterize exactly when p-adic annihilation is not preserved, and offer a notion of p-adic annihilation that seems to always be preserved under power maps. The average numerical rates of annihilation from Appendix A often do not decrease under power maps -we remark upon this briefly.
In certain situations, the compression formulas show that Hauptmodul p-adic annihilation is preserved under power maps, e.g. powering from the group ℓ+ to level 1, when ℓ is prime. When p ≥ 5, we may also explain via ordinary spaces.
Furthermore, in light of Lemma 4.8 and Section 4.2, we seek inclusions of the form S(Γ d , [λ] p ) 0 ֒→ S(Γ, [λ] p ) 0 for Γ being n|h-type with p ∤ nh. This would explain d-th power map preservation of p-adic annihilation from Γ λ to Γ d λ , and similarly for groups with the same ordinary spaces. The following proposition accomplishes this in certain situations. Proposition 4.14. Let p ≥ 5 be prime, (d, nh) = 1, Γ be an n|h-type group with eigenvalue map η, and Γ ′ be a dn|h-type group with eigenvalue map η ′ such that AL(Γ) = AL(Γ ′ ), (d, e) = 1 for all e ∈ AL(Γ), and η| Γ ′ = η ′ .

(a) There is an inclusion
Proof. Part (b) follows from part (a) using the description of S given in the proof of Proposition 3.6. Part (a) follows from the isomorphic inclusion Γ/Γ 0 (dnh) ֒→ Γ ′ /Γ 0 (nh) from Section 2.3. We now discuss when power maps do not preserve Hauptmodul p-adic annihilation.
Example 4.16. If T is the Hauptmodul on Γ λ for Γ an n|h-type group, then the Hauptmodul on Γ d λ need not be annihilated when d | h, which describe all the grey boxes in the figures of Appendix B. For example, the Hauptmodul for 6|3 is 5-adically annihilated, but the Hauptmodul for 2 is not.  In contrast with p-adic annihilation of Hauptmoduln, strong p-adic annihilation numerically appears to be always preserved under power maps. Furthermore, our numerical data indicate that rates of strong p-adic annihilation are also non-decreasing under power maps. We discussed polynomials in the Hauptmodul in Remark 3.5, and one finds that Γ η has strong p-adic annihilation if S p−1 (Γ, 1 t ) 0 = F p and S p−1 (Γ, [(λ σr ) t ] p ) 0 = 0 otherwise.
We can check that strong p-adic annihilation is now preserved for the groups of Example 4.16. Indeed, let T be the Hauptmodul on Γ λ for Γ an n|h-type group, and T ′ be the Hauptmodul on Γ d λ for d | h. We recall the V m operator, given by f |V m = f • A where we set A = m 0 0 1 as in Section 3.2. It acts on Fourier expansions by ( a(n)q n ) |V m = a(n)q mn . From [5, § 6], we know that T ′ |V d + c = T d for some constant c. In particular, if T ′ is not p-adically annihilated, T ′ |V d is a polynomial in T with no constant term and is also not p-adically annihilated. Thus, if Γ d λ does not have strong p-adic annihilation, neither does Γ λ . This reflects the following inclusions.
(b) For any α ∈ Z/(2p − 2)Z, this gives an injection V d : We observe empirically that strong p-adic annihilation is also preserved for the groups from Example 4.17, e.g. T 2 is not 2-adically annihilated when T is the Hauptmodul on 12.
Further aspects of this connection between p-adic properties of modular forms and the associated conjugacy classes of the monster group will be discussed in Section 5.

Moonshine
In this section, we will investigate groups with p-adic moonshine. Recall from the introduction that a moonshine module for a finite group G is a graded G-module V = ∞ n=−1 V n such that (i) For each g ∈ G, the McKay-Thompson series is the Hauptmodul of an order ord(g) conjugacy class of the Monster.
(ii) For any g ∈ G and any n ∈ Z, if T g is the Hauptmodul for Γ, then T g n is the Hauptmodul for Γ n .
If for some prime p, V also satisfies the following property, then we call V a p-adic moonshine module. Throughout this section, we assume the converse to Theorem 4.1. That is, we assume that Theorem 4.1 exactly characterizes which of the 171 Hauptmoduln appearing in monstrous moonshine are p-adically annihilated for each prime p.
We begin in Section 5.1 by stating basic facts about groups with p-adic moonshine, including the fact that for each prime p, only finitely many such groups exist. In Section 5.2, we illustrate p-adic moonshine with several examples of groups having p-adic moonshine modules. In Section 5.3, we find a surprising class of subgroups of the Monster having p-adic moonshine in a slightly more general sense.

Basic Facts
We begin by presenting an alternative formulation of p-adic moonshine that will allow us to make use of the results in Section 4.
Lemma 5.1. Let G be a finite group with a moonshine module V = ∞ n=−1 V n . For any prime p, V is a p-adic moonshine module if and only if the McKay-Thompson series T g is p-adically annihilated for each g ∈ G.
Proof. By the Schur orthogonality relations, we have If each M χ is p-adically annihilated, then for all n ∈ N, there exists N ∈ N such that M χ |U N p ≡ 0 (mod p n ). Since each χ(g) is an algebraic integer, it follows that T g |U N p ≡ 0 (mod p n ), so each T g is p-adically annihilated. Similarly, if each T g is p-adically annihilated, then each |G|M χ , and hence each M χ , is also p-adically annihilated.
As an immediate consequence, a group G has p-adic moonshine if and only if we can assign to each element g ∈ G a p-adically annihilated Hauptmodul T g in a way that agrees with power maps in G and so that the multiplicities defined by have positive integral coefficients for all irreducible characters χ. In particular, each McKay-Thompson series T g must be a Hauptmodul for one of the groups listed in Theorem 4.1. In fact, since we require that the assignment of group elements to Hauptmoduln agree with power maps, we can restrict our attention to only those congruence groups Γ with the property that Γ n is p-adically annihilated for every n ∈ Z. By Theorem 4.1, it follows that every McKay-Thompson series T g must be a Hauptmodul for one of the groups listed in Table 5.1.  2   3  5  7  11  1  12|3+  1  1  1  1  2+  12+3  2+  2+  2+  3|3  2  12|6  3+  3+  3|3  11+  3+  16|2+  3  3|3  4|2+  3|3 16 As a first application of this simpler description of p-adic moonshine, we bound the powers of primes dividing the orders of groups with p-adic moonshine for each prime p. Proof. First, note that if q | |G|, then by Cauchy's theorem, G contains an element of order q, so there must be some Hauptmodul corresponding to a q|h-type group in Table 5.1. This proves all of the cases with r = 0 above. For the remaining cases with r > 0, we follow the method given in [8]. Let J, T 1 , . . . , T n be the distinct Hauptmoduln given in Table 5.1 as candidate McKay-Thompson series for groups with p-adic moonshine. For each i ∈ {1, . . . , n}, let a i be the number of elements of G whose McKay-Thompson series is T i . Then, the multiplicity of the trivial character ǫ is (J + a 1 T 1 + · · · + a n T n ) .
In particular, since M ǫ must have integral coefficients, it follows that J + a 1 T 1 + · · · + a n T n ≡ 0 (mod |G|).
Therefore, if we choose r large enough so that there are no coefficients a 1 , . . . , a n such that J + a 1 T 1 + · · · + a n T n ≡ 0 (mod q r+1 ), then we have shown that v q (|G|) ≤ r. As in [8], this computation was carried out using Sage [29] by computing the kernel of the matrix of coefficients of J, T 1 , . . . , T n .

Examples of Groups with p-adic moonshine
We begin by illustrating the process of showing a group has p-adic moonshine using the group A 5 and the prime p = 5. The character table for A 5 is given below: The square of any element of the fourth conjugacy class listed in the character table is in the fifth conjugacy class, so any moonshine module must have the same McKay-Thompson series for elements of those two conjugacy classes. The only other non-trivial power relations come from the fact that g ord(g) = 1 for any g, so a possible assignment of 5-adically annihilated McKay-Thompson series that agrees with power maps in A 5 is given by assigning T 1 to the element of the conjugacy class of 1, T 2+ to the elements of the conjugacy class of (1 2)(3 4), T 3|3 to the elements of the conjugacy class of (1 2 3), and T 5 to the elements of the conjugacy classes of (1 2 3 4 5) and (1 3 4 5 2). The associated multiplicities are then given by M ǫ = q −1 + 4378q + 382380q 2 + 14714988q 3 + 340105628q 4 + O(q 5 ), In order to show that this gives a valid moonshine module, we must show that these multiplicities are both positive and integral. For positivity, we may use inequality (4.16) in [8]. Indeed, this inequality holds for n = 2, and hence for all n ≥ 2 since the left-hand side is monotonically increasing. Since the first coefficients of each multiplicity generating function are positive, this implies that all of them must be. For integrality, we may simply note that each multiplicity generating function is on Γ 0 (90), so we may use Sturm's bound to reduce the computation to checking only the coefficients up to q 216 . Using Sage [29], it turns out to indeed be the case that all of the coefficients are integers. Thus, A 5 has 5-adic moonshine with the McKay-Thompson series given above.
In fact, it turns out that p-adic moonshine is not such a rare phenomenon among groups with small orders. Indeed, using Sage [29], we have computed that for every prime p, every group G of order at most 25 for which there is some assignment of p-adically annihilated Hauptmoduln to elements of G obeying power maps has p-adic moonshine. In fact, out of all 45252 such feasible assignments, only 11 do not give rise to a p-adic moonshine module, and all 11 exceptions are for p = 2 and the group G = Z/2Z × (Z/3Z) 2 .
In certain special cases, these computations become somewhat simpler. For example, consider the case of a non-trivial group G in which we assign every non-identity element the same Hauptmodul T . In particular, this means that every non-identity element of G must have order q for some prime q. We will characterize exactly when G has a p-adic moonshine module under this assignment.
Under such an assignment, the multiplicity of the trivial character ǫ is given by and the multiplicity of any non-trivial character χ is given by These multiplicities are both integral if and only if |G| | J − T , and for checking positivity, we may once again use inequality (4.16) in [8]. After a computation in Sage [29], we have the following result.
Proposition 5.3. Let p and q be primes, G be a group of exponent q, and T be a Hauptmodul for one of the order q conjugacy class of the Monster. Then, G has a p-adic moonshine module in which the McKay-Thompson series for each non-identity element is T if and only if p and T appear in the following table and |G| ≤ q r where r is the corresponding entry in the third row.

Appendices A Table of Annihilation
The following table gives the precise congruences that numerically appear to be satisfied by each Hauptmodul. The notation a 1 , . . . , a m → b 1 , . . . , b n means the the sequence beginning a 1 , . . . , a m , and then each subsequent term is given by adding b k (mod n) from k = 1 to ∞. For example, 0, 1 → 0, 3 is the sequence 0, 1, 1, 4, 4, 7, 7, 10, 10, . . . . The entry under p for the group Γ indicates the sequence a 1 , a 2 , a 3 , . . . such that a n is the highest power of p dividing T Γ |U n p . If no such cyclic pattern is clear, then we simply list the first few terms of the sequence in parentheses.

B Power Maps
For each p ∈ {3, 5, 7, 11} we record here the structure of the collection of groups whose Hauptmoduln are p-adically annihilated by U p (the case p = 2 appears as Figure 1.1 in the introduction). In each diagram below, we write the groups Γ such that T Γ is p-adically annihilated by U p but T Γ |U p = 0, and all of the powers of such groups. Solid lines indicate power maps, groups in white boxes satisfy T Γ |U p = 0, and groups in black boxes are not p-adically annihilated by U p at all.