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

$\bm{p}$-Adic Properties of Hauptmoduln
with Applications to Moonshine††This paper is a contribution to the Special
Issue on Moonshine and String Theory. The full collection is available at
https://www.emis.de/journals/SIGMA/moonshine.html

Ryan C. CHEN, Samuel MARKS and Matthew TYLER

R.C. Chen, S. Marks and M. Tyler

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Received September 19, 2018, in final form April 10, 2019; Published online
April 29, 2019

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.

modular forms congruences; $p$-adic modular forms; moonshine

11F11; 11F22; 11F33

##  1 Introduction and statements of results

The theory of monstrous moonshine arose from the remarkable observation of
McKay and Thompson [41] that

$\displaystyle 196884=1+196883$

and its generalizations, including

$\displaystyle 21493760=1+196883+21296876,$ $\displaystyle 864299970=2\times
1+2\times 196883+21296876+842609326,$ $\displaystyle 20245856256=2\times
1+3\times 196883+2\times 21296876+842609326+19360062527.$

Here, the left-hand sides of the equations are the coefficients of the
normalized modular $j$-function

$\displaystyle
J(\tau)=j(\tau)-744=q^{-1}+196884q+21493760q^{2}+\cdots,\qquad\text{where}\quad
q={\rm e}^{2\pi{\rm i}\tau},$

and the right-hand sides are simple sums involving the dimensions of the
irreducible representations of the monster group $\mathbb{M}$:

$\displaystyle 1,\;196883,\;21296876,\;842609326,\;19360062527,\;\dots.$

Thompson conjectured [40] that these identities could be explained by the
existence of an infinite-dimensional graded monster module

$\displaystyle V^{\natural}=\bigoplus_{n=-1}^{\infty}V^{\natural}_{n}$

such that the graded dimension is given by $J$. More generally, the graded-
trace functions

$\displaystyle\mathcal{T}_{g}(\tau)=\sum_{n=-1}^{\infty}\operatorname{Tr}\big{(}g|V_{n}^{\natural}\big{)}q^{n}$

for the action of $\mathbb{M}$ on $V^{\natural}$ are known as the
McKay–Thompson series and depend only on the conjugacy class of
$g\in\mathbb{M}$. As part of their famous monstrous moonshine conjectures,
Conway and Norton computed for each monster conjugacy class $g$ a genus zero
group $\Gamma_{g}\leq\operatorname{GL}_{2}^{+}(\mathbb{R})$ on which they
conjectured $\mathcal{T}_{g}$ was a normalized Hauptmodul [10]. That is, each
$\mathcal{T}_{g}$ was conjectured to be the unique generator
$\mathcal{T}_{\Gamma_{g}}$ of the function field of the genus zero curve
$\Gamma_{g}\backslash\mathbb{H}^{*}$ having $q$-expansion of the form
$q^{-1}+O(q)$ at $\infty$. Since all of the Hautpmoduln appearing in this
paper will be normalized (meaning that they are bounded away from $\infty$ and
have $q$-expansion $q^{-1}+O(q)$ at $\infty$), we will henceforth omit the
word “normalized” and refer to such functions simply as Hautpmoduln.
Frenkel–Lepowsky–Meurman [18, 19] constructed $V^{\natural}$ with the correct
graded dimensions, and Borcherds [4] proved that the McKay–Thompson series
were Hauptmoduln for the $\Gamma_{g}$ given by Conway–Norton. After the proof
of monstrous moonshine, different incarnations of moonshine were shown for
other finite groups, such as the largest Mathieu group $M_{24}$ [20], and
later the other $22$ groups appearing in umbral moonshine [14]. There is also
a notion of generalized moonshine, conjectured by Norton [33] and recently
proved by Carnahan [8].

Thirty years before the observation of McKay and Thompson, Lehner [31, 32] and
Atkin [2] proved that the Fourier expansion of $J(\tau)=q^{-1}+\sum c(n)q^{n}$
satisfies the following congruences for all positive $\alpha$:

$\displaystyle c(2^{\alpha}n)\equiv 0\ \left({\rm mod}\ 2^{3\alpha+8}\right),$
$\displaystyle c(3^{\alpha}n)\equiv 0\ \left({\rm mod}\ 3^{2\alpha+3}\right),$
$\displaystyle c(5^{\alpha}n)\equiv 0\ \left({\rm mod}\ 5^{\alpha+1}\right),$
$\displaystyle c(7^{\alpha}n)\equiv 0\ \left({\rm mod}\ 7^{\alpha}\right),$
$\displaystyle c(11^{\alpha}n)\equiv 0\ \left({\rm mod}\ 11^{\alpha}\right).$
(1.1)

Viewed another way, these identities state that $J|U_{p}^{n}$ uniformly
converges to zero $p$-adically as $n\to\infty$, where $U_{p}$ is the operator
defined on $q$-expansions by

$\displaystyle\left(\sum a(n)q^{n}\right)|U_{p}=\sum a(pn)q^{n}.$

Such congruences led Serre, Katz, and others to develop a robust and fruitful
theory of $p$-adic modular forms [7, 22, 28, 29, 37].

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
$f|U_{p}^{n}$ uniformly converges to $0$ in the $p$-adic limit as
$n\rightarrow\infty$. Given that $J$ is $p$-adically annihilated for
$p\in\\{2,3,5,7,11\\}$, we can then ask if other Hauptmoduln appearing in
monstrous moonshine are as well.

There is some literature studying coefficient congruences of a related nature.
The papers [1, 26] discuss Hauptmoduln on $\Gamma_{0}(N)$, and [40] discusses
other coefficient congruences involving Hauptmoduln. However, there has not
been a systematic study of $p$-adic annihilation for all of the monstrous
moonshine Hauptmoduln.

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 $p$.

######  Theorem 1.1.

For primes $p\in\\{2,3,5,7,11\\}$, the Hauptmoduln $\mathcal{T}$ corresponding
to the genus zero groups of monstrous moonshine appearing in Table 4.1 are
$p$-adically annihilated.

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 $g\mapsto g^{m}$ of
the monster (or equivalently, the corresponding maps of conjugacy classes) to
$p$-adic annihilation of Hauptmoduln.

######  Theorem 1.2.

Let $\mathcal{T}_{g}$ be the Hauptmodul of a group appearing in Table 4.1, so
that $\mathcal{T}_{g}$ is $p$-adically annihilated by Theorem 1.1. Outside of
the exceptions discussed in Section 4.4, we also have that
$\mathcal{T}_{g^{m}}$ is $p$-adically annihilated for any $m\in\mathbb{N}$.

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 Fig. 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.

$1$$2$$2{+}$$3|3$$3{+}$$4$$4{+}$$6|3$$5{+}$$6{+}3$$6{+}$$11{+}$$8$$8{+}$$12|3{+}$$10{+}5$$10{+}$$12{+}$$22{+}11$$22{+}$$16$$16{+}$$12{+}3$$20{+}$$24{+}$$44{+}$$32{+}$
Figure 1.1: Conjugacy classes with $2$-adically annihilated Hauptmoduln and
their power maps.

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
$V=\bigoplus_{n=-1}^{\infty}V_{n}$ such that for each $g\in G$ the graded
trace $\mathcal{T}_{g}=\sum\operatorname{Tr}(g|V_{n})q^{n}$ associated to the
action of $g$ on $V$ is the Hauptmodul of an order $\operatorname{ord}(g)$
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 $\chi$ of $G$, we write $m_{\chi}(n)$ for the multiplicity of $\chi$
appearing in the character of $G$ acting on $V_{n}$, and define the
multiplicity generating function

$\displaystyle\mathcal{M}_{\chi}(\tau)=\sum_{n=-1}^{\infty}m_{\chi}(n)q^{n}.$

These series $\mathcal{M}_{\chi}$ were perhaps first studied in [23]. We say
that a moonshine module $V$ for $G$ is a $p$-adic moonshine module if
$\mathcal{M}_{\chi}$ is $p$-adically annihilated for each irreducible
character $\chi$. 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 1.1 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 $pA$. For other instances of moonshine modules for
centralizers of elements of the class $pA$, see Ryba’s modular moonshine
conjectures [35], which were proved by Borcherds and Ryba [5, 6].

$p$ | $2$ | $3$ | $5$ | $7$ | $11$
---|---|---|---|---|---
$C\big{(}pA^{2}\big{)}$ | $2^{2}\cdot{}^{2}\\!E_{6}(2)$ | $3^{2}\times O_{8}^{+}(3)$ | $5^{2}\times U_{3}(5)$ | $7^{2}\times L_{2}(7)$ |  $11^{2}$ 3pt3pt
$\\#C\big{(}pA^{2}\big{)}$ | $2^{38}{\cdot}3^{9}{\cdot}5^{2}{\cdot}7^{2}{\cdot}11{\cdot}13{\cdot}17{\cdot}19$ | $2^{12}{\cdot}3^{14}{\cdot}5^{2}{\cdot}7{\cdot}13$ | $2^{4}{\cdot}3^{2}{\cdot}5^{5}{\cdot}7$ | $2^{3}{\cdot}3{\cdot}7^{3}$ |  $11^{2}$3pt
Table 1.1: 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 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 1.1.

##  2 Preliminaries

In this section, we collect technical details and definitions that will be
used later. We first describe the types of groups $\Gamma$ 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 $\Gamma$ into modular forms on some
$\Gamma^{\prime}\geq\Gamma$. These facts will ultimately be used to interpret
Hauptmoduln as $p$-adic modular forms in Section 3.

###  2.1 $\bm{n|h}$-type groups

Monstrous moonshine associates to each $g\in\mathbb{M}$ a Hauptmodul
$\mathcal{T}_{g}$ for some genus zero group
$\Gamma_{g}\leq\operatorname{GL}_{2}(\mathbb{R})^{+}$. This means that
$\Gamma_{g}\backslash\mathbb{H}^{*}$ is a genus zero curve and that
$\mathcal{T}_{g}$ is a generator for the function field such that
$\mathcal{T}_{g}$ is bounded away from the cusp $\infty$; moreover the
$q$-expansion of $\mathcal{T}_{g}$ at infinity begins
$\mathcal{T}_{g}=q^{-1}+O(q)$. Conway and Norton described the groups
$\Gamma_{g}$ in [10], all of which take on a particular form which we
reproduce here.

First we describe the normalizer of $\Gamma_{0}(N)$ in
$\operatorname{PSL}_{2}(\mathbb{R})$. Let $h$ be the largest integer such that
$h^{2}|N$ and $h|24$, and set $n=N/h$. The normalizer of $\Gamma_{0}(N)$ is
given by $\bigcup_{e\|n/h}w_{e}$ where $w_{e}$ is the set of all matrices
$A=\left(\begin{smallmatrix}ae&b/h\\\ cn&de\end{smallmatrix}\right)$ such that
$a,b,c,d\in\mathbb{Z}$ 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
$e_{1}*e_{2}=\frac{e_{1}e_{2}}{\gcd(e_{1},e_{2})^{2}}$, and under $*$ the set
of exact divisors of any integer $N$ forms the abelian group
$(\mathbb{Z}/2\mathbb{Z})^{n}$ 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
$\Gamma_{0}(n|h)=w_{1}$. 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 $m^{2}\equiv 1\ \left({\rm mod}\ h\right)$ for all
$m$ coprime to $h$. For a subgroup $\\{1,e_{1},\dots,e_{n}\\}$ of the group of
exact divisors of $n/h$, we then define

$\displaystyle\Gamma_{0}(n|h){+}e_{1},e_{2},\dots,e_{n}=\langle\Gamma_{0}(n|h),w_{e_{1}},w_{e_{2}},\dots,w_{e_{n}}\rangle=w_{1}\cup
w_{e_{1}}\cup w_{e_{2}}\cup\dots\cup w_{e_{n}}.$

A group of this form is called an $n|h$-type group.

Setting $N=nh$, the group $\Gamma_{0}(n|h){+}e_{1},\dots,e_{n}$ normalizes
both $\Gamma_{0}(n|h)$ and $\Gamma_{0}(N)$, and the $w_{e_{i}}$ are cosets of
$\Gamma_{0}(n|h)$. When $h=1$ we have $\Gamma_{0}(n|1)=\Gamma_{0}(N)$ and we
denote the matrix

$\displaystyle w_{e}=\begin{pmatrix}ae&b\\\ cN&de\end{pmatrix}$

by $W_{e}$. The matrices $W_{e}$ for $e\|N$ are called Atkin–Lehner
involutions. Given an Atkin–Lehner involution $W_{E}$ on $\Gamma_{0}(N)$, we
can interpret this as an element of $\Gamma_{0}(n|h)$ via

$\displaystyle W_{E}=\begin{pmatrix}aE&b\\\
cN&dE\end{pmatrix}=\begin{pmatrix}aE/h_{E}&b/h_{E}\\\
cN/h_{E}&dE/h_{E}\end{pmatrix}=w_{e},$

where $h_{E}$ is the largest divisor of $h$ with $h_{E}^{2}|E$ and
$e=E/h_{E}^{2}$. In fact, setting

$\displaystyle\operatorname{AL}(\Gamma)=\\{e\|nh\colon W_{e}\in\Gamma\text{
and every prime dividing }e\text{ also divides }n/h\\}$

we have that this association gives a bijection

$\displaystyle\operatorname{AL}(\Gamma)\longleftrightarrow\\{e\colon
w_{e}\subset\Gamma\\}.$

For example, letting $\Gamma=\Gamma_{0}(8|2){+}4$ we have

$\displaystyle\operatorname{AL}(\Gamma)=\\{1,16\\}\longleftrightarrow\\{1,4\\}=\\{e\colon
w_{e}\subset\Gamma\\}.$

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
$\Gamma_{0}(n|1)=\Gamma_{0}(n)$, and when all $e\|n/h$ are included in a
group, we simply write $\Gamma_{0}(n|h){+}$ so that
$\Gamma_{0}(8|2){+}=\Gamma_{0}(8|2){+}4$. We will also sometimes use the
symbol $n|h{+}e,f,\dots$ to represent the group $\Gamma_{0}(n|h){+}e,f,\dots$
in order to save space, particularly in tables, so we might write $8|2{+}$
instead of $\Gamma_{0}(8|2){+}$.

The $n|h$-type groups appear in monstrous moonshine as eigengroups of the
Hauptmoduln. That is, the Hauptmodul $\mathcal{T}$ has an associated group
$\Gamma_{0}(n|h){+}e,\dots$ such that for all $A$ in this group,
$\mathcal{T}(A\tau)=\mu\mathcal{T}(\tau)$ for some $h^{\text{th}}$ root of
unity $\mu$. Conway–Norton [10] conjectured the following rule for computing
the eigenvalue $\lambda$ corresponding to an element of
$\Gamma_{0}(n|h){+}e,\dots$:

  1. (i)

$\lambda=1$ for any element of $\Gamma_{0}(N)$,

  2. (ii)

$\lambda=1$ for all $W_{e}$ with $e\in\operatorname{AL}(\Gamma),$

  3. (iii)

$\lambda={\rm e}^{-2\pi{\rm i}/h}$ for the element
$x=\left(\begin{smallmatrix}1&1/h\\\ 0&1\end{smallmatrix}\right)$,

  4. (iv)

$\lambda={\rm e}^{\pm 2\pi{\rm i}/h}$ for the element
$y=\left(\begin{smallmatrix}1&0\\\ n&1\end{smallmatrix}\right)$ where the sign
is $+$ if and only if $\tau\mapsto\frac{-1}{N\tau}$ is in
$\Gamma_{0}(n|h){+}e,\dots$.

This rule’s well-definedness and correctness follow from [17] and the
correctness of the monstrous moonshine conjectures, respectively.

Since the cosets $x$ and $y$ generate $\Gamma_{0}(n|h)$, we have
$\Gamma=\langle x,y,W_{e}\colon e\in\operatorname{AL}(\Gamma)\rangle$ for any
$n|h$-type group $\Gamma$. Hence Conway–Norton’s rule uniquely determines a
map $\lambda\colon\Gamma\rightarrow\bm{\mu}_{h}$ where $\bm{\mu}_{h}$ denotes
the group of $h^{\text{th}}$ roots of unity. We always use $\lambda$ to denote
this map.

More generally, let $\Gamma$ be an $n|h$-type group. An eigenvalue map is a
homomorphism $\eta\colon\\!\Gamma\\!\rightarrow\\!\bm{\mu}_{2h}$ such that
$\Gamma_{0}(nh)\subset\ker\eta$. Then we define

$\displaystyle\Gamma_{\eta}=\ker\eta.$

When $\lambda$ is the map given by Conway–Norton’s rule, we have that
$\Gamma_{\lambda}$ is an index $h$ subgroup of $\Gamma$ called the fixing
group of $\Gamma$. However, Conway–Norton’s rule does not always give a well-
defined map, so $\Gamma_{\lambda}$ does not exist for every $n|h$-type group
$\Gamma$; for example $\Gamma_{\lambda}$ doesn’t exist when
$\Gamma=\Gamma_{0}(21|3)$. Ferenbaugh [17, Theorem 2.8] classified the groups
$\Gamma$ for which Conway–Norton’s rule is consistent, and we call such
$\Gamma$ 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
$\Gamma/\Gamma_{0}(nh)$, and therefore also the structure of
$\Gamma_{\lambda}/\Gamma_{0}(nh)$. 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 [11].

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.

###  2.2 Action of $\bm{U_{p}}$ on Hauptmoduln

Given an $n|h$-type group $\Gamma$ with eigenvalue map
$\eta\colon\Gamma\rightarrow\bm{\mu}_{2h}$, we say a weight $k$ weakly
holomorphic modular form on $\Gamma_{0}(nh)$ is on $\Gamma$ with eigenvalue
map $\eta$ or on $(\Gamma,\eta)$ if

$\displaystyle f|_{k}\gamma=\eta(\gamma)f\qquad\text{for
all}\quad\gamma\in\Gamma,$

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 $M_{k}(\Gamma,\eta)$ for the space of weight
$k$ modular forms on $\Gamma$ with eigenvalue map $\eta$. Similarly, we denote
the space of weight $k$ modular forms on $\Gamma_{0}(nh)$ invariant under all
$\gamma\in\Gamma_{\eta}$ by $M_{k}(\Gamma_{\eta})$. Throughout, all our
weights will be integers.

Fix a prime $p$. In this section, we study $U_{p}$ applied to Hautpmoduln
$\mathcal{T}$, and more generally weakly holomorphic modular forms on
$\Gamma_{\eta}$ or on $\Gamma$ with eigenvalue map $\eta$. 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 $\Gamma_{0}(N){+}e,\dots$ should remember that this
corresponds to taking $h=1$ and ignoring eigenvalue maps in the following
results.

Recall that the weight $k$ slash operator $|_{k}\gamma$ for
$\gamma\in\operatorname{GL}_{2}^{+}(\mathbb{R})$ is defined by

$\displaystyle(f|_{k}\gamma)(\tau)=(\det\gamma)^{k/2}(c\tau+d)^{-k}f(\gamma\tau).$
(2.1)

If $f$ is on $\operatorname{SL}_{2}(\mathbb{Z})$ then for $N\in\mathbb{N}$, we
have $f(N\tau)$ on $\Gamma_{0}(N)$, and for $d|N$ and $e\|N$,

$\displaystyle
f(d\tau)|_{k}W_{e}=\left(\frac{d*e}{d}\right)^{k/2}f((d*e)\tau).$ (2.2)

In terms of the slash operator, $U_{p}$ is defined on weight $k$ modular forms
by

$\displaystyle
f|U_{p}=p^{k/2-1}\sum_{\mu=0}^{p-1}f|_{k}S_{\mu}=\frac{1}{p}\sum_{\mu=0}^{p-1}f(S_{\mu}\tau),$
(2.3)

where $S_{\mu}=\left(\begin{smallmatrix}1&\mu\\\ 0&p\end{smallmatrix}\right)$.
This operator is independent of $k$ and acts on Fourier expansions by

$\displaystyle\left(\sum a(n)q^{n}\right)|U_{p}=\sum a(pn)q^{n}.$

We first recall some basic facts about the $U_{p}$ operator (see [3, Section
2]).

######  Lemma 2.1.

Let $f$ be a weight $k$ meromorphic modular form on $\Gamma_{0}(p^{\alpha}N)$
where $p\nmid N$.

  1. $(a)$

If $e\|N$ then $f|U_{p}|_{k}W_{e}=f|_{k}W_{e}|U_{p}$.

  2. $(b)$

$f|U_{p}$ is modular on $\Gamma_{0}(p^{\beta}N)$ where
$\beta=\max\\{1,\alpha-1\\}$.

The following lemma extends these facts from $\Gamma_{0}(N)$ to $n|h$-type
groups.

######  Lemma 2.2.

Let $p\nmid nh$ be prime, and let $f$ be a weight $k$ meromorphic modular form
on a $p^{\alpha}n|h$-type group $\Gamma$ with eigenvalue map $\eta$.

  1. $(a)$

Let

$\displaystyle x=\begin{pmatrix}1&1/h\\\
0&1\end{pmatrix}\qquad\text{and}\qquad y=\begin{pmatrix}1&0\\\
p^{\alpha}n&1\end{pmatrix}$

be the matrices of Section 2.1. Then

$\displaystyle f|U_{p}|_{k}x^{p}$
$\displaystyle=f|_{k}x|U_{p}\qquad\text{and}\qquad
f|U_{p}|_{k}y^{p}=f|_{k}y|U_{p}.$

  2. $(b)$

Let $\Gamma^{\prime}$ be the $p^{\beta}n|h$-type group such that
$\beta=\max\\{1,\alpha-1\\}$ and $e\in\operatorname{AL}(\Gamma^{\prime})$ if
and only if $e\in\operatorname{AL}(\Gamma)$ and $p\nmid e$. Let
$\eta^{\prime}$ be an be an eigenvalue map on $\Gamma^{\prime}$ such that

$\displaystyle\eta^{\prime}(x^{p})=\eta(x),\qquad\eta^{\prime}(y^{p})=\eta(y),q\quad\eta^{\prime}(W_{e})=\eta(W_{e})\qquad\text{for}\quad
e\in\operatorname{AL}(\Gamma^{\prime}).$

Then $f|U_{p}$ is on $\Gamma^{\prime}$ with eigenvalue map $\eta^{\prime}$.

###### Proof.

For $0\leq\mu\leq p-1$, let $S_{\mu}=\left(\begin{smallmatrix}1&\mu\\\
0&p\end{smallmatrix}\right)$ denote the matrix appearing in the definition
(2.3) of $U_{p}$. The first identity of part (a) follows from the equation
$S_{\mu}x^{p}=xS_{\mu}$. The second identity follows from the equation

$\displaystyle S_{\mu}y^{p}=\begin{pmatrix}*&*\\\
p^{\alpha}n(-1+p^{2}+p^{\alpha+1}n\mu)&*\end{pmatrix}yS_{\mu},$ (2.4)

where each $*$ is an integer such that the matrix has determinant $1$. Since
$p\nmid h$, we have that $p^{2}\equiv 1\pmod{h}$ (since $h$ is a divisor of
$24$), so the matrix appearing in (2.4) is in $\Gamma_{0}(p^{\alpha}nh)$ and
therefore fixes $f$.

For part (b), note that since $p\nmid h$ and $\beta\leq\alpha+1$, the matrices
$x^{p}$ and $y^{p}$ generate $\Gamma_{0}\big{(}p^{\beta}n|h\big{)}$. Thus part
(b) follows from part (a) and Lemma 2.1. ∎

Let $\sigma_{a}\colon\mu\mapsto\mu^{a}$ be an endomorphism of $\bm{\mu}_{2h}$.
We set $\eta^{\sigma_{a}}=\sigma_{a}\circ\eta$. The preceding lemma says that
if $\Gamma$ is an $pn|h$-type group with $p\nmid nh$ then $U_{p}$ is a map
$M_{k}(\Gamma,\eta)\rightarrow M_{k}(\Gamma,\eta^{\sigma_{p}})$.

Analogous to the decomposition
$M_{k}(\Gamma_{1}(N))=\bigoplus_{\chi}M_{k}(\Gamma_{0}(N),\chi)$ over mod $N$
Dirichlet characters $\chi$, we have the following decomposition of
$M_{k}(\Gamma_{\eta})$.

######  Lemma 2.3.

Let $\Gamma$ be an $n|h$-type group with eigenvalue map $\eta$, such that
$\operatorname{im}\eta=\bm{\mu}_{h^{\prime}}\leq\bm{\mu}_{2h}$. There is a
decomposition

$\displaystyle
M_{k}(\Gamma_{\eta})=\bigoplus_{\begin{subarray}{c}\eta^{\prime}\colon\Gamma\rightarrow\bm{\mu}_{2h}\\\
\ker\eta^{\prime}\supseteq\Gamma_{\eta}\end{subarray}}M_{k}(\Gamma,\eta^{\prime})=\bigoplus_{a=0}^{h^{\prime}-1}M_{k}\big{(}\Gamma,\eta^{\sigma_{a}}\big{)}.$

###### Proof.

Since $\Gamma/\Gamma_{\eta}\cong\operatorname{im}\eta$ is finite and abelian,
the action of $\Gamma$ on $M_{k}(\Gamma_{\eta})$ can be simultaneously
diagonalized. ∎

Let $\Gamma$ be a $p^{\alpha}n|h$-type group with eigenvalue map $\eta$ such
that $\Gamma_{\eta}$ is genus zero and $p\nmid nh$. Lemma 2.2 tells us under
which group $\mathcal{T}|U_{p}$ is modular. We know that $\mathcal{T}|U_{p}$
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
$\Gamma_{0}(N)$ is given by

$\displaystyle\left\\{\frac{a}{b}\colon
b|N,a\in\mathbb{Z}\right\\}/{\sim},\qquad\text{where}\quad\frac{a}{b}\sim\frac{c}{d}\iff
b=d\quad\text{and}\quad a\equiv c\ \left({\rm mod}\ \gcd(b,N/b)\right).$

Moreover, if two cusps $\frac{a}{b}$ and $\frac{c}{d}$ (not necessarily
representatives of the form above) are equivalent under $\Gamma_{0}(N)$ and
$\gcd(a,b)=\gcd(c,d)=1$, then $\gcd(b,N)=\gcd(d,N)$. Both of these facts
follow from [13, Proposition 3.8.3].

In what follows, if $\Gamma\leq\operatorname{GL}_{2}^{+}(\mathbb{R})$ is
commensurable with $\operatorname{SL}_{2}(\mathbb{Z})$ and
$s,s^{\prime}\in\mathbb{P}^{1}(\mathbb{Q})$ are cusps, then we write

$\displaystyle s\stackrel{{\scriptstyle\Gamma}}{{\sim}}s^{\prime}$

to mean that $s$ and $s^{\prime}$ are equivalent under $\Gamma$. If
$\Gamma=\Gamma_{0}(N)$ then we simply write $s\stackrel{{\scriptstyle
N}}{{\sim}}s^{\prime}$.

######  Lemma 2.4.

Let $p\nmid nh$ be prime, and let $\Gamma$ be a $p^{\alpha}n|h$-type group
with eigenvalue map $\eta$. Let $\Gamma^{\prime}$ and $\eta^{\prime}$ be the
$p^{\beta}n|h$-type group and eigenvalue map defined in Lemma 2.2(b). Suppose
$f$ is a weakly holomorphic modular form on $\Gamma$ with eigenvalue map
$\eta$, so that $f|U_{p}$ is on $\Gamma^{\prime}$ with eigenvalue map
$\eta^{\prime}$ by Lemma 2.2.

  1. $(a)$

If $f$ is bounded away from $\infty$ then the poles of $f|U_{p}$ are supported
on the cusps

$\displaystyle\\{\infty\\}\cup\big{\\{}s\colon s\stackrel{{\scriptstyle
p}}{{\sim}}0\big{\\}}\quad\text{of}\quad\Gamma^{\prime}_{\eta^{\prime}}.$

  2. $(b)$

Suppose that $p\nmid e$ for all $e\in\operatorname{AL}(\Gamma)$. If the poles
of $f$ are supported on $\\{\infty\\}\cup\\{s\colon s\stackrel{{\scriptstyle
p}}{{\sim}}0\\}$ then the same is true of $f|U_{p}$.

  3. $(c)$

Suppose that $\alpha\geq 2$ and $p\nmid e$ for all
$e\in\operatorname{AL}(\Gamma)$. If $f$ is bounded away from $\infty$ and
$f|U_{p}$ is bounded at $\infty$, then $f|U_{p}$ is constant.

###### Proof.

We begin with part (a). Suppose that $f$ is bounded away from $\infty$, and
suppose that $f|U_{p}$ is unbounded at the cusp $s$. If
$s\stackrel{{\scriptstyle p}}{{\sim}}0$ then we are done. Thus we may assume
that $s\stackrel{{\scriptstyle p}}{{\sim}}\infty$. Also, by (2.3), we must
have $S_{\mu}\cdot s\stackrel{{\scriptstyle\Gamma_{\eta}}}{{\sim}}\infty$ for
some $\mu$. Equivalently, $s\stackrel{{\scriptstyle
p^{\alpha}nh}}{{\sim}}S_{\mu}^{-1}A\cdot\infty$ for some $A$ that can be
expressed as a word in the matrices

$\displaystyle x=\begin{pmatrix}1&1/h\\\ 0&1\end{pmatrix},\qquad
y=\begin{pmatrix}1&0\\\ p^{\alpha}n&1\end{pmatrix},\qquad\text{and}\qquad
W_{e}\in\Gamma_{\eta}$

of Section 2.1. Since $W_{e}$ normalizes $\Gamma_{0}(p^{\alpha}n|h)$ (see [17,
Theorem 2.7]), we may write all of the $W_{e}$’s on the right of $A$. By
cancelling $W_{p^{\alpha}}$’s, we can moreover demand that $p\nmid e$ for all
$W_{e}$ appearing in $A$, for otherwise we would have
$A\cdot\infty\stackrel{{\scriptstyle p}}{{\sim}}0$ while $S_{\mu}\cdot
s\stackrel{{\scriptstyle p}}{{\sim}}\infty$. As in the proof of Lemma 2.2, we
can write $S_{\mu}^{-1}A=A^{\prime}S_{\mu}^{-1}V$ for some
$V\in\Gamma_{0}(p^{\alpha}nh)$, where $A^{\prime}$ is obtained from $A$ by
replacing each $x$ with $x^{p}$ and each $y$ with $y^{p}$. Hence
$s\stackrel{{\scriptstyle
p^{\alpha}nh}}{{\sim}}A^{\prime}S_{\nu}^{-1}\cdot\infty$.

If $\alpha\leq 1$ and $\beta=1$ then either
$S_{\nu}^{-1}\cdot\infty\stackrel{{\scriptstyle p^{\beta}nh}}{{\sim}}\infty$
or $\frac{1}{nh}$. But since
$A^{\prime}\cdot\frac{1}{nh}\stackrel{{\scriptstyle p}}{{\sim}}0$, we must
have $S_{\nu}^{-1}\cdot\infty\stackrel{{\scriptstyle
p^{\beta}nh}}{{\sim}}\infty$. Hence $s\stackrel{{\scriptstyle
p^{\beta}nh}}{{\sim}}A^{\prime}\cdot\infty\stackrel{{\scriptstyle\Gamma^{\prime}_{\eta^{\prime}}}}{{\sim}}\infty$,
as desired. The case $\alpha\geq 2$ is dealt with similarly, completing the
proof of (a).

A similar argument gives part (b), where we must now use the fact that $p\nmid
e$ for $e\in\operatorname{AL}(\Gamma)$ to show that $p\nmid e$ for all $W_{e}$
appearing in $A$. For part (c), one finds that $f|U_{p}$ may only have a pole
at the cusp $\infty$; 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 $\mathcal{T}$ for which
$\mathcal{T}|U_{p}=0$. For the Hauptmoduln from monstrous moonshine to which
this applies, this property also follows from [30, Lemma 3.2]. Furthermore, if
$\Gamma$ is a $n|h$-type group and $\eta$ is an eigenvalue map with
$\eta(x)={\rm e}^{2\pi{\rm i}m/h}$, then any meromorphic modular form
$f=\sum\limits_{n=m}^{\infty}a(n)q^{n}$ on $\Gamma$ with eigenvalue map $\eta$
has $a(n)=0$ if $n\not\equiv m\ \left({\rm mod}\ h\right)$, since $x$ sends
$q\mapsto{\rm e}^{2\pi{\rm i}m/h}q$. In particular, if $h\equiv 0\ \left({\rm
mod}\ p\right)$ and $\mathcal{T}$ is the Hautpmodul on $\Gamma_{\lambda}$,
then $f|U_{p}=0$.

Inspection of Table LABEL:tbl:full_annihilation shows that the only monster
Hauptmoduln with $\mathcal{T}|U_{p}=0$ are those with $h\equiv 0\ \left({\rm
mod}\ p\right)$ and those coming from Lemma 2.4(c).

In Section 3 we will need a modular form $g$ on an $n|h$-type group $\Gamma$
such that the zeros of $g$ can cancel the poles of $\mathcal{T}|U_{p}$, 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
$\Delta(\tau)=q\prod\limits_{n\geq 1}(1-q^{n})^{24}$. If a modular form
$f=\sum\limits_{n=m}^{\infty}a(n)q^{n}$ has rational coefficients, we set
$v_{p}(f)=\inf\limits_{n}v_{p}(a_{n})$ where
$v_{p}(a_{n})=\sup\\{r\in\mathbb{Z}\colon p^{r}\mid a_{n}\\}$.

######  Lemma 2.6.

Let $\Gamma$ be a $pn|h$-type group where $p\nmid nh$ is prime and $p\nmid e$
for all $e\in\operatorname{AL}(\Gamma)$. Let $m=\\#\operatorname{AL}(\Gamma)$
and set $a=12m(p-1)$. Then there is a modular form $g$ on $\Gamma$ of weight
$a$ with rational coefficients such that

  1. $(a)$

$g\equiv 1\ \left({\rm mod}\ p\right)$,

  2. $(b)$

$v_{p}(g|_{a}W_{p})\geq 6m(p+1)$,

  3. $(c)$

As a function on $\Gamma_{0}(pnh)$, $g$ vanishes to order $\geq
m\big{(}p^{2}-1\big{)}$ at every cusp equivalent to $0$ under $\Gamma_{0}(p)$.

Specifically, $g$ can be chosen to be

$\displaystyle\prod_{e\in\operatorname{AL}(\Gamma)}\left.\frac{\Delta(h\tau)^{p}}{\Delta(ph\tau)}\right|_{12(p-1)}W_{e}.$

###### Proof.

First, let

$\displaystyle g_{p}(\tau)=\frac{\Delta(h\tau)^{p}}{\Delta(ph\tau)}$

and $a_{p}=12(p-1)$. Note that $g_{p}(\tau)$ is invariant under
$\Gamma_{0}(pn|h)$. For any $e\in\operatorname{AL}(\Gamma)$

$\displaystyle(g_{p}|_{a_{p}}W_{e})(\tau)=\frac{(\Delta(h\tau)|_{12}W_{e})^{p}}{\Delta(ph\tau)|_{12}W_{e}}=\left(\frac{h*e}{h}\right)^{6(p-1)}\frac{\Delta((h*e)\tau)^{p}}{\Delta(p(h*e)\tau)}$
(2.5)

by (2.2). On the other hand, we see that

$\displaystyle(g_{p}|_{a_{p}}W_{pe})(\tau)=p^{6(p+1)}\left(\frac{h*e}{h}\right)^{6(p-1)}\frac{\Delta(p(h*e)\tau)^{p}}{\Delta((h*e)\tau)}.$
(2.6)

Since $\Delta$ is nonzero on $\mathbb{H}$, (2.5) shows that
$g_{p}|_{a_{p}}W_{e}$ is a modular form on $\Gamma_{0}(pnh)$ with rational
coefficients. Moreover, (2.6) shows that each $g_{p}|_{a_{p}}W_{pe}$ for
$e\in\operatorname{AL}(\Gamma)$ vanishes to order $(h*e)\big{(}p^{2}-1\big{)}$
at $\infty$, so that each $g_{p}|_{a_{p}}W_{e}$ vanishes to order $\geq
p^{2}-1$ at the cusps $s\stackrel{{\scriptstyle p}}{{\sim}}0$.

Since $(1-q^{n})^{p}\equiv 1-q^{np}\ \left({\rm mod}\ p\right)$, we see that
$g_{p}|_{a_{p}}W_{e}\equiv\big{(}\frac{h*e}{h}\big{)}^{6(p-1)}\equiv 1\
\left({\rm mod}\ p\right)$. Moreover $v_{p}(g_{p}|_{a_{p}}W_{p})=6(p+1)$.
Thus, we may set

$\displaystyle g=\prod_{e\in\operatorname{AL}(\Gamma)}g_{p}|_{a_{p}}W_{e},$

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 [37,
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.

######  Lemma 2.8 (Sturm’s bound [39]).

Let $f\in M_{k}(\Gamma_{0}(N))$ with integer coefficients $a_{n}$. If $p\mid
a_{n}$ for $p\leq(k/12)[\operatorname{SL}_{2}(\mathbb{Z}):\Gamma_{0}(N)]$,
then $p\mid a_{n}$ for all $n$ .

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 $\mathcal{T}|U_{p}$.

######  Lemma 2.9.

Let $f$ be a weakly holomorphic function on $X_{0}(p^{\alpha}N)$, where
$p\nmid N$, and let $\beta=\max\\{\alpha,1\\}$. If $r$ is the maximum order of
a pole of $f$ on $X_{0}(p^{\alpha}N)$, then the poles of $f|U_{p}$ as a
function on $X_{0}\big{(}p^{\beta}nh\big{)}$ have order at most $rp^{2}$ when
$\alpha=0$, 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 $p$. Thus, for the case $\alpha=0$, 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

${\left(\Gamma_{0}(p^{\beta}nh)\cap\gamma^{-1}\Gamma_{0}(p^{\beta}nh)\gamma\right)\backslash\mathbb{H}^{*}}$${\left(\gamma\Gamma_{0}(p^{\beta}nh)\gamma^{-1}\cap\Gamma_{0}(p^{\beta}nh)\right)\backslash\mathbb{H}^{*}}$${X_{0}(p^{\beta}nh)}$${X_{0}(p^{\beta}nh)}$$\scriptstyle{\sim}$

where $\gamma=\left(\begin{smallmatrix}1&0\\\ 0&p\end{smallmatrix}\right)$.
The projections have degree $p$, so $\mathcal{T}$ pulls back to a function on
$\big{(}\gamma\Gamma_{0}\big{(}p^{\beta}nh\big{)}\gamma^{-1}$
$\cap\Gamma_{0}(p^{\beta}nh)\big{)}\backslash\mathbb{H}^{*}$ with poles at
most degree $rp^{2}$ when $\alpha=0$ and $rp$ else. The other maps of the
correspondence, which are pullback by the isomorphism and trace down to
$X_{0}(p^{\beta}nh)$, do not increase the maximum pole order, so the claim
follows. ∎

###  2.3 Trace formulas

Following Serre’s idea [37], 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 $\Gamma$ and $\Gamma^{\prime}$ are subgroups of
$\operatorname{GL}_{2}^{+}(\mathbb{R})$ with $\Gamma$ a finite index subgroup
of $\Gamma^{\prime}$. We define the trace
$\operatorname{Tr}_{\Gamma\backslash\Gamma^{\prime}}$ from $\Gamma$ to
$\Gamma^{\prime}$ to be the operation

$\displaystyle\operatorname{Tr}_{\Gamma\backslash\Gamma^{\prime}}f=\sum_{i=1}^{m}f|_{k}\gamma_{i},$
(2.7)

where $\\{\gamma_{1},\dots,\gamma_{m}\\}$ is a system of right coset
representatives for $\Gamma\backslash\Gamma^{\prime}$. If $f$ is modular on
$\Gamma$ then $\operatorname{Tr}_{\Gamma\backslash\Gamma^{\prime}}$ is modular
on the larger group $\Gamma^{\prime}$. When $\Gamma=\Gamma_{0}(N)$ and
$\Gamma^{\prime}=\Gamma_{0}(N^{\prime})$, we simply write
$\operatorname{Tr}_{N\backslash N^{\prime}}$ for
$\operatorname{Tr}_{\Gamma\backslash\Gamma^{\prime}}$.

First consider $\Gamma=\Gamma_{0}(pN)$ and $\Gamma^{\prime}=\Gamma_{0}(N)$.
The following generalizes [37, Lemma 7].

######  Lemma 2.10.

Let $p\nmid N$ be prime.

  1. $(a)$

A set of representatives for $\Gamma_{0}(pN)\backslash\Gamma_{0}(N)$ is given
by

$\displaystyle\left\\{\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\right\\}\cup\left\\{\begin{pmatrix}1&\lambda\\\
N&N\lambda+1\end{pmatrix}\colon 1\leq\lambda\leq p\right\\}.$

  2. $(b)$

If $f$ is a weight $k$ modular form on $\Gamma_{0}(pN)$ then

$\displaystyle\operatorname{Tr}_{pN\backslash
N}f=f+p^{1-k/2}(f|_{k}W_{p})|U_{p}$

where $W_{p}$ is the corresponding Atkin–Lehner involution on
$\Gamma_{0}(pN)$.

###### Proof.

Since $[\Gamma_{0}(N):\Gamma_{0}(pN)]=p+1$, part (a) follows upon checking
that the representatives are inequivalent modulo $\Gamma_{0}(pN)$. One can
also check that for any $1\leq\lambda\leq p$, if $\mu\equiv N^{-1}\ \left({\rm
mod}\ p\right)$, then

$\displaystyle\begin{pmatrix}1&\lambda\\\
N&N\lambda+1\end{pmatrix}=VW_{p}\begin{pmatrix}1/p&(\lambda+\mu)/p\\\
0&1\end{pmatrix}$

for some $V\in\Gamma_{0}(pN)$. Part (b) follows from this. ∎

The remainder of this section will extend Lemma 2.10 to the more general
context we need, for example tracing from $\Gamma_{0}(pN){+}e,\dots$ to
$\Gamma_{0}(N){+}e,\dots$ for $p\nmid N$. More precisely, for a prime $p\nmid
nh$, suppose that $\Gamma$ is a $pn|h$-type group with eigenvalue map $\eta$
with $p\nmid e$ for all $e\in\operatorname{AL}(\Gamma)$. Let $\Gamma^{\prime}$
be the $n|h$-type group such that
$\operatorname{AL}(\Gamma^{\prime})=\operatorname{AL}(\Gamma)$. We have
$\Gamma\subset\Gamma^{\prime}$, and can take $\eta^{\prime}$ to be the
eigenvalue map on $\Gamma^{\prime}$ with $\eta^{\prime}|\Gamma=\eta$. Since
$\Gamma^{\prime}$ is generated by

$\displaystyle x=\begin{pmatrix}1&1/h\\\ 0&1\end{pmatrix},\qquad
y=\begin{pmatrix}1&0\\\ n&1\end{pmatrix},\qquad W_{e}\ \text{such that}\quad
e\in\operatorname{AL}(\Gamma^{\prime})=\operatorname{AL}(\Gamma)$

as in Section 2.1, this uniquely determines $\eta^{\prime}$. Then [17, Lemma
2.3] and [17, Theorem 2.7] together imply that the inclusion of
representatives

$\displaystyle\iota\colon\
\Gamma/\Gamma_{0}(pnh)\hookrightarrow\Gamma^{\prime}/\Gamma_{0}(nh)$

is an isomorphism. We set
$H=\operatorname{im}(\iota|\Gamma_{\eta})\leq\Gamma^{\prime}$, and consider
the restricted isomorphism

$\displaystyle\iota|\Gamma_{\eta}\colon\
\Gamma_{\eta}/\Gamma_{0}(pnh)\stackrel{{\scriptstyle\sim}}{{\hookrightarrow}}H/\Gamma_{0}(nh).$
(2.8)

We have that $H\leq\Gamma^{\prime}_{\eta^{\prime}}$. Moreover,
$\operatorname{im}\eta^{\prime}=\operatorname{im}\eta$, so
$[\Gamma^{\prime}:\Gamma^{\prime}_{\eta^{\prime}}]=[\Gamma:\Gamma_{\eta}]=[\Gamma^{\prime}:H]$
and thus $\Gamma^{\prime}_{\eta^{\prime}}=H$. We have nearly proved the
following lemma.

######  Lemma 2.11.

Let $\Gamma$ be a $pn|h$-type group with eigenvalue map $\eta$ such that
$p\nmid e$ for all $e\in\operatorname{AL}(\Gamma)$. Let $\Gamma^{\prime}$ be
an $n|h$-type group with
$\operatorname{AL}(\Gamma^{\prime})=\operatorname{AL}(\Gamma)$ with eigenvalue
map $\eta^{\prime}$ such that $\eta^{\prime}|\Gamma=\eta$. Then for any weight
$k$ modular form on $\Gamma_{\eta}$ we have

$\displaystyle\operatorname{Tr}_{\Gamma_{\eta}\backslash\Gamma^{\prime}_{\eta^{\prime}}}f=\operatorname{Tr}_{pnh\backslash
nh}f=f+p^{1-k/2}(f|_{k}W_{p})|U_{p}.$

###### Proof.

We need to show
$\operatorname{Tr}_{\Gamma_{\eta}\backslash\Gamma_{\eta^{\prime}}}f=\operatorname{Tr}_{pnh\backslash
nh}f$. Let $\\{\gamma_{i}\\}$ be any set of representatives for
$\Gamma_{0}(pnh)\backslash\Gamma_{0}(nh)$. Then $\\{\gamma_{i}\\}$ is also a
set of representatives for
$\Gamma_{\eta}\backslash\Gamma^{\prime}_{\eta^{\prime}}$. Indeed, by the
isomorphism (2.8) we have
$[\Gamma^{\prime}_{\eta^{\prime}}:\Gamma_{\eta}]=[\Gamma_{0}(nh):\Gamma_{0}(pnh)]$
so it suffices to check that no two $\gamma_{i}$ are equivalent. Suppose
$\gamma_{i}\gamma_{j}^{-1}\in\Gamma_{\eta}$. Then
$\gamma_{i}\gamma_{j}^{-1}\in\Gamma\cap\Gamma_{0}(nh)=\Gamma_{0}(pnh)$ so that
$\gamma_{i}=\gamma_{j}$ as desired. The formula then follows from Lemma 2.10.
∎

######  Remark 2.12.

Lemma 2.11 only assumes that $f$ is on $\Gamma_{\eta}$. Under the stronger
assumption that $f$ is on $(\Gamma,\eta)$, we obtain the finer result that
$\operatorname{Tr}_{pnh\backslash nh}f$ is on
$(\Gamma^{\prime},\eta^{\prime})$. To see this, let $x$ and $Y=y^{p}$ be the
generators of $\Gamma_{0}(pn|h)/\Gamma_{0}(pnh)$, and choose appropriate
representatives of $W_{e}$ which normalize $\Gamma_{0}(nh)$. Then, apply $x$,
$Y$, and $W_{e}$ to both sides of (2.7).

##  3 $\bm{p}$-adic modular forms

In this section, we extend of Serre’s theory of $p$-adic modular forms from
[37] 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 $p\geq 5$, and let
$\Gamma$ be an $n|h$-type group with eigenvalue map $\eta$. We first study
$M_{k}(\Gamma,\eta)$ and its $p$-adic completion. In Section 3.1 we prove that
Hauptmoduln $\mathcal{T}$ become $p$-adic modular forms on some
$(\Gamma^{\prime},\eta^{\prime})$ 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 $\Gamma_{0}(N){+}e,\dots$ can take $h=1$ and let
eigenvalue maps be identically $1$.

For an $n|h$-type group $\Gamma$ with eigenvalue map
$\eta\colon\Gamma\rightarrow\bm{\mu}_{2h}$, we first define the spaces:

  1. (1)

$M_{k}^{\mathbb{Q}}(\Gamma,\eta)=M_{k}(\Gamma,\eta)\cap\mathbb{Q}[\\![q]\\!]$,
the $\mathbb{Q}$-vector space of modular forms with rational $q$-expansion;

  2. (2)

$M_{k}^{(p)}(\Gamma,\eta)=M_{k}(\Gamma,\eta)\cap\mathbb{Z}_{(p)}[\\![q]\\!]$,
the $\mathbb{Z}_{(p)}$-module of modular forms with $p$-integral
$q$-expansion; and

  3. (3)

$\tilde{M}_{k}(\Gamma,\eta)=M_{k}^{(p)}(\Gamma,\eta)\otimes\mathbb{F}_{p}$,
the $\mathbb{F}_{p}$-vector space obtained by reducing
$M_{k}^{(p)}(\Gamma,\eta)$ mod $p$.

Similarly define $M_{k}^{\mathbb{Q}}(\Gamma_{\eta})$,
$M_{k}^{(p)}(\Gamma_{\eta})$, and $\tilde{M}_{k}(\Gamma_{\eta})$. If $f$
reduces mod $p$ to a form in $\tilde{M}_{k}(\Gamma,\eta)$, we abuse notation
and write $f\in\tilde{M}_{k}(\Gamma,\eta)$, and similarly for $\Gamma_{\eta}$.
We focus on $M_{k}(\Gamma,\eta)$, and the corresponding results for
$M_{k}(\Gamma_{\eta})$ follow from Lemma 2.3.

Following [37], we define a $p$-adic modular form on $(\Gamma,\eta)$ to be a
$q$-expansion $f\in\mathbb{Q}_{p}[\\![q]\\!]$ admitting a sequence $f_{m}\in
M_{k_{m}}^{\mathbb{Q}}(\Gamma,\eta)$ that converges $p$-adically with
$v_{p}(f_{m}-f)\rightarrow\infty$, i.e.,

$\displaystyle\lim_{m\rightarrow\infty}f_{m}=f.$

Similarly, a $p$-adic modular form on $\Gamma$ is a $p$-adic modular form on
$(\Gamma,\bm{1})$.

For any $N$, if $f$, $f^{\prime}$ are modular forms on $\Gamma_{0}(N)$ of
weight $k$, $k^{\prime}$ and $f\equiv f^{\prime}\ \left({\rm mod}\
p^{n}\right)$ then $k\equiv k^{\prime}\ \left({\rm mod}\ p^{n-1}(p-1)\right)$
(see [28, Corollary 4.4.2]). It follows that the weight of a $p$-adic modular
form on $\Gamma_{0}(N)$, defined as the limit of the $k_{m}$ in the space

$\displaystyle\mathfrak{X}=\varprojlim\mathbb{Z}/p^{n}\mathbb{Z}\times\mathbb{Z}/(p-1)\mathbb{Z}=\mathbb{Z}_{p}\times\mathbb{Z}/(p-1)\mathbb{Z},$

exists and does not depend on the choice of sequence $(f_{m})$. The same is
true of $p$-adic modular forms on $(\Gamma,\eta)$, since
$M_{k}(\Gamma,\eta)\subseteq M_{k}(\Gamma_{0}(nh))$. In particular, forms in
$M_{k}(\Gamma,\eta)$ have trivial nebentypus, as required in [28].

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 $\Gamma$ be an $n|h$ type group and $p\nmid nh$ a prime. If $\eta$ is an
eigenvalue map on $\Gamma$, then the twist of $\eta$ is the map
$\eta_{t}\colon\Gamma\rightarrow\bm{\mu}_{2h}$ defined by

$\displaystyle\eta_{t}(\gamma)=\eta(\gamma)\qquad\text{for}\quad\gamma\in\Gamma_{0}(n|h)\qquad\text{and}\qquad\eta_{t}(W_{e})=\left(\frac{e}{p}\right)\eta(W_{e})\qquad\text{for}\quad
e\in\operatorname{AL}(\Gamma),$

where $\left(\frac{\cdot}{p}\right)$ denotes the Legendre symbol.

######  Remark 3.2.

If $\bm{1}$ denotes the trivial eigenvalue map $\bm{1}(\gamma)=1$ for all
$\gamma$, then $\eta_{t}=\eta\bm{1}_{t}$ for all $\eta$. Also, if
$\left(\frac{e}{p}\right)=1$ for all $e\in\operatorname{AL}(\Gamma)$ then
$\eta=\eta_{t}$.

Below, we collect some useful facts and begin to see the relationship between
eigenvalue map twists and the weight mod $2(p-1)$. Let $E_{k}(\tau)$ denote
the weight $k$ _Eisenstein series_ with constant term $1$.

######  Proposition 3.3.

Let $\Gamma$ be an $n|h$-type group with eigenvalue map $\eta$, and let $p\geq
5$ be prime with $p\nmid nh$.

  1. $(a)$

Let $F(\tau)=E_{p-1}(h\tau)$. Then

$\displaystyle\hat{F}=\sum_{e\in\operatorname{AL}(\Gamma)}\left(\frac{e}{p}\right)F|_{p-1}W_{e}\in
M_{p-1}(\Gamma,\bm{1}_{t})$

is congruent to $\\#\operatorname{AL}(\Gamma)$ mod $p$.

  2. $(b)$

For all $k$, we have the inclusions

$\displaystyle\tilde{M}_{k}(\Gamma,\eta)\subseteq\tilde{M}_{k+p-1}(\Gamma,\eta_{t})\subseteq\tilde{M}_{k+2(p-1)}(\Gamma,\eta)\subseteq
M_{k+3(p-1)}(\Gamma,\eta_{t})\subseteq\cdots.$

  3. $(c)$

Suppose $\eta\neq\eta_{t}$. Then for $f\in M_{k}^{\mathbb{Q}}(\Gamma,\eta)$
and $f^{\prime}\in M_{k^{\prime}}^{\mathbb{Q}}(\Gamma,\eta)$ we have that

$\displaystyle 0\not\equiv f\equiv
f^{\prime}\pmod{p^{n}}\qquad\text{implies}\qquad k\equiv
k^{\prime}\pmod{2p^{n-1}(p-1)}.$

  4. $(d)$

If $\eta\neq\eta_{t}$ then the weight of a $p$-adic modular form on
$(\Gamma,\eta)$ is well-defined as an element of
$\hat{\mathfrak{X}}=\mathbb{Z}_{p}\times\mathbb{Z}/(2p-2)\mathbb{Z}$.

###### Proof.

Since $F$ is invariant under $\Gamma_{0}(n|h)$, the statement that $\hat{F}\in
M_{p-1}(\Gamma,\bm{1}_{t})$ becomes

$\displaystyle\hat{F}|W_{e}=\left(\frac{e}{p}\right)F,$

which follows from multiplicativity of the Legendre symbol. We also have

$\displaystyle
F|_{p-1}W_{e}(\tau)=\left(\frac{h*eh^{2}}{h}\right)^{\frac{p-1}{2}}E_{p-1}\big{(}\big{(}h*eh^{2}\big{)}\tau\big{)}\equiv\left(\frac{e}{p}\right)\
\left({\rm mod}\ p\right)$

since $E_{p-1}\equiv 1\ \left({\rm mod}\ p\right)$ (see [37, Section 1]). Thus
$\hat{F}\equiv\sum\limits_{e\in\operatorname{AL}(\Gamma)}1\ \left({\rm mod}\
p\right)$, giving part (a). Since $p\neq 2$, this implies
$1\in\tilde{M}_{p-1}(\Gamma,\bm{1}_{t})$, giving part (b) since
$\eta\bm{1}_{t}=\eta_{t}$ and

$\displaystyle M_{k}(\Gamma,\eta)\cdot
M_{k^{\prime}}(\Gamma,\eta^{\prime})=M_{k+k^{\prime}}(\Gamma,\eta\eta^{\prime}).$

For part (c), we already know $k\equiv k^{\prime}\pmod{p^{n-1}(p-1)}$. Assume
without loss of generality that $k\leq k^{\prime}$. If $k\not\equiv
k^{\prime}\ \left({\rm mod}\ 2p-2\right)$ then by part (b) there exists $g\in
M_{k^{\prime}}^{\mathbb{Q}}(\Gamma,\eta_{t})$ with $g\equiv f$ $\ \left({\rm
mod}\ p\right)$. Since $\eta\neq\eta_{t}$, let $e\in\operatorname{AL}(\Gamma)$
be a quadratic nonresidue so that

$\displaystyle\eta(W_{e})f^{\prime}=f^{\prime}|_{k^{\prime}}W_{e}\equiv
g|_{k^{\prime}}W_{e}=-\eta(W_{e})g\equiv-\eta(W_{e})f^{\prime}\ \left({\rm
mod}\ p\right),$

which implies $f^{\prime}\equiv f\equiv 0\ \left({\rm mod}\ p\right)$. Part
(d) then follows. ∎

This motivates the following $\mathbb{F}_{p}$-spaces, which incorporate these
twisted inclusions. Set

$\displaystyle\tilde{M}(\Gamma,\eta)^{\alpha}=\bigcup_{k\equiv\alpha\
\left({\rm mod}\
2p-2\right)}\tilde{M}_{k}(\Gamma,\eta)\quad\cup\quad\bigcup_{k\equiv\alpha+p-1\
\left({\rm mod}\ 2p-2\right)}\tilde{M}_{k}(\Gamma,\eta_{t})$

for $\alpha\in\mathbb{Z}/(2p-2)\mathbb{Z}$. If every
$e\in\operatorname{AL}(\Gamma)$ is a quadratic residue mod $p$, we have
$\eta=\eta_{t}$, and $\tilde{M}(\Gamma,\eta)^{\alpha}$ only depends on
$\alpha$ mod $p-1$.

###  3.1 Producing $\bm{p}$-adic modular forms from Hautpmoduln

Let $\Gamma$ be a $p^{\alpha}n|h$-type group with eigenvalue map $\eta$ where
$p\nmid nh$ is prime. Suppose $\Gamma_{\eta}$ is genus zero, and its
Hautpmodul $\mathcal{T}$ is on $\Gamma$ with eigenvalue map $\eta$. We show
that for some $\beta$, $\mathcal{T}|U_{p}^{\beta}$ is a $p$-adic modular form
on $(\Gamma^{\prime},\eta^{\prime})$ for some specified $n|h$-type group
$\Gamma^{\prime}$ and eigenvalue map $\eta^{\prime}$. We can take $\beta=1$
whenever $\alpha\leq 3$.

######  Lemma 3.4.

Let $\Gamma$ be a $p^{\alpha}n|h$-type group for $p\nmid nh$ prime, and $\eta$
be an eigenvalue map on $\Gamma$. Let $f$ be a weight $0$ weakly holomorphic
modular form on $(\Gamma,\eta)$ that is holomorphic away from $\infty$. Let
$\Gamma^{\prime}$ be the $n|h$-type group with
$e\in\operatorname{AL}(\Gamma^{\prime})$ if and only if
$e\in\operatorname{AL}(\Gamma)$ and $p\nmid e$. Let
$\beta=\max\\{1,\alpha-1\\}$, and let $\eta^{\prime}$ be the eigenvalue map on
$\Gamma^{\prime}$ such that

$\displaystyle\eta^{\prime}(x)=\eta\big{(}x^{p^{\beta}}\big{)},\qquad\eta^{\prime}(y^{\prime})=\eta\big{(}y^{p^{\alpha+\beta}}\big{)},\qquad\eta^{\prime}(W_{e})=\eta(W_{e})\qquad\text{for}\quad
e\in\operatorname{AL}(\Gamma^{\prime}),$

where $x$, $y$, $y^{\prime}$ are the generators of $\Gamma_{0}(p^{\alpha}n|h)$
and $\Gamma_{0}(n|h)$ given by

$\displaystyle x=\begin{pmatrix}1&h\\\ 0&1\end{pmatrix},\qquad
y=\begin{pmatrix}1&0\\\ p^{\alpha}n&1\end{pmatrix},\qquad
y^{\prime}=\begin{pmatrix}1&0\\\ n&1\end{pmatrix}.$

Then, $f|U_{p}^{\beta}$ is $p$-adic modular form of weight $0$ on
$(\Gamma^{\prime},\eta^{\prime})$.

###### Proof.

By Lemma 2.2, $f|U_{p}^{\beta}$ is a weakly holomorphic modular form on
$(\Gamma,\nu)$ where $\Gamma$ is the $pn|h$-type group $\Gamma$ where
$e\in\operatorname{AL}(G)$ if and only if $e\in\operatorname{AL}(\Gamma)$ and
$p\nmid e$, and $\nu$ satisfies

$\displaystyle\nu(x)=\eta\big{(}x^{p^{\beta}}\big{)},\qquad\nu(Y)=\eta\big{(}y^{p^{\alpha+\beta-1}}\big{)},\qquad\nu(W_{e})=\eta(W_{e}),\qquad\text{where}\quad
Y=\begin{pmatrix}1&0\\\ pn&1\end{pmatrix}.$

The remainder of the proof follows [37, Theorem 10]. To show that
$f|U_{p}^{\beta}$ is a $p$-adic modular form on
$(\Gamma^{\prime},\eta^{\prime})$, we set for $m\geq 0$

$\displaystyle
f_{m}=\operatorname{Tr}_{G_{\nu}\backslash\Gamma^{\prime}_{\eta^{\prime}}}\big{(}f|U_{p}^{\beta}g^{p^{m}}\big{)}=\operatorname{Tr}_{pnh\backslash
nh}\big{(}f|U_{p}^{\beta}g^{p^{m}}\big{)},$

where $g$ is the modular form on $\Gamma$ given by Lemma 2.6. Since

$\displaystyle\eta^{\prime}(x)=\nu(x)=\eta\big{(}x^{p^{\beta}}\big{)}\qquad\text{and}\qquad\eta^{\prime}(y^{\prime})=\nu(Y^{p})=\eta\big{(}y^{p^{\alpha+\beta}}\big{)}\qquad\text{and}$
$\displaystyle\eta^{\prime}(W_{e})=\nu(W_{e})=\eta(W_{e})\qquad\text{for}\quad
e\in\operatorname{AL}(\Gamma^{\prime})$

we know $f_{m}$ is a weakly holomorphic modular form on
$(\Gamma^{\prime},\eta^{\prime})$ by Lemma 2.11. Lemma 2.4 shows that
$f|U_{p}^{\beta}$ has poles only at the cusps equivalent to $0$ on
$\Gamma_{0}(p)$, and since $g$ has zeros at all such cusps, we know
$f|U_{p}^{\beta}g^{m}$ 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 $f_{m}\rightarrow
f|U_{p}^{\beta}$ in the $p$-adic limit. We compute that

$\displaystyle
f_{m}-f|U_{p}^{\beta}=\big{(}f_{m}-f|U_{p}^{\beta}g^{p^{m}}\big{)}+f|U_{p}^{\beta}\big{(}g^{p^{m}}-1\big{)},$
$\displaystyle
v_{p}\big{(}f_{m}-f|U_{p}^{\beta}\big{)}\geq\min\big{\\{}v_{p}\big{(}f_{m}-f|U_{p}^{\beta}g^{p^{m}}\big{)},v_{p}\big{(}f|U_{p}^{\beta}\big{)}+v_{p}\big{(}g^{p^{m}}-1\big{)}\big{\\}}.$

Since $g^{p^{m}}\equiv 1\ \left({\rm mod}\ p^{m+1}\right)$, we have
$v_{p}\big{(}f|U_{p}^{\beta}\big{)}+v_{p}\big{(}g^{p^{m}}-1\big{)}\geq m+1$.
Lemma 2.11 implies

$\displaystyle
f_{m}-f|U_{p}^{\beta}g^{p^{m}}=p^{1-ap^{m}/2}\big{(}f|U_{p}^{\beta}g^{p^{m}}|_{ap^{m}}W_{p}\big{)}|U_{p},$

and since applying $U_{p}$ does not decrease the power of $p$ dividing a
$q$-expansion, we have

$\displaystyle v_{p}\big{(}f_{m}-f|U_{p}^{\beta}g^{p^{m}}\big{)}$
$\displaystyle\geq
1-\frac{ap^{m}}{2}+v_{p}\big{(}f|U_{p}^{\beta}|_{0}W_{p}\big{)}+p^{m}v_{p}(g|_{a}W_{p})$
$\displaystyle\geq
1+v_{p}\big{(}f|U_{p}^{\beta}|_{0}W_{p}\big{)}+p^{m}\left(v_{p}(g|_{a}W_{p})-\frac{a}{2}\right).$

Lemma 2.6 gives $v_{p}(g|_{a}W_{p})>\frac{a}{2}$. Hence,
$v_{p}\big{(}f_{m}-f|U_{p}^{\beta}\big{)}\rightarrow\infty$ as
$m\rightarrow\infty$, as desired. ∎

######  Remark 3.5.

If $\Gamma$ is a $p^{\alpha}n|h$-type group with eigenvalue map $\eta$ such
that $\Gamma_{\eta}$ is genus zero, and the Hauptmodul $\mathcal{T}$ on
$\Gamma_{\eta}$ is on $(\Gamma,\eta)$, then Lemma 3.4 applies. Moreover, since
$\mathcal{T}^{r}$ is on $(\Gamma,\eta^{\sigma_{r}})$, the lemma also applies
to powers of the Hauptmodul. In particular, polynomials in $\mathcal{T}$ are
$p$-adic modular forms on $\Gamma^{\prime}_{\eta^{\prime}}$ after enough
applications of $U_{p}$.

###  3.2 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 [37], 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 $p\geq 5$ prime with $p\nmid nh$. For the Hecke operator $T_{p}$,
we have

$\displaystyle
f|_{k}T_{p}=f|U_{p}+p^{k/2-1}f|_{k}A=f|U_{p}+p^{k-1}f(p\tau),\qquad\text{where}\quad
A=\begin{pmatrix}p&0\\\ 0&1\end{pmatrix},$

so $f|U_{p}\equiv f|_{k}T_{p}\ \left({\rm mod}\ p\right)$ for $k\geq 2$. Since
$T_{p}$ acts on $M_{k}(\Gamma_{0}(nh))$, we know $U_{p}$ acts on
$\tilde{M}_{k}(\Gamma_{0}(nh))$. Furthermore, let $\Gamma$ be an $n|h$-type
group and $\eta$ be an eigenvalue map. Since

$\displaystyle Ax=x^{p}A,\qquad Ay^{p}=yA,\qquad
AW_{e}=W_{e}A\qquad\text{for}\quad p\nmid e,$

Lemma 2.2 implies that $T_{p}\colon M_{k}(\Gamma,\eta)\rightarrow
M_{k}(\Gamma,\eta^{\sigma_{p}})$. Hence
$U_{p}\colon\tilde{M}_{k}(\Gamma,\eta)\rightarrow\tilde{M}_{k}(\Gamma,\eta^{\sigma_{p}})$,
so we consider the space

$\displaystyle\tilde{M}_{k}(\Gamma,[\eta]_{p})=\tilde{M}_{k}(\Gamma,\eta)+\tilde{M}_{k}(\Gamma,\eta^{\sigma_{p}}),$

which is stabilized by $U_{p}$. This sum is direct if and only if
$\eta\neq\eta^{\sigma_{p}}$. We thus set

$\displaystyle\tilde{M}(\Gamma,[\eta]_{p})^{\alpha}=\tilde{M}(\Gamma,\eta)^{\alpha}+\tilde{M}(\Gamma,\eta^{\sigma_{p}})^{\alpha}$

and remind the reader that $\tilde{M}(\Gamma,\eta)^{\alpha}$ already encodes
spaces with twisted eigenvalue map. We next show how $U_{p}$ contracts
$\tilde{M}(\Gamma,[\eta]_{p})^{\alpha}$ onto a finite-dimensional space called
the ordinary space. Ordinary spaces of $p$-adic modular forms were extensively
studied by Hida [25]. We describe our ordinary spaces in the language of
Serre’s $p$-adic modular forms.

######  Proposition 3.6 (ordinary decomposition).

Let $\Gamma$ be an $n|h$-type group with eigenvalue map $\eta$.

  1. $(a)$

We can write

$\displaystyle\tilde{M}(\Gamma,[\eta]_{p})^{\alpha}=\mathfrak{S}(\Gamma,[\eta]_{p})^{\alpha}\oplus\mathfrak{N}(\Gamma,[\eta]_{p})^{\alpha}$

so that $U_{p}$ is bijective on the ordinary space
$\mathfrak{S}(\Gamma,[\eta]_{p})^{\alpha}$ and locally nilpotent on
$\mathfrak{N}(\Gamma,[\eta]_{p})^{\alpha}$; that is, for any
$\mathfrak{N}(\Gamma,[\eta]_{p})^{\alpha}$, we have $f|U_{p}^{n}=0$ for $n$
sufficiently large.

  2. $(b)$

Let $4\leq k\leq p+1$ be such that $k\equiv\alpha\ \left({\rm mod}\
p-1\right)$. If $k\equiv\alpha\ \left({\rm mod}\ 2p-2\right)$ then

$\displaystyle\mathfrak{S}(\Gamma,[\eta]_{p})^{\alpha}\subseteq\tilde{M}_{k}(\Gamma,[\eta]_{p}).$

Otherwise,

$\displaystyle\mathfrak{S}(\Gamma,[\eta]_{p})^{\alpha}\subseteq\tilde{M}_{k}(\Gamma,[\eta_{t}]_{p}).$

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 $f\in\tilde{M}(\Gamma,[\eta]_{p})^{\alpha}$ with
respect to $(\Gamma,\eta)$ is

$\displaystyle w_{\Gamma,\eta}(f)=\min\big{\\{}k\colon
f\in\tilde{M}_{k}(\Gamma,[\eta]_{p})\text{ or
}\tilde{M}_{k}(\Gamma,[\eta_{t}]_{p})\big{\\}}.$

When $(\Gamma,\eta)$ is clear from context, we will simply write $w$ for
$w_{\Gamma,\eta}$. Similarly, the filtration $w_{\Gamma}$ of a modular form
mod $p$ with respect to $\Gamma$ is the filtration with respect to
$(\Gamma,\bm{1})$.

To prove Proposition 3.6 we generalize the following fact from [27].

######  Lemma 3.7.

Suppose $\Gamma=\Gamma_{0}(N)$. Then for modular forms $f$ mod $p$ on $\Gamma$
we have

$\displaystyle w_{\Gamma}(f|U_{p})\leq p+\frac{w_{\Gamma}(f)-1}{p}.$

In particular, $w_{\Gamma}(f|U_{p})<w_{\Gamma}(f)$ if $w_{\Gamma}(f)>p+1$.

We give a suitable modification of Lemma 3.7 for $(\Gamma,\eta)$.

######  Lemma 3.8.

Let $\Gamma$ be an $n|h$-type group with eigenvalue map $\eta$. For
$f\in\tilde{M}(\Gamma,[\eta]_{p})^{\alpha}$,

$\displaystyle w_{\Gamma,\eta}(f|U_{p})\leq p+\frac{w_{\Gamma,\eta}(f)-1}{p}.$

###### Proof.

We first consider the special case $\Gamma=\Gamma_{0}(n|h)$. We proceed by
induction on the finite index $[\Gamma:\Gamma_{0}(nh)]$. Suppose that for some
$\Gamma_{0}(nh)\leq\Gamma^{\prime}\lneq\Gamma_{0}(n|h)$ we have

$\displaystyle w_{\Gamma^{\prime},\eta|_{\Gamma^{\prime}}}(f|U_{p})\leq
p+\frac{w_{\Gamma^{\prime},\eta|_{\Gamma^{\prime}}}(f)-1}{p}.$ (3.1)

Let $T\in\Gamma_{0}(n|h)/\Gamma^{\prime}$ be a representative for a nontrivial
coset. Setting $\Gamma^{\prime\prime}=\langle\Gamma^{\prime},T\rangle$ and
$\eta^{\prime\prime}=\eta|\Gamma^{\prime\prime}$, we will prove that (3.1)
also holds for $(\Gamma^{\prime\prime},\eta^{\prime\prime})$.

Let $f\in
M_{k}^{(p)}(\Gamma^{\prime\prime},[\eta^{\prime\prime}]_{p})\subseteq
M_{k}^{(p)}(\Gamma^{\prime},[\eta|\Gamma^{\prime}]_{p})$ be a modular form of
filtration $k$. By assumption there is some $g\in
M_{k^{\prime}}^{(p)}(\Gamma^{\prime},[\eta|\Gamma^{\prime}]_{p})$ with
$g\equiv f|U_{p}\ \left({\rm mod}\ p\right)$ and of filtration

$\displaystyle k^{\prime}\leq p+\frac{k-1}{p}.$

Let $t$ be the order of $T$ in $\Gamma^{\prime\prime}/\Gamma_{0}(nh)$, and let
$\pi_{m}$ be the projection

$\displaystyle\pi_{m}(g)=\frac{1}{t}\sum_{\ell=0}^{t-1}{\rm e}^{-2\pi{\rm
i}\ell/t}g|_{k^{\prime}}T^{\ell}$

onto the ${\rm e}^{2\pi{\rm i}m/t}$ eigenspace of $T$. Since $p\geq 5$, we
know $p\nmid t$. Thus, if $\varphi$, $\varphi^{\prime}$ have the same weight
and $\varphi\equiv\varphi^{\prime}\ \left({\rm mod}\ p\right)$, then
$\pi_{m}(\varphi)\equiv\pi_{m}(\varphi^{\prime})\ \left({\rm mod}\ p\right)$.

By Proposition 3.3, multiplying by some power of $\hat{F}$ gives
$f^{\prime}\in M_{k}^{(p)}(\Gamma^{\prime},[\eta|\Gamma^{\prime}]_{p})$ with
$f^{\prime}\equiv g$ $\ \left({\rm mod}\ p\right)$. Since $\hat{F}$ is
invariant under $\Gamma_{0}(n|h)$, we have

$\displaystyle\pi_{m}(g)\equiv\pi_{m}(f^{\prime})\equiv\pi_{m}(f)\ \left({\rm
mod}\ p\right).$

Let $m$ be such that $\eta(T)={\rm e}^{2\pi{\rm i}m/t}$ and set

$\displaystyle\pi=\begin{cases}\pi_{m}&\text{if }\pi_{pm}=\pi_{m},\\\
\pi_{m}\oplus\pi_{pm}&\text{otherwise},\end{cases}$

which is projection onto the span of the eigenspaces of $T$ specified by
$[\eta]_{p}$. Thus, we have $\pi(g)\equiv\pi(f)=f\ \left({\rm mod}\ p\right)$.
Then $\pi(g)\in
M_{k^{\prime}}^{(p)}(\Gamma^{\prime\prime},[\eta^{\prime\prime}]_{p})$ with
$\pi(g)\equiv f\ \left({\rm mod}\ p\right)$ as desired.

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 function $\hat{F}$ is not necessarily fixed by $W_{e}$, but rather lies in
the $\big{(}\frac{e}{p}\big{)}$-eigenspace of $W_{e}$. Thus

$\displaystyle\pi_{m+\varepsilon}(g)=\pi_{m}(f^{\prime})\equiv\pi_{m}(f)\
\left({\rm mod}\ p\right)$

where $\varepsilon=\frac{1}{2}\big{(}1-\big{(}\frac{e}{p}\big{)}\big{)}$, and
we have $\pi(g)\in
M_{k^{\prime}}^{(p)}(\Gamma^{\prime\prime},[\eta^{\prime\prime}]_{p})$ or
$\pi(g)\in
M_{k^{\prime}}^{(p)}(\Gamma^{\prime\prime},[\eta^{\prime\prime}_{t}]_{p})$. ∎

###### Proof of Proposition 3.6.

Let $k$ be as in the statement of Proposition 3.6. Set

$\displaystyle\mathfrak{S}(\Gamma,[\eta]_{p})^{\alpha}=\begin{cases}\bigcap_{n\geq
1}\operatorname{im}U_{p}^{n}|\tilde{M}_{k}(\Gamma,[\eta]_{p})&\text{ if
}k\equiv\alpha\ \left({\rm mod}\ 2p-2\right),\\\ \bigcap_{n\geq
1}\operatorname{im}U_{p}^{n}|\tilde{M}_{k}(\Gamma,[\eta_{t}]_{p})&\text{
otherwise.}\end{cases}$

Also set

$\displaystyle\mathfrak{N}(\Gamma,[\eta]_{p})^{\alpha}=\bigcup_{n\geq 1}\ker
U_{p}^{n}|\tilde{M}(\Gamma,[\eta]_{p})^{\alpha}.$

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.

##  4 $\bm{p}$-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 $p$.

######  Theorem 4.1.

Let $p$ be a prime. If $\Gamma$ is an $n|h$-type group as specified in Table
4.1, then the Hauptmodul on $\Gamma_{\lambda}$ is $p$-adically annihilated by
$U_{p}$.

$2$ | $3$ | $5$ | $7$ | $11$
---|---|---|---|---
$1$ | $20|2{+}5$ | $1$ | $1$ | $1$ | $1$
$2{+}$ | $20|2{+}10$ | $2{+}$ | $2{+}$ | $2{+}$ | $3|3$
$2$ | $22{+}$ | $3{+}$ | $3{+}$ | $3|3$ | $4|2{+}$
$3{+}$ | $22{+}11$ | $3$ | $3|3$ | $4|2{+}$ | $11{+}$
$3|3$ | $24|2{+}$ | $3|3$ | $4|2{+}$ | $7{+}$ |
$4{+}$ | $24{+}$ | $6{+}$ | $5{+}$ | $7$ |
$4|2{+}$ | $24|2{+}3$ | $6{+}2$ | $5$ | $8|4{+}$ |
$4$ | $24|6{+}$ | $6|3$ | $6{+}$ | $14{+}$ |
$4|2$ | $24|4{+}6$ | $8|4{+}$ | $6|3$ | $21|3{+}$ |
$5{+}$ | $24|4{+}2$ | $9{+}$ | $7{+}$ | $28|2{+}$ |
$6{+}$ | $24|2{+}12$ | $9$ | $8|4{+}$ |  |
$6{+}3$ | $24|12$ | $12|3{+}$ | $8|4$ |  |
$6|3$ | $28|2{+}$ | $12|6$ | $10{+}$ |  |
$8{+}$ | $28{+}7$ | $15|3$ | $10{+}2$ |  |
$8|2{+}$ | $28|2{+}14$ | $18{+}2$ | $12|3{+}$ |  |
$8|4{+}$ | $32{+}$ | $18{+}$ | $15{+}$ |  |
$8|2$ | $32|2{+}$ | $18$ | $15|3$ |  |
$8$ | $36|2{+}$ | $21|3{+}$ | $16|2{+}$ |  |
$8|4$ | $40|4{+}$ | $24|6{+}$ | $20|2{+}$ |  |
$10{+}$ | $40|2{+}$ | $24|4{+}2$ | $21|3{+}$ |  |
$10{+}5$ | $40|2{+}20$ | $24|12$ | $24|4{+}6$ |  |
$11{+}$ | $44{+}$ | $27{+}$ | $25{+}$ |  |
$12{+}$ | $48|2{+}$ | $30|3{+}10$ | $30{+}$ |  |
$12|2{+}$ | $52|2{+}$ | $36{+}4$ | $30|3{+}10$ |  |
$12|3{+}$ | $52|2{+}26$ | $39|3{+}$ | $35{+}$ |  |
$12{+}3$ | $56|4{+}14$ | $42|3{+}7$ | $40|4{+}$ |  |
$12|2{+}6$ | $60|2{+}$ | $54{+}$ | $50{+}$ |  |
$12|2{+}2$ | $60|2{+}5,6,30$ | $57|3{+}$ |  |  |
$12$ | $60|6{+}10$ | $60|6{+}10$ |  |  |
$12|6$ | $68|2{+}$ | $84|3{+}$ |  |  |
$16|2{+}$ | $84|2{+}$ | $93|3{+}$ |  |  |
$16$ | $84|2{+}6,14,21$ |  |  |  |
$16{+}$ | $88|2{+}$ |  |  |  |
$20{+}$ | $104|4{+}$ |  |  |  |
$20|2{+}$ |  |  |  |  |
Table 4.1: Hauptmoduln with $p$-adic annihilation.

For all Hauptmoduln $\mathcal{T}$ appearing in monstrous moonshine, we
computed $\mathcal{T}|U_{p}^{n}$ for small values of $n$ and small primes $p$.
These data, as well as heuristics governing the sizes of the ordinary spaces
$\mathfrak{S}(\Gamma,[\lambda_{t}]_{p})^{0}$ for $n|h$-type groups $\Gamma$,
lead us to the following conjecture (the relevancy of the ordinary space to
$p$-adic annihilation is discussed in Section 4.2).

######  Conjecture 4.2.

The converse to Theorem 4.1 holds, meaning that the Hauptmoduln of Table 4.1
are the only Hauptmoduln appearing in monstrous moonshine with $p$-adic
annihilation. In particular, no Hauptmodul appearing in monstrous moonshine is
$p$-adically annihilated for $p\geq 13$.

######  Remark 4.3.

Given an admissible $n|h$-type group $\Gamma$, we remark here on a method for
showing $\mathcal{T}_{\Gamma}$ is not $p$-adically annihilated for a given
$p$. By Lemma 3.4, after applying $U_{p}$ enough times we may assume that
$p\nmid nh$. Then after further applying $U_{p}$, we have that
$\mathcal{T}|U_{p}^{\alpha}\in\tilde{M}_{p-1}(nh)$ for some $\alpha$. Since
$\tilde{M}_{p-1}(nh)$ is finite-dimensional, it is then straightforward to
verify that $\mathcal{T}$ 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 $J|U_{13}^{2}\equiv J|U_{13}\ \left({\rm
mod}\ 13\right)$. 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 space $\tilde{M}_{4}(nh)$.

######  Remark 4.4.

It is known that $J|U_{p}\not\equiv 0\ \left({\rm mod}\ p\right)$ for any
$p\geq 13$ [38], and see [16] for further study of such congruences.

In Section 4.1, generalizing formulas of Conway and Norton [10], we write down
_compression formulas_ which relate Hauptmoduln on different groups $n|h$-type
groups appearing in Moonshine. 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 $p\geq
5$. 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.

###  4.1 Compression formulas

Throughout this section, we use the following notation. If $\Gamma$ is the
group $\Gamma_{0}(n|h)+e_{1},e_{2},\dots$, then we write $\Gamma^{d}$ for the
group $\Gamma_{0}(n^{\prime}|h^{\prime})+e_{1}^{\prime},e_{2}^{\prime},\dots$
where $n^{\prime}=n/\gcd(n,d)$, $h^{\prime}=h/\gcd(h,d)$, and
$e_{1}^{\prime},e_{2}^{\prime},\dots$ are the divisors of
$n^{\prime}/h^{\prime}$ among $e_{1},e_{2},\dots$. This notation comes from
[10], where it is explained that for any element $g$ of the monster, if
$\mathcal{T}_{g}$ is the Hauptmodul for the group $\Gamma_{\lambda}$
corresponding to $g$ from moonshine, then $\mathcal{T}_{g^{d}}$ is the
Hauptmodul for $\Gamma^{d}_{\lambda}$. Additionally, if $\Gamma$ is the group
$\Gamma_{0}(n|h)+e_{1},e_{2},\dots$, then we write
$\left\langle\Gamma,w_{e}\right\rangle$ for the group
$\Gamma_{0}(n|h)+e_{1},e_{2},\dots,e_{1}*e,e_{2}*e,\dots$ generated by
$\Gamma$ and $w_{e}$, if it exists. In this section, we will also adopt the
notation $\mathcal{T}_{\Gamma}$ for the Hautpmodul on $\Gamma_{\lambda}$.

The formula

$\displaystyle
p\mathcal{T}_{\Gamma}|U_{p}=\mathcal{T}_{\Gamma^{p}}-\mathcal{T}_{\Gamma}\qquad\text{if}\quad
w_{p}\in\Gamma$ (4.1)

relating the Hauptmodul for $g^{p}$ with that of $g$ for $g\in\mathbb{M}$
appears in [10, Section 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.

######  Proposition 4.5.

Let $\Gamma$ be a $p^{r}n|h$-type group where $p$ is prime and $p\nmid nh$.
Then, whenever all of the relevant groups appear in monstrous moonshine,

  1. $(a)$

$p\mathcal{T}_{\Gamma}|U_{p}=\mathcal{T}_{\Gamma}-\mathcal{T}_{\left\langle\Gamma,w_{p}\right\rangle}$
if $r=1$ and $p\nmid e$ for $w_{e}\in\Gamma$,

  2. $(b)$

$p^{2}\mathcal{T}_{\Gamma}|U_{p}^{2}=\mathcal{T}_{\Gamma}-\mathcal{T}_{\Gamma^{p}}$
if $r=1$ and $p\nmid e$ for $w_{e}\in\Gamma$,

  3. $(c)$

$p\mathcal{T}_{\Gamma}|U_{p}=\mathcal{T}_{\left\langle\Gamma^{p},w_{p^{r-1}}\right\rangle}-\mathcal{T}_{\Gamma^{p}}$
if $r>1$ and $w_{p^{r}}\in\Gamma$, and

  4. $(d)$

$\mathcal{T}_{\Gamma}|U_{p}=-\mathcal{T}_{\Gamma^{p}}|U_{p}$ if $r=2$ and
$w_{p^{2}}\in\Gamma$.

######  Remark 4.6.

If $r=1$, $p\nmid e$ for $w_{e}\in\Gamma$, and
$\left\langle\Gamma,w_{p}\right\rangle$ appears in monstrous moonshine, then
part (b) follows from part (a) and (4.1). However, part (b) holds even when
$\left\langle\Gamma,w_{p}\right\rangle$ is not admissible. For example
$4\mathcal{T}_{6|3}|U_{2}^{2}=\mathcal{T}_{6|3}-\mathcal{T}_{3|3}$ even though
$6|3{+}$ is not admissible. Similarly, if $r=2$, $w_{p^{2}}\in\Gamma$, and
$\left\langle\Gamma^{p},w_{p}\right\rangle$ appears in monstrous moonshine,
then part (d) follows from parts (a) and (c), but part (d) holds even when
$\left\langle\Gamma^{p},w_{p}\right\rangle$ 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 zero ($2500$ coefficients
suffice in all cases). These relations reduce the number of Hauptmoduln one
needs to check to show that the Hauptmoduln in Theorem 4.1 are indeed
annihilated. For example, (a) implies that if $\Gamma$ is an $pn|h$-type group
with $p\nmid n$ and $\mathcal{T}_{\Gamma}$ is $p$-adically annihilated, then
so is $\mathcal{T}_{\left\langle\Gamma,w_{p}\right\rangle}$. Note that many of
these formulas allow us to prove that $\mathcal{T}_{\Gamma^{p}}$ is
$p$-adically annihilated from the fact that $\mathcal{T}_{\Gamma}$ 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
$\mathcal{T}$ is $p$-adically annihilated in the cases where
$\mathcal{T}|U_{p}=0$ (see Remark 2.5), we have altogether reduced the
verification of Theorem 4.1 to the following much smaller check.

######  Corollary 4.7.

In order to show that the Hauptmoduln in Theorem 4.1 are $p$-adically
annihilated, it suffices to check $p$-adic annihilation for the following
smaller set shown in Table 4.2.

$2$ | $3$ | $5$ | $7$ | $11$
---|---|---|---|---
$6{+}3$ | $6{+}2$ | $4|2{+}$ | $2{+}$ | $3|3$
$6|3$ | $24|4{+}2$ | $6{+}$ | $3|3$ | $4|2{+}$
$10{+}5$ |  | $6|3$ | $4|2{+}$ | $11{+}$
$22{+}11$ |  | $7{+}$ | $8|4{+}$ |
|  | $8|4{+}$ |  |
|  | $8|4$ |  |
|  | $12|3{+}$ |  |
|  | $16|2{+}$ |  |
|  | $21|3{+}$ |  |
|  | $24|4{+}6$ |  |
|  | $30|3{+}10$ |  |
Table 4.2: Hauptmoduln for which it suffices to prove Theorem 4.1.

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
$p\geq 5$ we will prove Theorem 4.1 directly in Section 4.2 rather than using
the reduction given here.

###  4.2 Annihilation via ordinary spaces

In this section, we will prove Theorem 4.1 for $p\geq 5$ 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.

######  Lemma 4.8.

Suppose that $\mathfrak{S}(\Gamma,[\eta]_{p})^{0}\subseteq\mathbb{F}_{p}$. If
$f$ is a weight $0$ $p$-adic modular form on $(\Gamma,\eta)$, we have
$f|U_{p}^{n}\rightarrow c$ in the $p$-adic limit, where $c$ is the constant
term of $f$.

When $f=\mathcal{T}|U_{p}$ for $\mathcal{T}$ a Hauptmodul, then $c=0$. By
Lemma 3.4, $f$ is a weight $0$ $p$-adic modular form on $(\Gamma,\eta)$ for
some $\Gamma$, $\eta$. Hence Lemma 4.8 applies.

Using the mfslashexpansion and mfatkin functions in Pari [34], one can compute
the actions of

$\displaystyle x=\begin{pmatrix}1&h\\\ 0&1\end{pmatrix},\qquad
y=\begin{pmatrix}1&0\\\ n&1\end{pmatrix},$

and all Atkin–Lehner involutions on $M_{p-1}(\Gamma,[\lambda_{t}]_{p})$. Then
using elementary linear algebra, it is easy to find a basis for
$\mathfrak{S}(\Gamma,[\lambda]_{p})^{0}\subseteq\tilde{M}_{p-1}(\Gamma,[\lambda_{t}]_{p})$.
We performed this computation for various $n|h$ groups $\Gamma$ appearing in
monstrous moonshine with small values of $n$ with $p\nmid n$ (specifically,
$n\leq 24$ for $p=5$, $n\leq 11$ for $p=7$, and $n\leq 7$ for $p=11$).

The $\Gamma$ for which
$\mathfrak{S}(\Gamma,[\lambda]_{p})^{0}\subseteq\mathbb{F}_{p}$ are given in
Table 4.3. Applying Lemma 3.4 to the Hauptmoduln of Table 4.1 then proves
Theorem 4.1 for $p\geq 5$. We note that every group from Table 4.3 corresponds
to at least one group from Theorem 4.1.

$p=5$ | $1$ | $2{+}$ | $3{+}$ | $3|3$ | $4|2{+}$ | $6{+}$ | $6|3$ | $7{+}$ | $8|4{+}$ | $8|4$ | $12|3{+}$ | $16|2{+}$ | $24|4{+}6$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$p=7$ | $1$ | $2{+}$ | $3|3$ | $4|2{+}$ | $8|4{+}$ |  |  |  |  |  |  |  |
$p=11$ | $1$ | $3|3$ | $4|2{+}$ |  |  |  |  |  |  |  |  |  |
Table 4.3: $\Gamma$ such that
$\mathfrak{S}(\Gamma,[\lambda]_{p})^{0}\subset\mathbb{F}_{p}$.

######  Remark 4.9.

Another method for computing $\mathfrak{S}(\Gamma,[\lambda]_{p})$ is as
follows. By drawing fundamental domains for $\Gamma_{\lambda_{t}}$, one can
compute the dimension of the space $M_{p-1}(\Gamma_{\lambda_{t}})$. One then
uses the $q$-expansions of the modular forms $\mathcal{T}^{r}|U_{p}^{m}$ to
compute a basis for $\tilde{M}_{p-1}(\Gamma,[\lambda_{t}]_{p})$. Computing
$\mathfrak{S}(\Gamma,[\lambda_{t}]_{p})^{0}$ then amounts to linear algebra.
We carried out this procedure in Sage [36], giving an alternate verification
of the results in Table 4.3. The Sturm’s bound calculations for this method
require checking $3500$ coefficients.

It is worth noting that these methods also apply to Hauptmoduln not appearing
in monstrous moonshine. For example, if $\Gamma=\Gamma_{0}(2|2)$ and
$\mathcal{T}$ is the Hauptmodul on $\Gamma_{\lambda}$ then Lemma 3.4 gives
that $\mathcal{T}|U_{5}$ is a $5$-adic modular form on $(\Gamma,\lambda)$.
Using the method above, one can compute that
$\mathfrak{S}(\Gamma,[\lambda]_{p})^{0}=0$, so that $\mathcal{T}$ 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 $10000$ coefficients.

######  Remark 4.10.

Let $\mathcal{T}$ be the Hauptmodul on some $\Gamma_{\lambda}$ appearing in
Table 4.1 for $p\geq 5$, $\Gamma^{\prime}$ from Table 4.3 as given by Lemma
3.4, and $\lambda^{\prime}$ the corresponding eigenvalue map specified by
Conway–Norton. Let $m$ be the smallest integer such that
$U_{p}^{m}\tilde{M}_{p-1}(\Gamma^{\prime},[\lambda^{\prime}]_{p})=\mathfrak{S}(\Gamma^{\prime},[\lambda^{\prime}]_{p})^{0}$.
Numerically, we observe $v_{p}\big{(}\mathcal{T}|U_{p}^{\ell+m+1}\big{)}\geq
v_{p}\big{(}\mathcal{T}|U_{p}^{\ell}\big{)}+1$, 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.

###  4.3 Additional $\bm{p}$-adic annihilation

The ordinary spaces of Section 3.2 and the annihilation verified in Section
4.2 were restricted to $p\geq 5$. 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 $p=2,3$.

######  Proposition 4.11.

We have

$\displaystyle v_{p}\big{(}\mathcal{T}|U_{p}^{n}\big{)}\geq\lfloor
n\alpha\rfloor,$

when $\mathcal{T}$ is the normalized Hauptmodul on $\Gamma_{\lambda}$, with
parameters given in Table 4.4.

$\Gamma$ | $6{+}2$ | $6{+}3$ | $10{+}5$ | $22{+}11$ | $6|3$ | $24|4{+}2$
---|---|---|---|---|---|---
$p$ | $3$ | $2$ | $2$ | $2$ | $2$ | $3$
$\alpha$ | $3/2$ | $1$ | $3/2$ | $1/2$ | $3/2$ | $1/2$
Table 4.4: Rates of annihilation.

###### Proof.

Throughout, we use $\mathfrak{T}$ to denote the normalized Hauptmodul
$\mathcal{T}$ plus some constant, and $\eta(\tau)=q^{1/24}\prod\limits_{n\geq
1}(1-q^{n})$ is the Dedekind eta function. The constant will be specified by
writing $\mathfrak{T}$ as an eta-quotient given in [10] and recorded in Table
4.5.

$\Gamma$ | $6{+}2$ | $6{+}3$ | $10{+}5$ | $22{+}11$ | $6|3$ | $24|4{+}2$
---|---|---|---|---|---|---
4pt$\mathfrak{T}$ | $\frac{\eta(\tau)^{4}\eta(2\tau)^{4}}{\eta(3\tau)^{4}\eta(6\tau)^{4}}$ | $\frac{\eta(\tau)^{6}\eta(3\tau)^{6}}{\eta(2\tau)^{6}\eta(6\tau)^{6}}$ | $\frac{\eta(\tau)^{4}\eta(5\tau)^{4}}{\eta(2\tau)^{4}\eta(10\tau)^{4}}$ | $\frac{\eta(\tau)^{2}\eta(11\tau)^{2}}{\eta(2\tau)^{2}\eta(22\tau)^{2}}$ | $\frac{\eta(3\tau)^{8}}{\eta(6\tau)^{8}}$ | $\frac{\eta(4\tau)\eta(8\tau)}{\eta(12\tau)\eta(24\tau)}$
Table 4.5: Eta quotients for unnormalized Hauptmoduln.

We follow the methods of [31, 32] for the $j$-function, and prove our six
cases in parallel.

We first compute expansions of $\mathfrak{T}$ at each cusp, and write
$\mathfrak{T}|U_{p}$ 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 may
also be verified via a Sturm’s bound coefficient check ($600$ suffice in all
cases). Let

$\displaystyle S_{\lambda}=\begin{pmatrix}1&\lambda\\\
0&p\end{pmatrix}\qquad\text{and}\qquad F_{2}=\begin{pmatrix}0&-1\\\
2&0\end{pmatrix}.$

For $W_{2}$ a suitable Atkin–Lehner matrix on $\Gamma_{0}(6)$, we have

$\displaystyle 3(\mathfrak{T}|U_{3})\left(-1/2\tau\right)$
$\displaystyle=\sum_{\lambda=0}^{2}\mathfrak{T}(S_{\lambda}F_{2})=\mathfrak{T}\left(-1/6\tau\right)+\sum_{\lambda=1}^{2}\mathfrak{T}(S_{\lambda}F_{2}\tau)$
$\displaystyle=\mathfrak{T}\left(-1/6\tau\right)+\sum_{\mu=1}^{2}\mathfrak{T}(W_{2}T_{\mu}\tau)=\mathfrak{T}\left(-1/6\tau\right)+3(\mathfrak{T}|U_{3})(\tau)-\mathfrak{T}(\tau/3),$

and upon substituting $\tau\mapsto 3\tau$ we obtain the cusp expansion

$\displaystyle 3(\mathfrak{T}|U_{3})\left(-1/6\tau\right)=-q^{-1}+O(1).$

We next subtract off this principal part with $-\mathfrak{T}(\tau)$. Sending
$\tau\mapsto-1/6\tau$ and applying the $\eta$ functional equation to the
quotients in Table 4.5, we find

$\displaystyle 3(\mathcal{T}|U_{3})(\tau)=-3^{4}\mathfrak{T}(\tau)^{-1}+c$

for some constant $c$. One can check that $\mathfrak{T}|U_{p}$ is bounded at
the other cusps; see Lemma 2.4. For appropriate constants $c$, similar
computations show

$\displaystyle p(\mathcal{T}|U_{p})(\tau)=-p^{e}\mathfrak{T}(\tau)^{-1}+c,$
(4.2)

where $e$ is as in Table 4.6.

$\Gamma$ | $6{+}2$ | $6{+}3$ | $10{+}5$ | $22{+}11$ | $6|3$ | $24|4{+}2$
---|---|---|---|---|---|---
$e$ | $4$ | $6$ | $4$ | $2$ | $4$ | $1$
Table 4.6: Parameters for functional equations.

These relations imply $v_{p}(\mathcal{T}|U_{p})\geq e-1$. Set
$Z=\mathfrak{T}^{-1}$ and $W=p^{e}Z(\tau/p)$. Expanding $Z$ at other cusps and
applying the principal part analysis as above, we find

$\displaystyle p(p^{e}Z|U_{p})(\tau)=f(p^{e}Z).$

where $f$ is a polynomial with
$f(\mathfrak{T}(\tau))=\mathfrak{T}(p\tau)+O(q)$ (different for each group).
Write

$\displaystyle(Z|U_{p})(\tau)=p^{\alpha}\sum_{j=1}^{p}b_{j}Z^{j}$ (4.3)

where the coefficients $b_{j}$ are listed in Table 4.7.

| $6{+}2$ | $6{+}3$ | $10{+}5$ | $22{+}11$ | $6|3$ | $24|4{+}2$
---|---|---|---|---|---|---
$b_{1}$ | $2\cdot 3^{1/2}$ | $3$ | $2^{1/2}$ | $2^{1/2}$ | 0 | 0
$b_{2}$ | $4\cdot 3^{5/2}$ | $2^{4}$ | $2^{3/2}$ | $2^{1/2}$ | $2^{3/2}$ | 0
$b_{3}$ | $3^{11/2}$ |  |  |  |  | $3^{1/2}$
Table 4.7: Polyomial coefficients.

Following Lehner [32, equation (2.2)] gives

$\displaystyle
W^{p}+\sum_{j=1}^{p}(-1)^{j}p_{j}W^{p-j}=0,\qquad\text{where}\quad(-1)^{j+1}p_{j}=p^{e+\alpha+1}\sum_{m=j}^{p}b_{m}Z^{m-j+1}.$

This equation has roots $W(\tau+\lambda)$ for $\lambda\in\\{0,\ldots,p-1\\}$.
If $S_{\ell}$ denotes the sum of the $\ell$-th power of these roots, then we
have $p^{-e\ell-1}S_{\ell}=Z^{\ell}|U_{p}$. We show that
$p^{\alpha(\ell-1)}(Z^{\ell}|U_{p})\in
p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$ for all $\ell\geq 1$.
Equivalently, we check that $S_{\ell}\in
p^{(e-\alpha)\ell+1+\alpha}p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$.

Lehner uses Newton sums to set up an induction, relating $S_{\ell}$ to $S_{j}$
for $j<\ell$. A similar computation works for our cases. The coefficients in
Table 4.7 imply the base case $\ell=1$. By Newton sums we have

$\displaystyle S_{\ell}=\sum_{j=1}^{\ell}(-1)^{j+1}p_{j}S_{\ell-j},$

where $b_{j},p_{h}=0$ for $j,h\geq p+1$ and $S_{0}=\ell$. For $\ell\leq p$ we
rewrite the Newton sum as

$\displaystyle
S_{\ell}=\sum_{j=1}^{\ell-1}(-1)^{j+1}p_{j}S_{\ell-j}-(-1)^{\ell}\ell
p_{\ell}.$

By construction we have that $p_{j}\in
p^{e+1+\alpha+v_{p}(b_{j})-\alpha}p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$
and applying the inductive hypothesis gives $p_{j}S_{\ell-j}\in
p^{e+1+\alpha+v_{p}(b_{j})-\alpha+(e-\alpha)(\ell-j)+1+\alpha}p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z]$.
We have the inequality $e+1+\alpha+v_{p}(b_{j})-\alpha\geq(e-\alpha)j$, which
one checks explicitly using the values of $b_{j}$. In particular,
$p_{j}S_{\ell-j}\in
p^{(e-\alpha)\ell+1+\alpha}p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$.
We also check that $\ell p_{\ell}\in
p^{(e-\alpha)\ell+1+\alpha}p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$
from the explicit values of $b_{j}$. Thus $S_{\ell}\in
p^{(e-\alpha)\ell+1+\alpha}p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$
for $\ell\leq p$. When $l\geq p+1$ we have

$\displaystyle S_{\ell}=\sum_{j=1}^{\ell-1}(-1)^{j+1}p_{j}S_{\ell-j}$

and we check that $p_{j}S_{\ell-j}\in
p^{(e-\alpha)\ell+1+\alpha}p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$
as above, which completes the induction. Thus, we have shown that

$\displaystyle\big{(}p^{\alpha(\ell-1)}Z^{\ell}|U_{p}\big{)}\in
p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$

so that if $g\in Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$, then $g|U_{p}\in
p^{\alpha}Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$. Note that
$\mathcal{T}|U_{p}=-p^{e-1}Z\in Z\cdot\mathbb{Z}[p^{\alpha}Z,p^{\alpha}]$.
Repeatedly applying the result of the induction thus gives the rates of
annihilation claimed in Proposition 4.11. ∎

######  Remark 4.12.

These explicit techniques also apply when $p\geq 5$, as long as
$\mathcal{T}|U_{p}$ is modular on a genus zero group. For example, if
$\mathcal{T}$ is the Hauptmodul for $2{+}$, then $\mathcal{T}|U_{5}$ is
modular on $10{+}2$, which is genus zero. There are many examples for $p\geq
5$ where this does not hold, e.g., if $\mathcal{T}$ the Hauptmodul for $2{+}$
then $\mathcal{T}|U_{7}$ 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.

###  4.4 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 4.13.

For a fixed Hauptmodul $\mathcal{T}_{g}$ and an integer $d$, we say that
$p$-adic annihilation is _preserved under the $d$-th power map_ if
$\mathcal{T}_{g}$ $p$-adically annihilated implies $\mathcal{T}_{g^{d}}$ is
$p$-adically annihilated.

These relationships are depicted in Appendix B for primes $p\leq 11$, which we
expect are the only primes with annihilation. Note that if $d$ is relatively
prime to the order of $g$, then $\mathcal{T}_{g^{d}}=\mathcal{T}_{g}$, so
$p$-adic annihilation is preserved under the $d$-th power map. We will
therefore restrict our attention to those $d$ that divide the order of $g$. 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
$\ell+$ to level $1$, when $\ell$ is prime. When $p\geq 5$, we may also
explain via ordinary spaces.

######  Example 4.14.

When $p=5$, the groups $50+$, $10+2$, $10+$, and $2+$ all have the same
ordinary space
$\mathbb{F}_{p}=\mathfrak{S}(\Gamma_{0}(2)+,\bm{1})^{0}\subseteq\tilde{M}_{p-1}(\Gamma_{0}(2)+,\bm{1}_{t})$,
and their Hauptmoduln share the same annihilation behavior. This explains
preservation of $p$-adic annihilation under $p$-th power maps. The compression
formulas of Proposition 4.5 capture these same relations.

Furthermore, in light of Lemma 4.8 and Section 4.2, we seek inclusions of the
form $\mathfrak{S}(\Gamma^{d},[\lambda]_{p})^{0}$
$\hookrightarrow\mathfrak{S}(\Gamma,[\lambda]_{p})^{0}$ for $\Gamma$ being
$n|h$-type with $p\nmid nh$. This would explain $d$-th power map preservation
of $p$-adic annihilation from $\Gamma_{\lambda}$ to $\Gamma^{d}_{\lambda}$,
and similarly for groups with the same ordinary spaces. The following
proposition accomplishes this in certain situations.

######  Proposition 4.15.

Let $p\geq 5$ be prime, $(d,nh)=1$, $\Gamma$ be an $n|h$-type group with
eigenvalue map $\eta$, and $\Gamma^{\prime}$ be a $dn|h$-type group with
eigenvalue map $\eta^{\prime}$ such that

$\displaystyle\operatorname{AL}(\Gamma)=\operatorname{AL}(\Gamma^{\prime}),\qquad(d,e)=1\qquad\text{for
all}\quad
e\in\operatorname{AL}(\Gamma),\qquad\text{and}\qquad\eta|_{\Gamma^{\prime}}=\eta^{\prime}.$

  1. $(a)$

There is an inclusion $M_{k}(\Gamma,\eta)\subseteq
M_{k}(\Gamma^{\prime},\eta^{\prime})$.

  2. $(b)$

For any $\alpha\in\mathbb{Z}/(2p-2)\mathbb{Z}$, there is an inclusion
$\mathfrak{S}(\Gamma,[\eta]_{p})^{\alpha}\subseteq\mathfrak{S}(\Gamma^{\prime},[\eta^{\prime}]_{p})^{\alpha}$.

###### Proof.

Part (b) follows from part (a) using the description of $\mathfrak{S}$ given
in the proof of Proposition 3.6. Part (a) follows from the isomorphic
inclusion
$\Gamma/\Gamma_{0}(dnh)\hookrightarrow\Gamma^{\prime}/\Gamma_{0}(nh)$ from
Section 2.3. ∎

######  Remark 4.16.

Inclusions of the form $M_{k}(\Gamma,\eta)\subseteq
M_{k}(\Gamma^{\prime},\eta^{\prime})$ may have further implications for
preserving average rates of $p$-adic annihilation via Remark 4.10.

We now discuss when power maps do not preserve Hauptmodul $p$-adic
annihilation.

######  Example 4.17.

If $\mathcal{T}$ is the Hauptmodul on $\Gamma_{\lambda}$ for $\Gamma$ an
$n|h$-type group, then the Hauptmodul on $\Gamma_{\lambda}^{d}$ need not be
annihilated when $d\mid h$, 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.

######  Example 4.18.

If $\mathcal{T}$ is the Hauptmodul on $\Gamma_{\lambda}$ and
$\mathcal{T}|U_{p}=0$, power maps from $\Gamma_{\lambda}$ may not preserve
$p$-adic annihilation. For example, if $\mathcal{T}$ is the Hauptmodul on
$12$, $\mathcal{T}|U_{p}=0$ while the Hauptmodul on $6$ is not $2$-adically
annihilated.

The classes of groups from Examples 4.17 and 4.18 exhaust the situations for
which power maps do not preserve Hauptmodul $p$-adic annihilation. If one
further excludes the groups $\Gamma$ with $\mathcal{T}_{\Gamma}|U_{p}=0$, then
rates of $p$-adic annihilation are preserved under power maps. These examples
motivate a notion of strong $p$-adic annihilation for genus zero groups. This
strong notion of annihilation will not be referenced outside of this section.

######  Definition 4.19.

Let $\Gamma$ be a genus zero group with Hauptmodul $\mathcal{T}$. We say that
$\Gamma$ has _strong $p$-adic annihilation_ if every polynomial
$f(\mathcal{T})$ in the Hauptmodul with no constant term is $p$-adically
annihilated.

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
$\Gamma_{\eta}$ has strong $p$-adic annihilation if
$\mathfrak{S}_{p-1}(\Gamma,\bm{1}_{t})^{0}=\mathbb{F}_{p}$ and
$\mathfrak{S}_{p-1}(\Gamma,[(\lambda^{\sigma_{r}})_{t}]_{p})^{0}=0$ otherwise.

We can check that strong $p$-adic annihilation is now preserved for the groups
of Example 4.17. Indeed, let $\mathcal{T}$ be the Hauptmodul on
$\Gamma_{\lambda}$ for $\Gamma$ an $n|h$-type group, and
$\mathcal{T}^{\prime}$ be the Hauptmodul on $\Gamma_{\lambda}^{d}$ for $d\mid
h$. We recall the $V_{m}$ operator, given by $f|V_{m}=f\circ A$ where we set
$A=\left(\begin{smallmatrix}m&0\\\ 0&1\end{smallmatrix}\right)$ as in Section
3.2. It acts on Fourier expansions by $\big{(}\sum a(n)q^{n}\big{)}|V_{m}=\sum
a(n)q^{mn}$. From [10, Section 6], we know that
$\mathcal{T}^{\prime}|V_{d}+c=\mathcal{T}^{d}$ for some constant $c$. In
particular, if $\mathcal{T}^{\prime}$ is not $p$-adically annihilated,
$\mathcal{T}^{\prime}|V_{d}$ is a polynomial in $\mathcal{T}$ with no constant
term and is also not $p$-adically annihilated. Thus, if $\Gamma_{\lambda}^{d}$
does not have strong $p$-adic annihilation, neither does $\Gamma_{\lambda}$.
This reflects the following inclusions.

######  Proposition 4.20.

Let $p\geq 5$ and $\Gamma$ be an $n|h$-type group with eigenvalue map $\eta$,
and let $\Gamma^{\prime}$ be a $nd|hd$ type group with eigenvalue map
$\eta^{\prime}$ such that

$\displaystyle\operatorname{AL}(\Gamma)=\operatorname{AL}(\Gamma^{\prime}),\qquad\eta^{\prime}(x^{\prime})=\eta(x),\qquad\eta^{\prime}(y^{\prime})=\eta(y),\qquad\text{and}$
$\displaystyle\eta^{\prime}(W_{e})=\eta(W_{e})\qquad\text{for all}\quad
e\in\operatorname{AL}(\Gamma)$

for $x$, $y$ and $x^{\prime}$, $y^{\prime}$ the usual generators of
$\Gamma_{0}(n|h)/\Gamma_{0}(nh)$ and
$\Gamma_{0}(nd|hd)/\Gamma_{0}\big{(}nhd^{2}\big{)}$.

  1. $(a)$

There is an injection given by the operator $V_{d}\colon
M_{k}(\Gamma,\eta)\hookrightarrow M_{k}(\Gamma^{\prime},\eta^{\prime})$.

  2. $(b)$

For any $\alpha\in\mathbb{Z}/(2p-2)\mathbb{Z}$, this gives an injection
$V_{d}\colon\mathfrak{S}(\Gamma,[\eta]_{p})^{\alpha}\hookrightarrow\mathfrak{S}(\Gamma^{\prime},[\eta^{\prime}]_{p})^{\alpha}.$

We observe empirically that strong $p$-adic annihilation is also preserved for
the groups from Example 4.18, e.g., $\mathcal{T}^{2}$ is not $2$-adically
annihilated when $\mathcal{T}$ 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.

##  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 $V=\bigoplus_{n=-1}^{\infty}V_{n}$ such that

  1. $(i)$

For each $g\in G$, the McKay–Thompson series

$\displaystyle\mathcal{T}_{g}(\tau)=\sum_{n=-1}^{\infty}\operatorname{Tr}(g\big{|}V_{n})q^{n}$

is the Hauptmodul of an order $\operatorname{ord}(g)$ conjugacy class of the
monster.

  2. $(ii)$

For any $g\in G$ and any $n\in\mathbb{Z}$, if $\mathcal{T}_{g}$ is the
Hauptmodul for $\Gamma$, then $\mathcal{T}_{g^{n}}$ is the Hauptmodul for
$\Gamma^{n}$.

If for some prime $p$, $V$ also satisfies the following property, then we call
$V$ a $p$-adic moonshine module.

  1. (iii)

For each irreducible character $\chi$ of $\Gamma$, if $m_{\chi}(n)$ is the
multiplicity of $\chi$ in the character of $V_{n}$, then the multiplicity
generating function

$\displaystyle\mathcal{M}_{\chi}(\tau)=\sum_{n=-1}^{\infty}m_{\chi}(n)q^{n}$

associated to $\chi$ is $p$-adically annihilated.

Throughout this section, we assume that Conjecture 4.2 holds. 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.

###  5.1 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
$V=\bigoplus_{n=-1}^{\infty}V_{n}$. For any prime $p$, $V$ is a $p$-adic
moonshine module if and only if the McKay–Thompson series $\mathcal{T}_{g}$ is
$p$-adically annihilated for each $g\in G$.

###### Proof.

By the Schur orthogonality relations, we have

$\displaystyle\mathcal{T}_{g}=\sum_{\chi\in\operatorname{Irrep}(G)}\chi(g)\mathcal{M}_{\chi}\qquad\text{for
each}\quad g\in G$

and

$\displaystyle\mathcal{M}_{\chi}=\frac{1}{|G|}\sum_{g\in
G}\overline{\chi(g)}\mathcal{T}_{g}\qquad\text{for
each}\quad\chi\in\operatorname{Irrep}(G).$

If each $\mathcal{M}_{\chi}$ is $p$-adically annihilated, then for all
$n\in\mathbb{N}$, there exists $N\in\mathbb{N}$ such that
$\mathcal{M}_{\chi}|U_{p}^{N}\equiv 0$ $\ \left({\rm mod}\ p^{n}\right)$.
Since each $\chi(g)$ is an algebraic integer, it follows that
$\mathcal{T}_{g}|U_{p}^{N}\equiv 0\ \left({\rm mod}\ p^{n}\right)$, so each
$\mathcal{T}_{g}$ is $p$-adically annihilated. Similarly, if each
$\mathcal{T}_{g}$ is $p$-adically annihilated, then each
$|G|\mathcal{M}_{\chi}$, and hence each $\mathcal{M}_{\chi}$, 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 $g\in G$ a $p$-adically annihilated Hauptmodul
$\mathcal{T}_{g}$ in a way that agrees with power maps in $G$ and so that the
multiplicities defined by

$\displaystyle\mathcal{M}_{\chi}=\frac{1}{|G|}\sum_{g\in
G}\overline{\chi(g)}\mathcal{T}_{g}$

have positive integral coefficients for all irreducible characters $\chi$. In
particular, each McKay–Thompson series $\mathcal{T}_{g}$ 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 $\Gamma$ with the
property that $\Gamma^{n}$ is $p$-adically annihilated for every
$n\in\mathbb{Z}$. By Theorem 4.1, it follows that every McKay–Thompson series
$\mathcal{T}_{g}$ must be a Hauptmodul for one of the groups listed in Table
5.1.

$2$ | $3$ | $5$ | $7$ | $11$
---|---|---|---|---
$1$ | $12|3{+}$ | $1$ | $1$ | $1$ | $1$
$2{+}$ | $12{+}3$ | $2{+}$ | $2{+}$ | $2{+}$ | $3|3$
$2$ | $12|6$ | $3{+}$ | $3{+}$ | $3|3$ | $11{+}$
$3{+}$ | $16|2{+}$ | $3$ | $3|3$ | $4|2{+}$ |
$3|3$ | $16$ | $3|3$ | $4|2{+}$ | $7{+}$ |
$4{+}$ | $16{+}$ | $6{+}$ | $5{+}$ | $7$ |
$4|2{+}$ | $20{+}$ | $6{+}2$ | $5$ | $8|4{+}$ |
$4$ | $20|2{+}$ | $9{+}$ | $6{+}$ | $14{+}$ |
$4|2$ | $20|2{+}5$ | $9$ | $7{+}$ | $21|3{+}$ |
$5{+}$ | $22{+}$ | $18{+}2$ | $8|4{+}$ | $28|2{+}$ |
$6{+}$ | $22{+}11$ | $18{+}$ | $10{+}$ |  |
$6{+}3$ | $24|2{+}$ | $27{+}$ | $10{+}2$ |  |
$6|3$ | $24{+}$ | $54{+}$ | $15{+}$ |  |
$8{+}$ | $24|2{+}3$ |  | $15|3$ |  |
$8|2{+}$ | $24|6{+}$ |  | $20|2{+}$ |  |
$8|4{+}$ | $24|12$ |  | $21|3{+}$ |  |
$8|2$ | $32{+}$ |  | $25{+}$ |  |
$8$ | $32|2{+}$ |  | $30{+}$ |  |
$8|4$ | $40|4{+}$ |  | $35{+}$ |  |
$10{+}$ | $40|2{+}$ |  | $40|4{+}$ |  |
$10{+}5$ | $44{+}$ |  | $50{+}$ |  |
$11{+}$ | $48|2{+}$ |  |  |  |
$12{+}$ | $88|2{+}$ |  |  |  |
$12|2{+}$ |  |  |  |  |
Table 5.1: Candidate McKay–Thompson series for groups with $p$-adic moonshine.

As a first application of this simpler description of $p$-adic moonshine, the
following proposition bounds the powers of primes dividing the orders of
groups with $p$-adic moonshine for each prime $p$.

######  Proposition 5.2.

Let $p$ be a prime and $G$ be a group with $p$-adic moonshine. Assuming that
Conjecture 4.2 holds, the following table gives for each prime $q$ a value $r$
such that every group $G$ with $p$-adic moonshine satisfies $v_{q}(|G|)\leq
r$.

$\begin{array}[]{l|c c c c c c}&p=2&p=3&p=5&p=7&p=11&p>11\\\ \hline\cr
q=2&46&12&15&15&0&0\\\ q=3&8&21&8&3&3&0\\\ q=5&3&0&9&0&0&0\\\
q=7&0&0&2&6&0&0\\\ q=11&2&0&0&0&2&0\\\ q>11&0&0&0&0&0&0\end{array}$

###### Proof.

First, note that if $q\mid|G|$, 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 [12]. Let
$J,\mathcal{T}_{1},\dots,\mathcal{T}_{n}$ be the distinct Hauptmoduln given in
Table 5.1 as candidate McKay–Thompson series for groups with $p$-adic
moonshine. For each $i\in\\{1,\dots,n\\}$, let $a_{i}$ be the number of
elements of $G$ whose McKay–Thompson series is $\mathcal{T}_{i}$. Then, the
multiplicity of the trivial character $\epsilon$ is

$\displaystyle\mathcal{M}_{\epsilon}=\frac{1}{|G|}\sum_{g\in
G}\mathcal{T}_{g}=\frac{1}{|G|}\left(J+a_{1}\mathcal{T}_{1}+\dots+a_{n}\mathcal{T}_{n}\right).$

In particular, since $\mathcal{M}_{\epsilon}$ must have integral coefficients,
it follows that

$\displaystyle J+a_{1}\mathcal{T}_{1}+\dots+a_{n}\mathcal{T}_{n}\equiv 0\
\left({\rm mod}\ |G|\right).$

Therefore, if we choose $r$ large enough so that there are no coefficients
$a_{1},\dots,a_{n}$ such that

$\displaystyle J+a_{1}\mathcal{T}_{1}+\dots+a_{n}\mathcal{T}_{n}\equiv 0\
\left({\rm mod}\ q^{r+1}\right),$

then we have shown that $v_{q}(|G|)\leq r$. As in [12], this computation was
carried out using Sage [36] by computing the kernel of the matrix of
coefficients of $J,\mathcal{T}_{1},\dots,\mathcal{T}_{n}$. ∎

###  5.2 Examples of groups with $\bm{p}$-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:

$\begin{array}[]{c|ccccc}A_{5}&1&(1\;2)(3\;4)&(1\;2\;3)&(1\;2\;3\;4\;5)&(1\;3\;4\;5\;2)\\\
\hline\cr\epsilon&1&1&1&1&1\\\ \chi_{1}&4&0&1&-1&-1\\\ \chi_{2}&5&1&-1&0&0\\\
\chi_{3}&3&-1&0&\frac{1+\sqrt{5}}{2}&\frac{1-\sqrt{5}}{2}\\\
\chi_{4}&3&-1&0&\frac{1-\sqrt{5}}{2}&\frac{1+\sqrt{5}}{2}\end{array}$

Table 5.2: Character Table for $A_{5}$.

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
$g^{\operatorname{ord}(g)}=1$ 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 $\mathcal{T}_{1}$ to the element of the
conjugacy class of $1$, $\mathcal{T}_{2{+}}$ to the elements of the conjugacy
class of $(1\;2)(3\;4)$, $\mathcal{T}_{3|3}$ to the elements of the conjugacy
class of $(1\;2\;3)$, and $\mathcal{T}_{5}$ 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

$\displaystyle\mathcal{M}_{\epsilon}=q^{-1}+4378q+382380q^{2}+14714988q^{3}+340105628q^{4}+O\big{(}q^{5}\big{)},$
$\displaystyle\mathcal{M}_{\chi_{1}}=13122q+1432996q^{2}+57620010q^{3}+1349723748q^{4}+O\big{(}q^{5}\big{)},$
$\displaystyle\mathcal{M}_{\chi_{2}}=17500q+1815128q^{2}+72334998q^{3}+1689829376q^{4}+O\big{(}q^{5}\big{)},$
$\displaystyle\mathcal{M}_{\chi_{3}}=8753q+1050626q^{2}+42904992q^{3}+1009618126q^{4}+O\big{(}q^{5}\big{)},\qquad\text{and}$
$\displaystyle\mathcal{M}_{\chi_{4}}=8753q+1050626q^{2}+42904992q^{3}+1009618126q^{4}+O\big{(}q^{5}\big{)}.$

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 [12]. Indeed, this inequality holds for $n=2$, and
hence for all $n\geq 2$ 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 $\Gamma_{0}(90)$,
so we may use Sturm’s bound to reduce the computation to checking only the
coefficients up to $q^{216}$. Using Sage [36], 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 [36], 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
$G=\mathbb{Z}/2\mathbb{Z}\times\left(\mathbb{Z}/3\mathbb{Z}\right)^{2}$.

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 $\mathcal{T}$. 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 $\epsilon$
is given by

$\displaystyle\mathcal{M}_{\epsilon}=\frac{1}{|G|}\sum_{g\in
G}\mathcal{T}_{g}=\frac{1}{|G|}(J-\mathcal{T})+\mathcal{T},$

and the multiplicity of any non-trivial character $\chi$ is given by

$\displaystyle\mathcal{M}_{\chi}=\frac{1}{|G|}\sum_{g\in
G}\overline{\chi(g)}\mathcal{T}_{g}=\frac{1}{|G|}\left(\overline{\chi(e)}J+\sum_{g\in
G-\\{e\\}}\overline{\chi(g)}\mathcal{T}\right)=\frac{\dim\chi}{|G|}(J-\mathcal{T}).$

These multiplicities are both integral if and only if
$|G|\mid(J-\mathcal{T})$, and for checking positivity, we may once again use
inequality (4.16) in [12]. After a computation in Sage [36], we have the
following result.

######  Proposition 5.3.

Let $p$ and $q$ be primes, $G$ be a group of exponent $q$, and $\mathcal{T}$
be a Hauptmodul for one of the order $q$ conjugacy class of the monster. Then,
assuming that Conjecture 4.2 holds, $G$ has a $p$-adic moonshine module in
which the McKay–Thompson series for each non-identity element is $\mathcal{T}$
if and only if $p$ and $\mathcal{T}$ appear in the following table and
$|G|\leq q^{r}$ where $r$ is the corresponding entry in the third row.

$\begin{array}[]{c|l l l l l l l l l l l}p&2&2&2&3&3&3&5&5&7&7&11\\\
\hline\cr\mathcal{T}&2{+}&2&3|3&3{+}&3&3|3&5{+}&5&7{+}&7&11{+}\\\ \hline\cr
r&12&13&1&6&9&3&3&5&2&4&2\end{array}$

###  5.3 Centralizers in the monster

Other examples of $p$-adic moonshine come from subgroups of the monster
meeting only those conjugacy classes whose Hauptmoduln are $p$-adically
annihilated, though Proposition 5.3 shows that not every group with $p$-adic
moonshine is of this form. In this section, we will exhibit a surprising class
of subgroups of the monster having $p$-adic moonshine in a slightly more
general sense.

Specifically, we say that a modular function $f$ is weakly $p$-adically
annihilated if $f|U_{p}^{n}\equiv 0$ $\ \left({\rm mod}\ p\right)$ for some
$n\geq 0$, and that a moonshine module is a weakly $p$-adic moonshine module
if each McKay–Thompson series is weakly $p$-adically annihilated. For $p\geq
5$, every McKay–Thompson series in monstrous moonshine that is weakly
$p$-adically annihilated is also $p$-adically annihilated, so weakly $p$-adic
moonshine and $p$-adic moonshine coincide. For $p\in\\{2,3\\}$, however, these
notions diverge. In addition to those in Table 4.1, the McKay–Thompson series
given in Table 5.3 are weakly $p$-adically annihilated. In each case, this may
be verified with a finite check using Sturm’s bound (with at most $3500$
coefficients).

$2$ | $3$
---|---
$3$ | $15{+}$ | $30{+}$ | $2$ | $13{+}$ | $28{+}$
$5$ | $15{+}5$ | $30{+}3,5,15$ | $4{+}$ | $14{+}$ | $30{+}6,10,15$
$6{+}6$ | $15|3$ | $30{+}5,6,30$ | $4|2{+}$ | $14{+}7$ | $30{+}$
$6{+}2$ | $17{+}$ | $30|3{+}10$ | $4|2$ | $15{+}$ | $31{+}$
$6$ | $18{+}$ | $33{+}$ | $5{+}$ | $15{+}5$ | $34{+}$
$7{+}$ | $18{+}9$ | $34{+}$ | $5$ | $15{+}15$ | $36{+}$
$7$ | $19{+}$ | $35{+}$ | $6{+}6$ | $16|2{+}$ | $36|2{+}$
$9{+}$ | $20{+}4$ | $36{+}$ | $6{+}3$ | $17{+}$ | $39{+}$
$10{+}2$ | $20{+}20$ | $38{+}$ | $6$ | $18{+}9$ | $40|4{+}$
$10{+}10$ | $21{+}$ | $41{+}$ | $7{+}$ | $18{+}18$ | $40|2{+}$
$10$ | $21{+}21$ | $42{+}$ | $8|2{+}$ | $19{+}$ | $42{+}$
$12{+}4$ | $24{+}8$ | $42{+}6,14,21$ | $8|4$ | $20|2{+}$ | $45{+}$
$12{+}12$ | $24{+}24$ | $51{+}$ | $10{+}$ | $20|2{+}5$ | $48|2{+}$
$13{+}$ | $26{+}$ | $56{+}$ | $10{+}10$ | $20|2{+}10$ | $51{+}$
$13$ | $26{+}26$ | $60{+}$ | $12{+}$ | $21{+}$ | $60|2{+}$
$14{+}$ | $28{+}$ | $66{+}$ | $12{+}4$ | $24|2{+}$ | $60|2{+}5,6,30$
$14{+}7$ | $29{+}$ | $70{+}$ | $12|2{+}$ | $24|4{+}6$ | $62{+}$
$14{+}14$ |  |  | $12|2{+}6$ | $26{+}$ | $78{+}$
|  |  | $12|2{+}2$ | $28|2{+}$ | $84|2{+}$
Table 5.3: Additional weakly $p$-adically annihilated McKay–Thompson series.

Using GAP [21], we have found that for each $p\in{2,3,5,7,11}$, the
centralizer of a $pA$-pure elementary abelian subgroup of the monster of order
$p^{2}$ has weakly $p$-adic moonshine given by restricting the monster module.
The ATLAS names of these groups, which were found using [9, 42], are given in
Table 5.4. In fact, each group in Table 5.4 intersects only those conjugacy
classes whose Hauptmoduln $\mathcal{T}$ satisfy $\mathcal{T}|U_{p}\equiv 0\
\left({\rm mod}\ p\right)$, which is somewhat stronger than weakly $p$-adic
annihilation.

$p$ | $2$ | $3$ | $5$ | $7$ | $11$
---|---|---|---|---|---
$C\big{(}pA^{2}\big{)}$ | $2^{2}\cdot{}^{2}\\!E_{6}(2)$ | $3^{2}\times O_{8}^{+}(3)$ | $5^{2}\times U_{3}(5)$ | $7^{2}\times L_{2}(7)$ |  $11^{2}$ 2pt2pt
2pt $\\#C\big{(}pA^{2}\big{)}$ | $2^{38}{\cdot}3^{9}{\cdot}5^{2}{\cdot}7^{2}{\cdot}11{\cdot}13{\cdot}17{\cdot}19$ | $2^{12}{\cdot}3^{14}{\cdot}5^{2}{\cdot}7{\cdot}13$ | $2^{4}{\cdot}3^{2}{\cdot}5^{5}{\cdot}7$ | $2^{3}{\cdot}3{\cdot}7^{3}$ | $11^{2}$
Table 5.4: Subgroups of the monster with weakly $p$-adic moonshine.

In light of this, it is natural to ask whether there are other natural
subgroups of the monster having $p$-adic or weakly $p$-adic moonshine and
whether there is an explanation intrinsic to the monster for the existence of
weakly $p$-adic moonshine for these subgroups. More generally, we pose the
question of whether the results of this paper extend to other known cases of
moonshine, such as Conway moonshine [15], umbral moonshine [14], and Thompson
moonshine [24]. Do analogues of the groups in Table 5.4 exist for these other
groups?

##  Appendix A Table of annihilation

The following table gives the precise congruences that numerically appear to
be satisfied by each Hauptmodul. The notation $a_{1},\dots,a_{m}\to
b_{1},\dots,b_{n}$ means the sequence beginning $a_{1},\dots,a_{m}$, and then
each subsequent term is given by adding $b_{k\ \left({\rm mod}\ n\right)}$
from $k=1$ to $\infty$. For example, $0,1\to 0,3$ is the sequence
$0,1,1,4,4,7,7,10,10,\dots$. The entry under $p$ for the group $\Gamma$
indicates the sequence $a_{1},a_{2},a_{3},\dots$ such that $a_{n}$ is the
highest power of $p$ dividing $\mathcal{T}_{\Gamma}|U_{p}^{n}$. If no such
cyclic pattern is clear, then we simply list the first few terms of the
sequence in parentheses.

Class | Group | $p=2$ | $p=3$ | $p=5$ | $p=7$ | $p=11$
---|---|---|---|---|---|---
1A | $1$ | $11\to 3$ | $5\to 2$ | $2\to 1$ | $1\to 1$ | $1\to 1$
2A | $2{+}$ | $11\to 3$ | $(3,5,9,9,11)$ | $(1,2,3,5,5)$ | $1\to 1$ | $0\to 0$
2B | $2$ | $11\to 3$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
3A | $3{+}$ | $5\to 1$ | $5\to 2$ | $(1,2,3,5,5)$ | $0\to 0$ | $0\to 0$
3B | $3$ | $2\to 0$ | $5\to 2$ | $0\to 0$ | $0\to 0$ | $0\to 0$
3C | $3|3$ | $3\to 3,0$ | $\infty$ | $0\to 1,0$ | $1\to 1$ | $0\to 1,0$
4A | $4{+}$ | $11\to 3$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
4B | $4|2{+}$ | $\infty$ | $1\to 0$ | $1\to 1$ | $1\to 1$ | $0\to 1,0$
4C | $4$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
4D | $4|2$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
5A | $5{+}$ | $3\to 2,1,3,0$ | $1\to 0$ | $2\to 1$ | $0\to 0$ | $0\to 0$
5B | $5$ | $1\to 0$ | $1\to 0$ | $2\to 1$ | $0\to 0$ | $0\to 0$
6A | $6{+}$ | $5\to 1$ | $(3,5,9,9,11)$ | $1\to 1$ | $0\to 0$ | $0\to 0$
6B | $6{+}6$ | $2\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
6C | $6{+}3$ | $5\to 1$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
6D | $6{+}2$ | $2\to 0$ | $(3,5,9,9,11)$ | $0\to 0$ | $0\to 0$ | $0\to 0$
6E | $6$ | $2\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
6F | $6|3$ | $3\to 3,0$ | $\infty$ | $0\to 1,0$ | $0\to 0$ | $0\to 0$
7A | $7{+}$ | $2\to 0$ | $1\to 0$ | $1\to 1$ | $1\to 1$ | $0\to 0$
7B | $7$ | $2\to 0$ | $0\to 0$ | $0\to 0$ | $1\to 1$ | $0\to 0$
8A | $8{+}$ | $7\to 3$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
8B | $8|2{+}$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
8C | $8|4{+}$ | $\infty$ | $0\to 1,0$ | $1\to 1$ | $0\to 1,0$ | $0\to 0$
8D | $8|2$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
8E | $8$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
8F | $8|4$ | $\infty$ | $1\to 0$ | $0\to 1,0$ | $0\to 0$ | $0\to 0$
9A | $9{+}$ | $1\to 0$ | $5\to 2$ | $0\to 0$ | $0\to 0$ | $0\to 0$
9B | $9$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
10A | $10{+}$ | $3\to 2,1,3,0$ | $1\to 0$ | $(1,2,3,5,5)$ | $0\to 0$ | $0\to 0$
10B | $10{+}5$ | $3\to 2,1,3,0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
10C | $10{+}2$ | $1\to 0$ | $0\to 0$ | $(1,2,3,5,5)$ | $0\to 0$ | $0\to 0$
10D | $10{+}10$ | $1\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
10E | $10$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
11A | $11{+}$ | $1\to 1,0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $1\to 1$
12A | $12{+}$ | $5\to 1$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12B | $12{+}4$ | $2\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12C | $12|2{+}$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12D | $12|3{+}$ | $3\to 3,0$ | $\infty$ | $0\to 1,0$ | $0\to 0$ | $0\to 0$
12E | $12{+}3$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12F | $12|2{+}6$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12G | $12|2{+}2$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12H | $12{+}12$ | $2\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12I | $12$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
12J | $12|6$ | $\infty$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
13A | $13{+}$ | $2\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
13B | $13$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
14A | $14{+}$ | $2\to 0$ | $1\to 0$ | $0\to 0$ | $1\to 1$ | $0\to 0$
14B | $14{+}7$ | $2\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
14C | $14{+}14$ | $2\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
15A | $15{+}$ | $1\to 0$ | $1\to 0$ | $(1,2,3,5,5)$ | $0\to 0$ | $0\to 0$
15B | $15{+}5$ | $1\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
15C | $15{+}15$ | $0\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
15D | $15|3$ | $1\to 0$ | $\infty$ | $0\to 1,0$ | $0\to 0$ | $0\to 0$
16A | $16|2{+}$ | $\infty$ | $0,1\to 0$ | $0\to 1,0$ | $0\to 0$ | $0\to 0$
16B | $16$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
16C | $16{+}$ | $4,6\to 3$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
17A | $17{+}$ | $1\to 0$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
18A | $18{+}2$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
18B | $18{+}$ | $1\to 0$ | $(3,5,9,9,11)$ | $0\to 0$ | $0\to 0$ | $0\to 0$
18C | $18{+}9$ | $1\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
18D | $18$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
18E | $18{+}18$ | $0\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
19A | $19{+}$ | $1,2\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
20A | $20{+}$ | $3\to 2,1,3,0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
20B | $20|2{+}$ | $\infty$ | $1\to 0$ | $1\to 1$ | $0\to 0$ | $0\to 0$
20C | $20{+}4$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
20D | $20|2{+}5$ | $\infty$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
20E | $20|2{+}10$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
20F | $20{+}20$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
21A | $21{+}$ | $1\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
21B | $21{+}3$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
21C | $21|3{+}$ | $0\to 0$ | $\infty$ | $0\to 1,0$ | $1\to 1$ | $0\to 0$
21D | $21{+}21$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
22A | $22{+}$ | $1\to 1,0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
22B | $22{+}11$ | $1\to 1,0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
23AB | $23{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24A | $24|2{+}$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24B | $24{+}$ | $3\to 1$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24C | $24{+}8$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24D | $24|2{+}3$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24E | $24|6{+}$ | $\infty$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24F | $24|4{+}6$ | $\infty$ | $1\to 0$ | $(0,0,1,2,2)$ | $0\to 0$ | $0\to 0$
24G | $24|4{+}2$ | $\infty$ | $0\to 1,0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24H | $24|2{+}12$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24I | $24{+}24$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
24J | $24|12$ | $\infty$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
25A | $25{+}$ | $0\to 0$ | $0\to 0$ | $2\to 1$ | $0\to 0$ | $0\to 0$
26A | $26{+}$ | $2\to 0$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
26B | $26{+}26$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
27AB | $27{+}$ | $0\to 0$ | $2\to 2$ | $0\to 0$ | $0\to 0$ | $0\to 0$
28A | $28|2{+}$ | $\infty$ | $0,1\to 0$ | $0\to 0$ | $1\to 1$ | $0\to 0$
28B | $28{+}$ | $2\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
28C | $28{+}7$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
28D | $28|2{+}14$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
29A | $29{+}$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
30A | $30{+}6,10,15$ | $0\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
30B | $30{+}$ | $1\to 0$ | $1\to 0$ | $1\to 1$ | $0\to 0$ | $0\to 0$
30C | $30{+}3,5,15$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
30D | $30{+}5,6,30$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
30E | $30|3{+}10$ | $1\to 0$ | $\infty$ | $0\to 1,0$ | $0\to 0$ | $0\to 0$
30F | $30{+}2,15,30$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
30G | $30{+}15$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
31AB | $31{+}$ | $0\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
32A | $32{+}$ | $2,3,5\to 3$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
32B | $32|2{+}$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
33A | $33{+}11$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
33B | $33{+}$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
34A | $34{+}$ | $1\to 0$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
35A | $35{+}$ | $1\to 0$ | $0\to 0$ | $1\to 1$ | $0\to 0$ | $0\to 0$
35B | $35{+}35$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
36A | $36{+}$ | $1\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
36B | $36{+}4$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
36C | $36|2{+}$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
36D | $36{+}36$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
38A | $38{+}$ | $1,2\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
39A | $39{+}$ | $0\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
39B | $39|3{+}$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
39CD | $39{+}39$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
40A | $40|4{+}$ | $\infty$ | $0,1\to 0$ | $(1,2,3,5,5)$ | $0\to 0$ | $0\to 0$
40B | $40|2{+}$ | $\infty$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
40CD | $40|2{+}20$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
41A | $41{+}$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
42A | $42{+}$ | $1\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
42B | $42{+}6,14,21$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
42C | $42|3{+}7$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
42D | $42{+}3,14,42$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
44AB | $44{+}$ | $1\to 1,0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
45A | $45{+}$ | $0\to 0$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
46AB | $46{+}23$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
46CD | $46{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
47AB | $47{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
48A | $48|2{+}$ | $\infty$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
50A | $50{+}$ | $0\to 0$ | $0\to 0$ | $(1,2,3,5,5)$ | $0\to 0$ | $0\to 0$
51A | $51{+}$ | $1\to 0$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
52A | $52|2{+}$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
52B | $52|2{+}26$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
54A | $54{+}$ | $0\to 0$ | $(1,2,4,8,8)$ | $0\to 0$ | $0\to 0$ | $0\to 0$
55A | $55{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
56A | $56{+}$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
56BC | $56|4{+}14$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
57A | $57|3{+}$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
59AB | $59{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
60A | $60|2{+}$ | $\infty$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
60B | $60{+}$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
60C | $60{+}4,15,60$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
60D | $60{+}12,15,20$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
60E | $60|2{+}5,6,30$ | $\infty$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
60F | $60|6{+}10$ | $\infty$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
62AB | $62{+}$ | $0\to 0$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
66A | $66{+}$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
66B | $66{+}6,11,66$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
68A | $68|2{+}$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
69AB | $69{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
70A | $70{+}$ | $1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
70B | $70{+}10,14,35$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
71AB | $71{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
78A | $78{+}$ | $0\to 0$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
78BC | $78{+}6,26,39$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
84A | $84|2{+}$ | $\infty$ | $0,1\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
84B | $84|2{+}6,14,21$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
84C | $84|3{+}$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
87AB | $87{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
88AB | $88|2{+}$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
92AB | $92{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
93AB | $93|3{+}$ | $0\to 0$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$
94AB | $94{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
95AB | $95{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
104AB | $104|4{+}$ | $\infty$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
105A | $105{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
110A | $110{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$
119AB | $119{+}$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$ | $0\to 0$

##  Appendix B Power maps

For each $p\in\\{3,5,7,11\\}$ 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 Fig. 1.1 in the introduction). In each diagram below, we
write the groups $\Gamma$ such that $\mathcal{T}_{\Gamma}$ is $p$-adically
annihilated by $U_{p}$ but $\mathcal{T}_{\Gamma}|U_{p}\neq 0$, and all of the
powers of such groups. Solid lines indicate power maps, groups in white boxes
satisfy $\mathcal{T}_{\Gamma}|U_{p}=0$, and groups in black boxes are not
$p$-adically annihilated by $U_{p}$ at all.

$1$$3$$2{+}$$3{+}$$9$$9{+}$$6{+}2$$4|2{+}$$6{+}$$27{+}$$18{+}2$$18{+}$$12|2{+}2$$8|4{+}$$54{+}$$24|4{+}2$
Figure B.1: Power maps for $3$-adically annihilated Hauptmoduln.
$1$$3{+}$$5{+}$$2{+}$$7{+}$$5$$3|3$$2$$3$$6{+}$$15{+}$$10{+}$$4|2{+}$$35{+}$$10{+}2$$25{+}$$15|3$$21|3{+}$$6|3$$10{+}10$$4{+}$$4$$4|2$$6{+}6$$30{+}$$20|2{+}$$8|4{+}$$50{+}$$30|3{+}10$$12|3{+}$$8{+}$$8|4$$12|2{+}6$$40|4{+}$$16|2{+}$$24|4{+}6$
Figure B.2: Power maps for $5$-adically annihilated Hauptmoduln.

$1$$3|3$$7{+}$$2{+}$$7$$21|3{+}$$14{+}$$4|2{+}$$28|2{+}$$8|4{+}$ Figure B.3:
Power maps for $7$-adically annihilated Hauptmoduln

$1$$3|3$$2{+}$$11{+}$$4|2{+}$ Figure B.4: Power maps for $11$-adically
annihilated Hauptmoduln

### Acknowledgements

We wish to thank Ken Ono, John Duncan, and Larry Rolen for their support and
guidance, as well as Frank Calegari for useful information regarding lifts of
characteristic $p$ modular forms for the primes $p=2,3$. We thank our referees
for helpful comments. We also thank Emory University, Princeton University,
the Asa Griggs Candler Fund, and NSF grant DMS-1557960.

## References

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