#  $2$-Elongated Plane Partitions and Powers of 7: The Localization Method
Applied to a Genus 1 Congruence Family

Koustav Banerjee  and  Nicolas Allen Smoot

###### Abstract.

Over the last century, a large variety of infinite congruence families have
been discovered and studied, exhibiting a great variety with respect to their
difficulty. Major complicating factors arise from the topology of the
associated modular curve: classical techniques are sufficient when the
associated curve has cusp count 2 and genus 0. Recent work has led to new
techniques that have proven useful when the associated curve has cusp count
greater than 2 and genus 0. We show here that these techniques may be adapted
in the case of positive genus. In particular, we examine a congruence family
over the 2-elongated plane partition diamond counting function $d_{2}(n)$ by
powers of 7, for which the associated modular curve has cusp count 4 and genus
1. We compare our method with other techniques for proving genus 1 congruence
families, and present a second congruence family by powers of 7 which we
conjecture, and which may be amenable to similar techniques.

###### Key words and phrases:

Partition congruences, infinite congruence family, modular functions, plane
partitions, partition analysis, modular curve, Riemann surface

###### 2010 Mathematics Subject Classification:

Primary 11P83, Secondary 30F35

##  1\. Introduction

###  1.1. Background

The first great breakthrough in the study of the arithmetic properties of
partitions began with the justifiably celebrated infinite congruence families
discovered by Ramanujan in 1918 [15] and refined by Watson in 1938 [20]:

######  Theorem 1.1.

$\displaystyle\text{If }n,\alpha\in\mathbb{Z}_{\geq 1}\text{ and }24n\equiv
1\pmod{\ell^{\alpha}},$ $\displaystyle\text{ then }p(n)\equiv
0\pmod{\ell^{\beta}}\text{, with}$ (1.1)
$\displaystyle\beta:=\begin{cases}\alpha&\text{ if }\ell\in\\{5,11\\},\\\
\left\lfloor\frac{\alpha}{2}\right\rfloor+1&\text{ if }\ell=7.\end{cases}$
(1.2)

The cases of this theorem for $\ell=5,7$ are comparatively easy to understand.
Ramanujan may have understood the proof of the case of $\ell=5$ as early as
1918 [6], and the proof of the case of $\ell=7$, published by Watson in 1938
[20], is similar in form. The case for $\ell=11$ is far more difficult, and a
proof was not found before Atkin’s work in 1967 [3].

In the decades since, an enormous variety of partition functions and
generalizations have been shown to exhibit divisibility properties which at
least superficially resemble those of Ramanujan’s congruences. Moreover—again
as with Ramanujan’s congruences—the difficulty of understanding these
congruence families varies enormously.

The reason that such congruence families occur at all is not fully understood,
nor is the substantial range in the difficulty of different families. We do
know that each congruence family for a given partition function $a(n)$ is
generally associated with a sequence of meromorphic functions over a compact
Riemann surface. Indeed, if $a(n)$ is enumerated by an eta quotient, then this
surface is a classical modular curve $\mathrm{X}_{0}(N)$ for some $N\geq 1$.
The topology of that curve appears critical in understanding why some
congruence families are easier to prove than others.

There are two topological numbers of $\mathrm{X}_{0}(N)$ which are important
to us: the genus and the cusp count. In the case of Theorem 1.1 for $\ell=5,7$
the genus is 0. For $\ell=11$ the genus is 1. The cusp count is 2 in all three
cases.

In recent years the study of congruence families associated with modular
curves of a more complex topology have given rise to new techniques,
especially those embodied in the localization method. These techniques have
been useful when studying congruence families in which the associated modular
curve has genus 0 and cusp count 4.

On his first approach using these methods, Smoot argued [18] that the
techniques of localization would not succeed when applied to a congruence
family associated with a curve of nonzero genus. In the central result of this
paper we delightfully acknowledge that we were mistaken. We have successfully
applied localization to the following congruence family for the 2-elongated
plane partition diamond function $d_{2}(n)$:

######  Theorem 1.2.

Let $n,\alpha\in\mathbb{Z}_{\geq 1}$ such that $8n\equiv 1\pmod{7^{\alpha}}$.
Then $d_{2}(n)\equiv 0\pmod{7^{\left\lfloor\alpha/2\right\rfloor}}$.

This family was conjectured by Banerjee, and is associated with the modular
curve $\mathrm{X}_{0}(14)$. This curve has cusp count 4 and genus 1.

The implications for this breakthrough are substantial. Unlike other methods
for proving congruence families on curves of nontrivial topology, the
techniques of localization are made to give an algebraic structure which
arises naturally from the curve itself. These techniques provide a
straightforward procedure for anyone who wishes to prove a given congruence
family “from scratch,” as it were.

To date, some families have been shown to be resistant to such techniques. We
have already attempted to adapt such techniques to a proof of the celebrated
Andrews–Sellers congruence family [12]—whose associated modular curve has
genus 1 and cusp count 6. Our attempts have thus far failed. Comparing the
difficulty of proving this case to our success in proving Theorem 1.2, we can
confidently predict that the most important contributor to the overall
difficulty of a congruence family is the cusp count—the genus appears to play
an insignificant role with respect to the accessibility of an associated
congruence family to proof by our methods.

This is not to say that the genus has no effect at all—rather, it appears that
the genus significantly complicates the “bookkeeping” aspect of the problem.
An enormous amount of information must be carefully tracked and recorded. Of
particular interest to us are problems related to the existence and proper
representation for the associated reference functions, as well as some
peculiarities in the algebraic structure of the associated localization rings.
These problems are shared to some extent with those of Theorem 1.1 in the case
that $\ell=11$.

###  1.2. $k$-Elongated Plane Partition Diamonds

In previous papers, the authors have examined a generalization of $p(n)$ in
the form of the counting function for $k$-elongated plane partition diamonds,
$d_{k}(n)$. These counting functions are enumerated by the generating function

$\displaystyle
D_{k}(q):=\sum_{n=0}^{\infty}d_{k}(n)q^{n}=\prod_{m=1}^{\infty}\frac{(1-q^{2m})^{k}}{(1-q^{m})^{3k+1}}.$
(1.3)

Notice that this is a class of generalizations for the unrestricted partition
function, as $d_{0}(n)=p(n)$. One appeal to studying these functions is that
they are interesting arithmetic objects in and of themselves.

However, what makes them particularly interesting to us is the fact that when
congruence families arise, they appear without exception to be associated with
modular curves of cusp count 4.

Such families have been proved for $d_{2}(n)$ over powers of 3 by Smoot [17],
$d_{5}(n)$ over powers of 5 by Banerjee and Smoot [5], and $d_{7}(n)$ over
powers of 2 by Sellers and Smoot [16]. In these cases, the associated modular
curves ($\mathrm{X}_{0}(6)$, $\mathrm{X}_{0}(10)$, and $\mathrm{X}_{0}(8)$
respectively) have genus 0 and cusp count 4. The case for $d_{7}(n)$ in [16]
is peculiar, in that it admits a much simpler proof than is expected. However,
our proof of Theorem 1.2 will resemble the proofs in [17] and [5] in form.

As in these cases, we begin by building a sequence of generating functions
$\mathcal{L}:=\left(L_{\alpha}\right)_{\alpha\geq 1}$, each with a prefactor
designed to make each member of the sequence a modular function over the
associated congruence subgroup $\Gamma_{0}(14)$ (equivalently, a meromorphic
function over $\mathrm{X}_{0}(14)$).

$\displaystyle L_{\alpha}:=\Phi_{\alpha}\cdot\sum_{8n\equiv
1\bmod{7^{\alpha}}}d_{2}(n)q^{\left\lfloor n/7^{\alpha}\right\rfloor+1},$
(1.4)

with $q:=e^{2\pi i\tau}$, $\tau\in\mathbb{H}$. We also construct an
alternating sequence of linear operators, $U^{(0)}$ and $U^{(1)}$, such that

$\displaystyle L_{\alpha+1}=U^{(\alpha)}\left(L_{\alpha}\right),$ (1.5)

with $U^{(\alpha)}\in\left\\{U^{(0)},U^{(1)}\right\\}$ depending on the parity
of $\alpha$.

The idea is to represent each $L_{\alpha}$ in terms of some convenient
reference functions that arise naturally from the space, and then to study the
application of $U^{(\alpha)}$ on these functions.

To do this, we note that because $\mathrm{X}_{0}(14)$ is compact, any function
which is holomorphic along the whole curve must be a constant. Thus, our most
interesting functions must have a pole somewhere. We consider the space of
functions on $\mathrm{X}_{0}(14)$ which have a pole only at a single point of
the curve—say, at the cusp $[0]$ (see the next section for details on the
construction of $\mathrm{X}_{0}(14)$).

The Weierstraß gap theorem [13] dictates that the space of all such functions
has the form

$\displaystyle\mathcal{M}^{0}\left(\mathrm{X}_{0}(14)\right)=\mathbb{C}[x]\oplus
y\mathbb{C}[x],$ (1.6)

for some functions $x,y$. In particular, $x$ ought to have order $-2$ at
$[0]$, while $y$ has order $-3$ at the same.

Now, in the case that the cusp count is 2, the functions $L_{\alpha}$ must not
have negative order at any points in $\mathrm{X}_{0}(14)$ besides $[0]$.
However, when the cusp count is larger than 2, this is no longer true.
Therefore, to kill any other poles that $L_{\alpha}$ may have, we consider
multiplying $L_{\alpha}$ by some function $z$ which lives in
$\mathcal{M}^{0}\left(\mathrm{X}_{0}(14)\right)$ and which has positive order
at all other poles of $L_{\alpha}$. It is useful that algorithms exist for
which we may construct a suitable eta quotient. One such quotient has the form

$\displaystyle z:=z(\tau)$
$\displaystyle=\frac{(q^{2};q^{2})_{\infty}^{7}(q^{7};q^{7})_{\infty}}{(q;q)_{\infty}^{7}(q^{14};q^{14})_{\infty}}.$
(1.7)

One can immediately notice that $z\equiv 1\pmod{7}$. This is useful in that
multiplying $L_{\alpha}$ by $z$ will not affect divisibility (or lack thereof)
by 7. Moreover, $z$ has order $-2$ at $[0]$. If we define

$\displaystyle x:=\frac{z-1}{7},$ (1.8)

then $x$ satisfies the condition of (1.6). Thus,

$\displaystyle(1+7x)^{n}L_{\alpha}$

can be made a member of $\mathcal{M}^{0}\left(\mathrm{X}_{0}(14)\right)$ for
an appropriately chosen $n$. There are different options for the function $y$.
One notable option is

$\displaystyle y_{0}:=y_{0}(\tau)$
$\displaystyle=\frac{(q^{2};q^{2})_{\infty}^{4}(q^{7};q^{7})_{\infty}^{8}}{(q;q)_{\infty}^{8}(q^{14};q^{14})_{\infty}^{4}}.$
(1.9)

It will be slightly more useful to choose

$\displaystyle y:=\frac{y_{0}-1}{8},$ (1.10)

which follows from $y_{0}\equiv 1\pmod{8}$. We should expect that each
$L_{\alpha}$ has a useful representation as a member of the localization ring
$\mathbb{Z}[x]_{\mathcal{S}}\oplus y\mathbb{Z}[x]_{\mathcal{S}}$, in which

$\displaystyle\mathcal{S}:=\left\\{(1+7x)^{n}:n\geq 0\right\\}.$ (1.11)

The coefficients in the numerator of each expression should then give us some
information regarding the divisibility of $L_{\alpha}$ by powers of 7.

Aside from the comparatively easy problem of finding functions $x,y$ which are
suitable for us, this procedure so far is very straightforward. Various
complications will indeed emerge from this approach, but they do so in a
manner which allows us to very precisely state where these complications
arise.

Let us examine $L_{1}$, which we define in the following manner

$\displaystyle L_{1}$
$\displaystyle=\frac{(q^{7};q^{7})^{7}_{\infty}}{(q^{14};q^{14})^{2}_{\infty}}\cdot\sum_{n=0}^{\infty}d_{2}(7n+1)q^{n+1}$
(1.12)

(see Section 2 for a precise definition of every $L_{\alpha}$). Notice that
$L_{1}\in\mathbb{Z}[[q]]$. Now in this case we do not expect to find
divisibility by 7, since $\left\lfloor 1/2\right\rfloor=0$. Generally, we
expect that for some integer sequence $\left(\psi(\alpha)\right)_{\alpha\geq
1}$

$\displaystyle\frac{(1+7x)^{\psi}}{7^{\left\lfloor\alpha/2\right\rfloor}}\cdot
L_{\alpha}\in\mathbb{Z}[x]\oplus y\mathbb{Z}[x].$

In particular, we expect

$\displaystyle(1+7x)^{\psi}\cdot L_{1}\in\mathbb{Z}[x]\oplus y\mathbb{Z}[x].$

However, a strange complication emerges, as we see in examining $L_{1}$:

$\displaystyle L_{1}=$
$\displaystyle\frac{1}{(1+7x)^{3}}\Bigg{(}\frac{320013737}{7}x+\frac{29164229489}{7}x^{2}+\frac{1226655768017}{7}x^{3}+4505536916704x^{4}$
(1.13)
$\displaystyle+79044206825472x^{5}+999877459130368x^{6}+9391378522824704x^{7}+66411983644131328x^{8}$
$\displaystyle+354409645379944448x^{9}+1415208166316048384x^{10}+4140177110624894976x^{11}$
$\displaystyle+8532124891883765760x^{12}+11539756946659737600x^{13}+8913467434661314560x^{14}$
$\displaystyle+2773078757450186752x^{15}-\frac{320013688}{7}y-\frac{28844055074}{7}xy-171156188528x^{2}y-4337927987008x^{3}y$
$\displaystyle-74846829673728x^{4}y-928384597776384x^{5}y-8516830910414848x^{6}y-58508210959679488x^{7}y$
$\displaystyle-300982634640572416x^{8}y-1145123381897592832x^{9}y-3131903931035156480x^{10}y$
$\displaystyle-5830893280174276608x^{11}y-6623201496588615680x^{12}y-3466348446812733440x^{13}y\Bigg{)}.$

The presence of rational coefficients—indeed, the presence of rational
coefficients containing 7 in the denominators—demands our attention and
rectification. Given that $x,y,$ and indeed $L_{1}$ all have integer series
expansions in $q$, we must also have an integer $q$-series expansion for the
combination of coefficients of $x$, $x^{2}$, $x^{3}$, $y$, and $xy$. Examining
the value of the numerators of these coefficients modulo 7, we can conclude
that

$\displaystyle
r_{L}:=\frac{1}{7}\left(x+3x^{2}+x^{3}+6y+5xy\right)\in\mathbb{Z}[[q]]$ (1.14)

In particular,

$\displaystyle x+3x^{2}+x^{3}+6y+5xy\equiv 0\pmod{7}.$

The resolution of this problem lies in the behavior of the coefficients of
$x^{m}$, $x^{m}y$ modulo 7. Taking the presence of $r_{L}$ into account, there
is no other complication for the representation of $L_{\alpha}$ in terms of
$x$ and $y$. The function $1+7x$ can annihilate the poles of $L_{\alpha}$
without interfering with divisibility of $L_{\alpha}$ by powers of 7. We now
have enough information to state our main theorem:

######  Theorem 1.3.

Let

$\displaystyle\psi:=$
$\displaystyle\psi(\alpha)=\left\lfloor\frac{7^{\alpha+1}}{16}\right\rfloor,\text{
and }\beta:=\beta(\alpha)=\left\lfloor\frac{\alpha}{2}\right\rfloor.$ (1.15)

There exists an integer sequence $(k_{\alpha})_{\alpha\geq 1}$ such that, for
all $\alpha\geq 1$,

$\displaystyle\frac{(1+7x)^{\psi}}{7^{\beta}}\cdot L_{\alpha}+k_{\alpha}\cdot
r_{L}\in\mathbb{Z}[x]\oplus y\mathbb{Z}[x].$ (1.16)

This theorem was proved by an extension of previous applications of the
localization method. We were surprised by this success, as well as the
peculiar complications which emerge.

In the first place, the functions $x,y$ are indeed modular, and may be
expressed in terms of eta quotients over $\Gamma_{0}(14)$. However, neither
function is itself an eta quotient.

A second complication emerges in establishing the recurrence relations
necessary to complete an induction argument through arbitrary powers of $x$
and $1+7x$. The presence of a second variable complicates the otherwise
straightforward process of deriving a modular equation for $1+7x$. This is
resolved by an important theorem on the relationship between any two
meromorphic functions on a compact Riemann surface subject to certain
conditions on their respective orders.

The third and most interesting complication emerged right away as we examined
$L_{1}$—the presence of negative powers of 7 which must be accounted for. We
could resolve it in the very first case easily enough. However, one may expect
that a general resolution is more difficult. Indeed, this is the first
indication of the complex interactions between coefficients of $L_{\alpha}$
which manifest in what we refer to as the congruence ideal associated with
$L_{\alpha}$. We give precise definitions for this below; suffice for now to
say that a given function on $\mathrm{X}_{0}(14)$ need not behave nicely as we
apply the associated $U^{(\alpha)}$ operators.

In the case of Theorem 1.3, we observe an additional factor of 7 every two
steps, so that it is natural to expect any reference function to gain
divisibility by 7 after applying $U^{(0)}\circ U^{(1)}$. This is not generally
true. So in order for such a sequence to converge 7-adically to 0 per the
conditions of the congruence family, the basis functions
$y^{j}x^{m}/(1+7x)^{n}$ which represent each function must interact together
in some manner such that all terms not divisible by the predicted powers of 7
must cancel out.

There are different ways of expressing this idea. In an earlier paper, we
expressed this in terms of constructing a certain linear transformation
$\Omega$ between vector spaces and studying the kernel of $\Omega$. Here we
take an alternative approach. We can express this “cancelation” behavior by
constructing an ideal $I(\alpha)$ of a certain polynomial ring, associated
with $L_{\alpha}$ (indeed, for this specific problem, $I(\alpha)$ is
associated with $L_{2\alpha-1}$). If we substitute the indeterminates in this
polynomial ring with the coefficients of the deviant monomials $x^{m}$ and
$yx^{m}$, the ideal $I(\alpha)$ should vanish in order for all deviant terms
of $L_{\alpha}$ to cancel out.

Moreover, we need to establish stability of the sequence
$\mathcal{I}:=\left(I(\alpha)\right)_{\alpha\geq 1}$, i.e., we must show that
$I(\alpha+1)\subseteq I(\alpha)$ for every $\alpha\geq 1$. This ensures that
the cancelation behavior which causes a given $L_{2\alpha}$ to gain
divisibility by 7 is inherited by $L_{2(\alpha+1)}$.

This third complication is all the more important, given that stability of the
congruence ideal sequence $\mathcal{I}$ is the very last step that can be
checked, and must be so examined before completion of the proof. If
$\mathcal{I}$ cannot be properly expressed or understood, then the immensity
of work done on a proof prior to this last step may be rendered worthless. For
example, the second author has attempted to give a proof of the
Andrews–Sellers congruences using this method. So far, he has been
unsuccessful. However, he was able to determine all of the steps up to
determining stability of $\mathcal{I}$. As such, it is extremely useful to
understand when stability of $\mathcal{I}$ may be reliably determined and when
it may not, before one begins a straightforward but admittedly tedious process
of checking every other step.

What makes this work so important is that it provides evidence that the
congruence ideals $I(\alpha)$, and therefore the overall sequence, can be
properly understood in the case that
$\epsilon_{\infty}\left(\mathrm{X}_{0}(N)\right)=4$, even when the genus is
nonzero. That is, if one is faced with a conjectured infinite congruence
family in which the associated modular curve is classical with cusp count 4,
then the localization method appears to always work.

This also helps to contextualize why some congruence families are so much more
difficult than others. Once the cusp count reaches 6, the interaction of the
basis functions for each associated member of $\mathcal{L}$ becomes more
complicated, and precisely determining how all terms not divisible by the
associated prime power cancel out becomes more difficult.

To express our confidence, we state the following conjecture, due to Banerjee,
which ought to be amenable to the techniques in this paper, for the
3-elongated plane partition diamond function $d_{3}(n)$:

######  Conjecture 1.4.

Let $n,\alpha\in\mathbb{Z}_{\geq 1}$ such that $6n\equiv 1\pmod{7^{\alpha}}$.
Then $d_{3}(n)\equiv 0\pmod{7^{\left\lfloor\alpha/2\right\rfloor}}$.

As in the case of Theorem 1.2, the associated modular curve is
$\mathrm{X}_{0}(14)$, with genus 1 and cusp count 4. The reader is happily
invited to test our assertion by applying our techniques to this congruence
family.

###  1.3. Outline

The remainder of our article is organized as follows:

In Section 2 we define the key generating functions $L_{\alpha}$ which
enumerate the cases of Theorem 1.2, and the operators $U^{(\alpha)}$ which
allow us to map each $L_{\alpha}$ to $L_{\alpha+1}$. We also define the
modular equations for $z$ and $x$. In Section 3 we define our localization
rings $\mathcal{V}^{(\alpha)}$ of modular functions which contain the even-
and odd-indexed functions $L_{\alpha}$. We use our modular equations to derive
general relations which detail how $U^{(\alpha)}$ acts on elements of
$\mathcal{V}^{(\alpha)}$. These relations feature certain auxiliary functions
$h^{(\alpha)}_{\beta\gamma}$, for which we prove some useful arithmetic
properties.

Sections 4-6 give us the key steps to the proof of Theorem 1.3. In Section 4
we show that $U^{(0)}$ takes elements of $\mathcal{V}^{(0)}$ to
$\mathcal{V}^{(1)}$. We confront the more difficult matter of mapping elements
of $\mathcal{V}^{(1)}$ to $\mathcal{V}^{(0)}$ in Section 5. We show that
$U^{(1)}$ generally does not send every element from $\mathcal{V}^{(1)}$ to
$\mathcal{V}^{(0)}$, and we record the deviant terms, which may be bounded to
a finite range of variables.

We show in Section 6 how the deviant terms may be shown to cancel out through
construction and manipulation of the congruence ideals associated with each
$L_{\alpha}$, thus finishing the proof of Theorem 1.3. The implications,
questions, and future research directions of our work are discussed in Section
7.

Our proof encompasses a great deal of arithmetic, analytic, and topological
considerations, as well as a surprising application of commutative algebra. It
also demands computer algebra. The environment of function spaces associated
with this proof is somewhat hostile. An enormity of calculations was necessary
to complete the proof, and it is unreasonable to expect these to be done by
hand. To this end, we do not place all of our computations in this article.
Instead, we include a well-organized Mathematica supplement, which may be
found online at https://www3.risc.jku.at/people/nsmoot/d7congsuppG.nb, or
through direct contact with either author, and which includes the more tedious
calculations. In addition, we also have a second supplementary notebook,
https://www3.risc.jku.at/people/nsmoot/FullInitialCases014.nb, in which some
of the more tedious computations in calculating the initial relations (taking
an hour or more) are already completed.

##  2\. Setup

For want of space, we will only give an extremely short discussion of the
theory of modular functions and Riemann surfaces. The reader can consult [8]
and [9] for a more comprehensive discussion.

###  2.1. A Very Brief Overview of the Theory

Define the congruence subgroup

$\displaystyle\Gamma_{0}(N)=\Bigg{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z}):N|c\Bigg{\\}}\leq\mathrm{SL}(2,\mathbb{Z}),$

for $N\in\mathbb{Z}_{\geq 1}$. If we define the extended complex plane

$\displaystyle\hat{\mathbb{H}}:=\mathbb{H}\cup\mathbb{Q}\cup\\{\infty\\},$

then $\Gamma_{0}(N)$ acts on $\hat{\mathbb{H}}$ such that

$\displaystyle\left(\begin{pmatrix}a&b\\\ c&d\end{pmatrix},\tau\right)$
$\displaystyle\longrightarrow\frac{a\tau+b}{c\tau+d}.$

We denote this action

$\displaystyle\gamma\tau:=\frac{a\tau+b}{c\tau+d},$

for $\tau\in\hat{\mathbb{H}}$, $\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$. For a given $\tau$, we define the orbits of this group
action as

$\displaystyle[\tau]_{N}:=\left\\{\gamma\tau:\gamma\in\Gamma_{0}(N)\right\\}.$

######  Definition 2.1.

For $N\in\mathbb{Z}_{\geq 1}$, the classical modular curve of level $N$ is the
set of all orbits of $\Gamma_{0}(N)$ applied to $\hat{\mathbb{H}}$:

$\displaystyle\mathrm{X}_{0}(N):=\left\\{[\tau]_{N}:\tau\in\hat{\mathbb{H}}\right\\}.$

For $\tau\in\mathbb{Q}\cup\\{\infty\\}$, $[\tau]_{N}$ is a cusp of
$\mathrm{X}_{0}(N)$.

Since we are working almost exclusively with $\mathrm{X}_{0}(14)$, we will
sometimes denote

$\displaystyle[\tau]:=[\tau]_{14}.$ (2.1)

The curve $\mathrm{X}_{0}(N)$ is a compact 1-dimensional complex manifold,
i.e., a compact Riemann surface.

######  Definition 2.2.

Let $f:\mathbb{H}\longrightarrow\mathbb{C}$ be holomorphic on $\mathbb{H}$.
Then $f$ is a modular function over $\Gamma_{0}(N)$ if the following
properties are satisfied for every $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb{Z})$:

  1. (1)

If $\gamma\in\Gamma_{0}(N)$, we have
$\displaystyle{f\left(\gamma\tau\right)=f(\tau)}.$

  2. (2)

We have

$\displaystyle{f\left(\gamma\tau\right)=\sum_{n=n_{\gamma}}^{\infty}\alpha_{\gamma}(n)q^{n\gcd(c^{2},N)/N}},$

with $n_{\gamma}\in\mathbb{Z}$, and $\alpha_{\gamma}(n_{\gamma})\neq 0$. If
$n_{\gamma}\geq 0$, then $f$ is holomorphic at the cusp $[a/c]_{N}$.
Otherwise, $f$ has a pole of order $n_{\gamma}$, and principal part

$\displaystyle\sum_{n=n_{\gamma}}^{-1}\alpha_{\gamma}(n)q^{n\gcd(c^{2},N)/N},$
(2.2)

at the cusp $[a/c]_{N}$.

We refer to $\mathrm{ord}_{a/c}^{(N)}(f):=n_{\gamma}(f)$ as the order of $f$
at the cusp $[a/c]_{N}$.

######  Definition 2.3.

Let $\mathcal{M}\left(\Gamma_{0}(N)\right)$ be the set of all modular
functions over $\Gamma_{0}(N)$, and
$\mathcal{M}^{a/c}\left(\Gamma_{0}(N)\right)\subset\mathcal{M}\left(\Gamma_{0}(N)\right)$
to be those modular functions over $\Gamma_{0}(N)$ with a pole only at the
cusp $[a/c]_{N}$. These are commutative algebras with 1, and standard addition
and multiplication [14, Section 2.1].

As might be expected from the way modular functions are defined, there is a
one-to-one correspondence between $\mathcal{M}\left(\Gamma_{0}(N)\right)$ and
the set of meromorphic functions on $\mathrm{X}_{0}(N)$ which are holomorphic
everywhere except for the cusps [10, Chapter VI, Theorem 4A]. Moreover, the
notions of poles, zeros, and orders match between the different sets.

The correspondence itself is the natural

$\displaystyle f\longrightarrow\hat{f}:\mathrm{X}_{0}(N)$
$\displaystyle\longrightarrow\mathbb{C}\cup\\{\infty\\}$
$\displaystyle:[\tau]_{N}$ $\displaystyle\longrightarrow f(\tau).$

######  Theorem 2.4.

Let $\mathrm{X}$ be a compact Riemann surface, and let
$\hat{f}:\mathrm{X}\longrightarrow\mathbb{C}$ be analytic on the entirety of
$\mathrm{X}$. Then $\hat{f}$ is be a constant function.

######  Corollary 2.5.

For any $N\in\mathbb{Z}_{\geq 1}$, if
$f\in\mathcal{M}\left(\Gamma_{0}(N)\right)$ has nonnegative order at every
cusp of $\Gamma_{0}(N)$, then $f$ is a constant.

###  2.2. Constructing $L_{\alpha}$

We want to construct the sequence of functions
$\mathcal{L}=\left(L_{\alpha}\right)_{\alpha\geq 1}$, each of which has the
form (1.4), together with a suitable sequence of modified Atkin $U_{7}$
operators $U^{(\alpha)}$ such that (1.5) is satisfied. To this end, we
construct

$\displaystyle\mathcal{A}$
$\displaystyle:=q^{6}\frac{D_{2}(q)}{D_{2}(q^{49})}=q^{6}\frac{(q^{2};q^{2})^{2}_{\infty}(q^{49};q^{49})^{7}_{\infty}}{(q;q)^{7}_{\infty}(q^{98};q^{98})^{2}_{\infty}}.$
(2.3)

This is a modular function over $\Gamma_{0}(98)$.

A useful bookeeping notation for us will be to denote

$\displaystyle(\alpha)\in\\{0,1\\}:\ (\alpha)\equiv\alpha\pmod{2},$

for any $\alpha\in\mathbb{Z}$. With this in mind, we define the sequence of
linear operators $\left(U^{(\alpha)}\right)_{\alpha\geq 0}$ by

$\displaystyle U^{(1)}(f):=$ $\displaystyle U_{7}(f),$ (2.4) $\displaystyle
U^{(0)}(f):=$ $\displaystyle U_{7}(\mathcal{A}\cdot f),$ (2.5)

where $U_{7}$ is the following classical operator:

######  Definition 2.6.

Let $f(q)=\sum_{m\geq M}a(m)q^{m}$. Then

$\displaystyle U_{\ell}\left(f(q)\right):=\sum_{\ell m\geq M}a(\ell m)q^{m}.$
(2.6)

Many of the key properties of $U_{\ell}$ are listed in [9, Chapter 8]. We note
in particular that $U_{\ell}$ is linear, and that for any two integer power
series $f,g$ in $q=e^{2\pi i\tau}$, we have

$\displaystyle
U_{\ell}\left(f(q^{\ell})g(q)\right)=f(q)U_{\ell}\left(g(q)\right).$ (2.7)

We can now define our main function sequence
$\left(L_{(\alpha)}\right)_{\alpha\geq 0}$ by

$\displaystyle L_{2\alpha-1}(\tau)$
$\displaystyle=\frac{(q^{7};q^{7})^{7}_{\infty}}{(q^{14};q^{14})^{2}_{\infty}}\cdot\sum_{n=0}^{\infty}d_{2}(7^{2\alpha-1}n+\lambda_{2\alpha-1})q^{n+1},$
(2.8) $\displaystyle L_{2\alpha}(\tau)$
$\displaystyle=\frac{(q;q)^{7}_{\infty}}{(q^{2};q^{2})^{2}_{\infty}}\cdot\sum_{n=0}^{\infty}d_{2}(7^{2\alpha}n+\lambda_{2\alpha})q^{n+1}.$
(2.9)

Here, each $\lambda_{\alpha}$ represents the minimal positive inverse of 8
modulo $7^{\alpha}$. In particular, we compute that

$\displaystyle\lambda_{2\alpha-1}$ $\displaystyle:=\frac{1+7^{2\alpha-1}}{8},$
(2.10) $\displaystyle\lambda_{2\alpha}$
$\displaystyle:=\frac{1+7^{2\alpha+1}}{8}=\lambda_{2\alpha+1}.$ (2.11)

######  Lemma 2.7.

For all $\alpha\geq 1$, we have

$\displaystyle L_{\alpha+1}$
$\displaystyle=U^{(\alpha)}\left(L_{\alpha}\right).$ (2.12)

###### Proof.

Using the standard properties of $U_{\ell}$ for any prime $\ell$, we have the
following for $\alpha\geq 1$:

$\displaystyle U^{(2\alpha-1)}\left(L_{2\alpha-1}\right)$
$\displaystyle=U_{7}\left(L_{2\alpha-1}\right)$
$\displaystyle=U_{7}\left(\frac{1}{D_{2}(q^{7})}\sum_{n\geq
0}d_{2}\left(7^{2\alpha-1}n+\lambda_{2\alpha-1}\right)q^{n+1}\right)$
$\displaystyle=\frac{1}{D_{2}(q)}\cdot U_{7}\left(\sum_{n\geq
1}d_{2}\left(7^{2\alpha-1}(n-1)+\lambda_{2\alpha-1}\right)q^{n}\right)$
$\displaystyle=\frac{1}{D_{2}(q)}\cdot\sum_{7n\geq
1}d_{2}\left(7^{2\alpha-1}(7n-1)+\lambda_{2\alpha-1}\right)q^{n}$
$\displaystyle=\frac{1}{D_{2}(q)}\cdot\sum_{n\geq
1}d_{2}\left(7^{2\alpha}n-7^{2\alpha-1}+\lambda_{2\alpha-1}\right)q^{n}$
$\displaystyle=\frac{1}{D_{2}(q)}\cdot\sum_{n\geq
0}d_{2}\left(7^{2\alpha}n+7^{2\alpha}-7^{2\alpha-1}+\frac{1+7^{2\alpha-1}}{8}\right)q^{n+1}$
$\displaystyle=\frac{1}{D_{2}(q)}\cdot\sum_{n\geq
0}d_{2}\left(7^{2\alpha}n+\lambda_{2\alpha}\right)q^{n+1}.$ $\displaystyle
U^{(2\alpha)}\left(L_{2\alpha}\right)$
$\displaystyle=U_{7}\left(\mathcal{A}\cdot L_{2\alpha}\right)$
$\displaystyle=U_{7}\left(q^{6}\frac{D_{2}(q)}{D_{2}(q^{49})}\frac{1}{D_{2}(q)}\sum_{n\geq
0}d_{2}\left(7^{2\alpha}n+\lambda_{2\alpha}\right)q^{n+1}\right)$
$\displaystyle=\frac{1}{D_{2}(q^{7})}\cdot U_{7}\left(\sum_{n\geq
7}d_{2}\left(7^{2\alpha}(n-7)+\lambda_{2\alpha}\right)q^{n}\right)$
$\displaystyle=\frac{1}{D_{2}(q^{7})}\cdot\sum_{7n\geq
7}d_{2}\left(7^{2\alpha}(7n-7)+\lambda_{2\alpha}\right)q^{n}$
$\displaystyle=\frac{1}{D_{2}(q^{7})}\cdot\sum_{n\geq
1}d_{2}\left(7^{2\alpha+1}n-7^{2\alpha+1}+\lambda_{2\alpha}\right)q^{n}$
$\displaystyle=\frac{1}{D_{2}(q^{7})}\cdot\sum_{n\geq
0}d_{2}\left(7^{2\alpha+1}(n+1)-7^{2\alpha+1}+\lambda_{2\alpha}\right)q^{n+1}$
$\displaystyle=\frac{1}{D_{2}(q^{7})}\cdot\sum_{n\geq
0}d_{2}\left(7^{2\alpha+1}n+\lambda_{2\alpha+1}\right)q^{n+1}.$

∎

We note also that

$\displaystyle U^{(2\alpha)}\left(1\right)$
$\displaystyle=U_{7}\left(\mathcal{A}\right)=U_{7}\left(q^{6}\frac{D_{2}(q)}{D_{2}(q^{49})}\right)=\frac{1}{D_{2}(q^{7})}\cdot
U_{7}\left(q^{6}D_{2}(q)\right)$ $\displaystyle=\frac{1}{D_{2}(q^{7})}\cdot
U_{7}\left(\sum_{n=0}^{\infty}d_{k}(n)q^{n+6}\right)=\frac{1}{D_{2}(q^{7})}\cdot
U_{7}\left(\sum_{n\geq 6}d_{k}(n-6)q^{n}\right)$
$\displaystyle=\frac{1}{D_{2}(q^{7})}\cdot\sum_{7n\geq
6}d_{k}(7n-6)q^{n}=\frac{1}{D_{2}(q^{7})}\cdot\sum_{n\geq
0}d_{2}\left(7n+1\right)q^{n+1}$ $\displaystyle U^{(2\alpha)}\left(1\right)$
$\displaystyle=L_{1}.$ (2.13)

This will be useful in proving (1.13).

Now that we have a well-defined function sequence, as well as a means of going
from one member of the sequence to its successor, we now need to consider the
functions $L_{\alpha}$ in terms of some useful reference functions. We defined
$z,x,y$ in the introduction. We now build a modular equation for $z$ that will
allow us to build recurrence relations to describe how $U^{(\alpha)}$ affects
rational polynomials in $x,y$:

######  Theorem 2.8.

For $0\leq k\leq 14$, let $b_{k}\in\mathbb{Z}[X]$ be defined as in Appendix I.
We have

$\displaystyle z^{14}+\sum_{k=0}^{13}b_{k}(z(7\tau))z^{k}=0.$ (2.14)

Notice that by (2.7), $U_{7}\left(b_{k}(z(7\tau))\right)=b_{k}(z(\tau))$.

By substituting $z=1+7x$ into (2.14) and simplifying, we immediately have the
following:

######  Corollary 2.9.

For $0\leq j\leq 13$, let $a_{j}\in\mathbb{Z}[X]$ be defined as in Appendix I.
We have

$\displaystyle x^{14}+\sum_{j=0}^{13}a_{j}(x(7\tau))x^{j}=0.$ (2.15)

We prove Theorem 2.8 in our Mathematica supplement.

##  3\. Algebra Structure

###  3.1. Function Spaces

We may express the space of modular functions over $\mathrm{X}_{0}(14)$ with a
pole only at the cusp $[0]$ as $\mathbb{C}[x]\oplus y\mathbb{C}[x]$. We have
seen from the form of $L_{1}$ that we need to work over a broader space. To do
this, we define the following two sets (remembering that
$(\alpha)\in\\{0,1\\}$):

$\displaystyle\mathcal{V}^{(\alpha)}_{n}:=\left\\{\frac{1}{(1+7x)^{n}}\sum_{\begin{subarray}{c}1\leq\beta\leq
0\\\ m\geq 1-\beta\end{subarray}}s_{\beta}(m)\cdot
7^{\theta^{(\alpha)}_{\beta}(m)}\cdot y^{\beta}x^{m}:\text{ $s_{\beta}$ is
$\mathbb{Z}$-valued and has finite support}\right\\}.$ (3.1)

We have a total of four functions $\theta^{(\alpha)}_{\beta}$ which we define
in the Appendix. At times we will also refer to the more general spaces

$\displaystyle\mathcal{V}^{(\alpha)}:=\bigcup_{n\geq
1}\mathcal{V}^{(\alpha)}_{n}.$ (3.2)

Having defined both the operators $U^{(\alpha)}$ which take $L_{\alpha}$ to
$L_{\alpha+1}$ and the parent spaces $\mathcal{V}^{(\alpha)}$ of $L_{\alpha}$,
we now need to study how the former acts on the latter. We have the following
general relation:

######  Theorem 3.1.

Let $\beta\in\\{0,1\\}$, $m,n\in\mathbb{Z}$ such that $n\geq 1$, $m\geq
1-\beta$. Then there exists arrays $h^{(\alpha)}_{\beta\gamma}(m,n,r)$ which
have finite support in $r$, such that

$\displaystyle U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=$
$\displaystyle\frac{1}{(1+7x)^{7n+\kappa}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1\\\ r\geq
1-\gamma\end{subarray}}h^{(\alpha)}_{\beta\gamma}(m,n,r)7^{\pi^{(\alpha)}_{\beta\gamma}(m,r)}y^{\gamma}x^{r}.$
(3.3)

In particular, the functions $\pi^{(\alpha)}_{\beta\gamma}$ are defined in the
Appendix, and

$\displaystyle\kappa=\begin{cases}0,&(\alpha)=1,\\\ 3,&(\alpha)=0.\end{cases}$

We show that this general relation may be proved by explicit computation for
$1\leq m\leq 14$, $1\leq n\leq 14$, beyond which the relation may be proved
via recurrence properties incurred by our modular equations.

######  Lemma 3.2.

Suppose that we define

$\displaystyle\hat{\pi}^{(\alpha)}_{\beta\gamma}(m,r):=\left\lfloor\frac{7r-m+\epsilon^{(\alpha)}_{\beta\gamma}}{9}\right\rfloor$
(3.4)

for some $\epsilon^{(\alpha)}_{\beta\gamma}\in\mathbb{Z}$. If the relation

$\displaystyle U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=$
$\displaystyle\frac{1}{(1+7x)^{7n+\kappa}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1\\\ r\geq
1-\gamma\end{subarray}}h^{(\alpha)}_{\beta\gamma}(m,n,r)7^{\hat{\pi}^{(\alpha)}_{\beta\gamma}(m,r)}y^{\gamma}x^{r}$
(3.5)

holds for $1\leq m\leq 14$, $1\leq n\leq 14$, then it must also hold for all
$m,n\geq 15$.

###### Proof.

We begin by taking (2.8):

$\displaystyle
z^{14}+\sum_{k=0}^{13}b_{k}(z(7\tau))z^{k}=\sum_{k=1}^{14}b_{k}(z(7\tau))z^{k}+b_{0}(z(7\tau)).$
(3.6)

Rearranging and dividing through by $b_{0}(z(7\tau))$, we have

$\displaystyle 1=$
$\displaystyle\frac{-1}{b_{0}(z(7\tau))}\sum_{k=1}^{14}b_{k}(7\tau)\cdot
z^{k}.$ (3.7)

We now divide both sides by $z^{n}$:

$\displaystyle\frac{1}{z^{n}}=$
$\displaystyle\frac{-1}{z(7\tau)^{14}}\sum_{k=1}^{14}b_{k}(7\tau)\cdot\frac{1}{z^{n-k}}.$
(3.8)

We now multiply by a given power of $x$ and $y$:

$\displaystyle\frac{y^{\beta}x^{m}}{z^{n}}=$
$\displaystyle\frac{-1}{z(7\tau)^{14}}\sum_{k=1}^{14}b_{k}(7\tau)\frac{y^{\beta}x^{m}}{z^{n-k}}.$
(3.9)

We can go still further by

Applying $U^{(\alpha)}$ to both sides and and substituting $z=1+7x$, we have

$\displaystyle U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=$
$\displaystyle\frac{-1}{(1+7x)^{14}}\sum_{k=1}^{14}b_{k}(\tau)U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n-k}}\right).$
(3.10)

Next, we recall (2.15):

$\displaystyle x^{14}+\sum_{j=0}^{13}a_{j}(x(7\tau))x^{j}=0.$ (3.11)

If we multiple both sides by $\frac{y^{\beta}x^{m-14}}{z^{n-k}}$ for some
$m\geq 14$, we have

$\displaystyle\frac{y^{\beta}x^{m}}{z^{n-k}}+\sum_{j=0}^{13}a_{j}(x(7\tau))\frac{y^{\beta}x^{m+j-14}}{z^{n-k}}=0.$
(3.12)

Applying $U^{(\alpha)}$, rearranging, and again substituting $z=1+7x$, we have

$\displaystyle
U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n-k}}\right)=-\sum_{j=0}^{13}a_{j}(x(\tau))U^{(\alpha)}\left(\frac{y^{\beta}x^{m+j-14}}{(1+7x)^{n-k}}\right).$
(3.13)

Substituting this into (3.10), we now take $m\geq 14$, $n\geq 14$. We show
that if (3.5) applies to
$U^{(\alpha)}\left(\frac{y^{\beta}x^{m+j-14}}{(1+7x)^{n-k}}\right)$ for $0\leq
j\leq 13$, $1\leq k\leq 14$, then the relation must apply to
$U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)$. Thus, if we
verify the relation for the first 14 consecutive integral values of $m,n$,
then the relation will apply for all higher $m,n$. To this end, we have

$\displaystyle U^{(\alpha)}$
$\displaystyle\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=\frac{-1}{(1+7x)^{14}}\sum_{k=1}^{14}b_{k}(\tau)\cdot
U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n-k}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{14}}\sum_{j=0}^{13}\sum_{k=1}^{14}a_{j}(\tau)b_{k}(\tau)\cdot
U^{(\alpha)}\left(\frac{y^{\beta}x^{m+j-14}}{(1+7x)^{n-k}}\right)$ (3.14)
$\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{14}}\sum_{j=0}^{13}\sum_{k=1}^{14}a_{j}(\tau)b_{k}(\tau)\cdot\frac{1}{(1+7x)^{7(n-k)+\kappa}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1,\\\ r\geq
1-\gamma\end{subarray}}h^{(\alpha)}_{\beta\gamma}(m+j-14,n-k,r)7^{\pi^{(\alpha)}_{\beta\gamma}(m+j-14,r)}y^{\gamma}x^{r}$
(3.15) $\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{7n+\kappa}}\sum_{j=0}^{13}\sum_{k=1}^{14}w(j,k)\sum_{\begin{subarray}{c}0\leq\gamma\leq
1,\\\ r\geq
1-\gamma\end{subarray}}h^{(\alpha)}_{\beta\gamma}(m+j-14,n-k,r)7^{\pi^{(\alpha)}_{\beta\gamma}(m+j-14,r)}y^{\gamma}x^{r},$
(3.16)

where we define

$\displaystyle w(j,k):=$ $\displaystyle a_{j}(\tau)b_{k}(\tau)(1+7x)^{7(k-2)}$
$\displaystyle=$ $\displaystyle\sum_{l=1}^{L}v(j,k,l)\cdot
7^{\left\lfloor\frac{7l+j-6}{9}\right\rfloor}\cdot x^{l}.$ (3.17)

We verify that $w(j,k)$ has this form in our Mathematica supplement. This
gives us

$\displaystyle U^{(\alpha)}$
$\displaystyle\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{7n+\kappa}}\sum_{\begin{subarray}{c}0\leq j\leq
13,\\\ 1\leq k\leq 14\\\ 0\leq l\leq L\\\ 0\leq\gamma\leq 1\\\ r\geq
1-\gamma\end{subarray}}v(j,k,l)h^{(\alpha)}_{\beta\gamma}(m+j-14,n-k,r)7^{\pi^{(\alpha)}_{\beta\gamma}(m+j-14,r)+\left\lfloor\frac{7l+j-6}{9}\right\rfloor}y^{\gamma}x^{r+l}$
(3.18) $\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{7n+\kappa}}\sum_{\begin{subarray}{c}0\leq j\leq
13,\\\ 1\leq k\leq 14\\\ 0\leq l\leq L\\\ 0\leq\gamma\leq 1\\\ r\geq
1+l-\gamma\end{subarray}}v(j,k,l)h^{(\alpha)}_{\beta\gamma}(m+j-14,n-k,r-l)7^{\pi^{(\alpha)}_{\beta\gamma}(m+j-14,r-l)+\left\lfloor\frac{7l+j-6}{9}\right\rfloor}y^{\gamma}x^{r},$
(3.19)

with the latter line coming from adjusting $r$. Examining the powers of 7, we
note that

$\displaystyle\pi^{(\alpha)}_{\beta\gamma}(m+j-14,r)+\left\lfloor\frac{7l+j-6}{9}\right\rfloor$
$\displaystyle=\left\lfloor\frac{7r-(m+j-14)+\epsilon^{(\alpha)}_{\beta\gamma}}{9}\right\rfloor+\left\lfloor\frac{7l+j-6}{9}\right\rfloor$
(3.20)
$\displaystyle\geq\left\lfloor\frac{7(r+l)-m+\epsilon^{(\alpha)}_{\beta\gamma}}{9}\right\rfloor$
(3.21) $\displaystyle=\pi^{(\alpha)}_{\beta\gamma}(m,r+l).$ (3.22)

That is, for $r\geq l$,

$\displaystyle\pi^{(\alpha)}_{\beta\gamma}(m+j-14,r-l)+\left\lfloor\frac{7l+j-6}{9}\right\rfloor$
$\displaystyle\geq\pi^{(\alpha)}_{\beta\gamma}(m,r).$ (3.23)

Therefore, the power of 7 necessary for (3.5) is achieved in the coefficient
of $y^{\gamma}x^{r}$. We now note that we can define

$\displaystyle
h^{(\alpha)}_{\beta\gamma}(m,n,r):=\sum_{\begin{subarray}{c}0\leq j\leq 13,\\\
1\leq k\leq 14\\\ 0\leq l\leq
L\end{subarray}}v(j,k,l)h^{(\alpha)}_{\beta\gamma}(m+j-14,n-k,r-l)7^{\pi^{(\alpha)}_{\beta\gamma}(m+j-14,r-l)+\left\lfloor\frac{7l+j-6}{9}\right\rfloor-\pi^{(\alpha)}_{\beta\gamma}(m,r)},$
(3.24)

since the latter also has finite support in $r$. ∎

######  Lemma 3.3.

Let $\pi^{(\alpha)}_{\beta\gamma}$ be defined as in the Appendix. If the
relation

$\displaystyle U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=$
$\displaystyle\frac{1}{(1+7x)^{7n+\kappa}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1\\\ r\geq
1-\gamma\end{subarray}}h^{(\alpha)}_{\beta\gamma}(m,n,r)7^{\pi^{(\alpha)}_{\beta\gamma}(m,r)}y^{\gamma}x^{r}$
(3.25)

holds for a fixed $m$ and $1\leq n\leq 14$, then it must hold for all $n\geq
15$.

###### Proof.

We suppose that (3.25) holds for
$U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n-k}}\right)$ for some fixed
$m\geq 1$ and $n$ with $1\leq k\leq 14$. We show that the relation will then
apply to $U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)$.

Returning to (3.10), we have

$\displaystyle U^{(\alpha)}$
$\displaystyle\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)$ $\displaystyle=$
$\displaystyle\frac{-1}{(1+7x)^{14}}\sum_{k=1}^{14}b_{k}(\tau)\cdot
U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n-k}}\right)$ (3.26)
$\displaystyle=$
$\displaystyle\frac{-1}{(1+7x)^{14}}\sum_{k=1}^{14}\frac{b_{k}(\tau)}{(1+7x)^{7(n-k)+\kappa}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1,\\\ r\geq 1-\gamma\end{subarray}}h^{(\alpha)}_{\beta,\gamma}(m,n-k,r)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r)}\cdot y^{\gamma}x^{r}$ (3.27)
$\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{7n+\kappa}}\sum_{k=1}^{14}\hat{w}(k)\sum_{\begin{subarray}{c}0\leq\gamma\leq
1,\\\ r\geq 1-\gamma\end{subarray}}h^{(\alpha)}_{\beta,\gamma}(m,n-k,r)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r)}\cdot y^{\gamma}x^{r},$ (3.28)

in which we can expand

$\displaystyle\hat{w}(k):$ $\displaystyle=-b_{k}(\tau)(1+7x)^{7(k-2)}$
$\displaystyle=\begin{cases}&\displaystyle{\sum_{l=0}^{84}\hat{v}(k,l)\cdot
7^{\phi(l)}\cdot x^{l}},\ k\neq 7,14\\\
&\displaystyle{25398809+\sum_{l=1}^{84}\hat{v}(7,l)\cdot 7^{\phi(l)}\cdot
x^{l}},\ k=7,\\\ &\displaystyle{-1+\sum_{l=1}^{84}\hat{v}(14,l)\cdot
7^{\phi(l)}\cdot x^{l}},\ k=14,\end{cases}$ (3.29) $\displaystyle\phi(l):$
$\displaystyle=\left\lfloor\frac{7l+17}{9}\right\rfloor.$ (3.30)

Notice that $25398809\equiv 2\bmod{49}$. We can now express

$\displaystyle U^{(\alpha)}$
$\displaystyle\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=\frac{1}{(1+7x)^{7n+\kappa}}$
$\displaystyle\times$
$\displaystyle\Bigg{(}\sum_{\begin{subarray}{c}0\leq\gamma\leq 1,\\\ k\neq
7,14,\\\ 0\leq l\leq 84,\\\ r\geq 1-\gamma\end{subarray}}\hat{v}(k,l)\cdot
h^{(\alpha)}_{\beta,\gamma}(m,n-k,r)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r)+\phi(l)}\cdot y^{\gamma}x^{r+l}$ (3.31)
$\displaystyle+\sum_{\begin{subarray}{c}0\leq\gamma\leq 1\\\ k=7,14,\\\ 1\leq
l\leq 84,\\\ r\geq 1-\gamma\end{subarray}}\hat{v}(k,l)\cdot
h^{(\alpha)}_{\beta,\gamma}(m,n-k,r)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r)+\phi(l)}\cdot y^{\gamma}x^{r+l}$ (3.32)
$\displaystyle+\sum_{\begin{subarray}{c}0\leq\gamma\leq 1\\\ r\geq
1-\gamma\end{subarray}}\left(25398809h^{(\alpha)}_{\beta,\gamma}(m,n-7,r)-h^{(\alpha)}_{\beta,\gamma}(m,n-14,r)\right)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r)}\cdot y^{\gamma}x^{r}\Bigg{)}.$ (3.33)

Relabeling our powers of $x$, we have

$\displaystyle U^{(\alpha)}$
$\displaystyle\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=\frac{1}{(1+7x)^{7n+\kappa}}$
$\displaystyle\times$
$\displaystyle\Bigg{(}\sum_{\begin{subarray}{c}0\leq\gamma\leq 1,\\\ k\neq
7,14,\\\ 0\leq l\leq 84,\\\ r\geq 1+l-\gamma\end{subarray}}\hat{v}(k,l)\cdot
h^{(\alpha)}_{\beta,\gamma}(m,n-k,r-l)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r-l)+\phi(l)}\cdot y^{\gamma}x^{r}$ (3.34)
$\displaystyle+\sum_{\begin{subarray}{c}0\leq\gamma\leq 1\\\ k=7,14,\\\ 1\leq
l\leq 84,\\\ r\geq 1+l-\gamma\end{subarray}}\hat{v}(k,l)\cdot
h^{(\alpha)}_{\beta,\gamma}(m,n-k,r-l)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r-l)+\phi(l)}\cdot y^{\gamma}x^{r}$ (3.35)
$\displaystyle+\sum_{\begin{subarray}{c}0\leq\gamma\leq 1\\\ r\geq
1-\gamma\end{subarray}}\left(25398809h^{(\alpha)}_{\beta,\gamma}(m,n-7,r)-h^{(\alpha)}_{\beta,\gamma}(m,n-14,r)\right)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r)}\cdot y^{\gamma}x^{r}\Bigg{)}.$ (3.36)

For any fixed $m$, $\pi^{(\alpha)}_{\beta\gamma}(m,r)$ has the form

$\displaystyle\pi^{(\alpha)}_{\beta\gamma}(m,r)=\left\lfloor\frac{7r+M}{9}\right\rfloor$
(3.37)

for some $M\in\mathbb{Z}$. With this in mind, we have

$\displaystyle\pi^{(\alpha)}_{\beta\gamma}(m,r)+\phi(l)=$
$\displaystyle\left\lfloor\frac{7r+M}{9}\right\rfloor+\left\lfloor\frac{7l+17}{9}\right\rfloor\geq\left\lfloor\frac{7(r+l)+M+9}{9}\right\rfloor\geq\pi^{(\alpha)}_{\beta\gamma}(m,r+l)+1.$
(3.38)

Relabeling, we have

$\displaystyle\pi^{(\alpha)}_{\beta\gamma}(m,r-l)+\phi(l)\geq\pi^{(\alpha)}_{\beta\gamma}(m,r)+1,$
(3.39)

and the dominating power of 7 is $\pi^{(\alpha)}_{\beta\gamma}(m,r)$ in
(3.36). ∎

###### Proof of Theorem 3.1.

Every function $\pi^{(\alpha)}_{\beta\gamma}$ in the Appendix has an
associated form $\hat{\pi}^{(\alpha)}_{\beta\gamma}$ (3.4) for $m\geq 4$. For
$1\leq m\leq 3$,

$\displaystyle\pi^{(\alpha)}_{\beta\gamma}(m,r)\geq\hat{\pi}^{(\alpha)}_{\beta\gamma}(m,r).$

Therefore, we can first use Lemma 3.2 by checking the cases $1\leq m\leq 14$,
$1\leq n\leq 14$, thus proving Theorem 3.1 in all but some cases for $1\leq
m\leq 3$, and $m=0$. For $1\leq m\leq 3$ we verify these cases for $1\leq
n\leq 14$. By Lemma 3.3, we have verified these exceptional cases.

We similarly check the exceptional cases for $m=0$. ∎

###  3.2. Auxiliary Coefficients

As a consequence of Lemma 3.3, we have the following very important result:

######  Theorem 3.4.

The congruence

$\displaystyle h^{(\alpha)}_{\beta\gamma}(m,n,r)\equiv
h^{(\alpha)}_{\beta\gamma}(m,n+7,r)\pmod{49}.$ (3.40)

applies in the following cases:

$\displaystyle h^{(1)}_{00}(m,n,r),\ 1\leq m\leq 4,\ 1\leq r\leq 14,$ (3.41)
$\displaystyle h^{(1)}_{00}(m,n,r),\ m=5,\ r=1,$ (3.42) $\displaystyle
h^{(1)}_{01}(m,n,r),\ 1\leq m\leq 4,\ 1\leq r\leq 14,$ (3.43) $\displaystyle
h^{(1)}_{10}(m,n,r),\ 0\leq m\leq 2,\ 1\leq r\leq 14,$ (3.44) $\displaystyle
h^{(1)}_{10}(m,n,r),\ m=3,\ r=1$ (3.45) $\displaystyle h^{(1)}_{11}(m,n,r),\
1\leq m\leq 14,\ 1\leq r\leq 14,$ (3.46) $\displaystyle
h^{(0)}_{\beta\gamma}(m,n,r),\ 1\leq m\leq 14,\ 1\leq r\leq 14.$ (3.47)

###### Proof.

Reexamining (3.34), (3.35), (3.36), we already know that

$\displaystyle\pi^{(\alpha)}_{\beta\gamma}(m,r-l)+\phi(l)\geq\pi^{(\alpha)}_{\beta\gamma}(m,r)+1,$

so that in (3.34), (3.35) we end up with at least 1 additional power of 7.
This already tells us that

$\displaystyle
h^{(\alpha)}_{\beta\gamma}(m,n,r)\equiv\hat{v}(7,0)h^{(\alpha)}_{\beta,\gamma}(m,n-7,r)+\hat{v}(14,0)h^{(\alpha)}_{\beta,\gamma}(m,n-14,r)\pmod{7}.$

As $\hat{v}(7,0)=25398809\equiv 2\pmod{49}$ and $\hat{v}(14,0)=-1$, if we have

$\displaystyle h^{(\alpha)}_{\beta\gamma}(m,n-7,r)\equiv
h^{(\alpha)}_{\beta\gamma}(m,n-14,r)\pmod{7},$

then the congruence will persist. We go even further in our Mathematica
supplement and show that for the restrictions in (3.41)-(3.47), we have

$\displaystyle h^{(\alpha)}_{\beta\gamma}(m,n,r)\equiv
h^{(\alpha)}_{\beta\gamma}(m,n+1,r)\pmod{7}.$ (3.48)

That is, for a fixed $m,r$ in our range, $h^{(\alpha)}_{\beta\gamma}$ keeps
the same congruence value mod 7 no matter how we vary $n$. Thus, we certainly
have the congruence mod 7.

To account for the additional power of 7, we examine our sum over $k$ in
(3.34). Notice that we can therefore factor out our power
$7^{\pi^{(\alpha)}_{\beta\gamma}(m,r-l)+\phi(l)}$:

$\displaystyle\sum_{\begin{subarray}{c}0\leq\gamma\leq 1,\\\ 0\leq l\leq
84,\\\ r\geq 1+l-\gamma\end{subarray}}\sum_{\begin{subarray}{c}1\leq k\leq
14,\\\ k\neq 7,14\end{subarray}}\hat{v}(k,l)\cdot
h^{(\alpha)}_{\beta,\gamma}(m,n-k,r-l)\cdot
7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r-l)+\phi(l)}$
$\displaystyle=\sum_{\begin{subarray}{c}0\leq\gamma\leq 1,\\\ 0\leq l\leq
84,\\\ r\geq
1+l-\gamma\end{subarray}}7^{\pi^{(\alpha)}_{\beta,\gamma}(m,r-l)+\phi(l)}\sum_{\begin{subarray}{c}1\leq
k\leq 14,\\\ k\neq 7,14\end{subarray}}\hat{v}(k,l)\cdot
h^{(\alpha)}_{\beta,\gamma}(m,n-k,r-l).$

If we take the internal sum over $k$ while holding $r$, $l$ fixed, we have

$\displaystyle\sum_{\begin{subarray}{c}1\leq k\leq 14,\\\ k\neq
7,14\end{subarray}}\hat{v}(k,l)\cdot
h^{(\alpha)}_{\beta,\gamma}(m,n-k,r-l)\equiv
h^{(\alpha)}_{\beta,\gamma}(m,n,r-l)\sum_{\begin{subarray}{c}1\leq k\leq
14,\\\ k\neq 7,14\end{subarray}}\hat{v}(k,l)\pmod{7},$

since $h^{(\alpha)}_{\beta\gamma}$ has the same congruence value mod 7.
Finally, we compute in our Mathematica supplement that for any fixed $l$, we
have

$\displaystyle\sum_{\begin{subarray}{c}1\leq k\leq 14,\\\ k\neq
7,14\end{subarray}}\hat{v}(k,l)\equiv 0\pmod{7}.$ (3.49)

Similarly, we compute that for any fixed $l$,

$\displaystyle\hat{v}(7,l)+\hat{v}(14,l)\equiv 0\pmod{7}.$ (3.50)

So we know that the sums in (3.34), (3.35) are both divisible by 49 (in
addition to the factor $7^{\pi^{(\alpha)}_{\beta\gamma}(m,r)}$), and that
therefore

$\displaystyle h^{(\alpha)}_{\beta\gamma}(m,n,r)\equiv
2h^{(\alpha)}_{\beta,\gamma}(m,n-7,r)-h^{(\alpha)}_{\beta,\gamma}(m,n-14,r)\pmod{49}.$

We therefore need to directly check

$\displaystyle h^{(\alpha)}_{\beta\gamma}(m,n,r)\equiv
h^{(\alpha)}_{\beta,\gamma}(m,n+7,r)\pmod{49}.$ (3.51)

for $m,r$ in our range, and $1\leq n\leq 7$. We check this computationally in
the supplement. ∎

###  3.3. Initial Values

The relations of Theorem 3.1 for $1\leq m\leq 14$, $1\leq n\leq 14$ may be
directly checked computationally. Notably, these 196 different relations may
be reduced to a smaller number of relations in the following manner:

$\displaystyle U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)=$
$\displaystyle\frac{1}{7^{m}}\cdot
U^{(\alpha)}\left(\frac{y^{\beta}(z-1)^{m}}{z^{n}}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{7^{m}}\sum_{r=0}^{m}(-1)^{m-r}{{m}\choose{r}}\cdot
U^{(\alpha)}\left(z^{r-n}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{7^{m}}\sum_{r=0}^{m}(-1)^{m-r}{{m}\choose{r}}\cdot
U^{(\alpha)}\left((1+7x)^{r-n}\right).$

Moreover, by Theorem 2.8, we have

$\displaystyle U^{(\alpha)}\left(y^{\beta}(1+7x)^{n}\right)$
$\displaystyle=-\sum_{k=0}^{13}b_{k}(\tau)\cdot
U^{(\alpha)}\left(y^{\beta}(1+7x)^{k+n-14}\right).$

This allows us to ultimately express
$U^{(\alpha)}\left(\frac{y^{\beta}x^{m}}{(1+7x)^{n}}\right)$ in terms of
$U^{(\alpha)}\left(y^{\beta}(1+7x)^{n}\right)$ for positive $n$. Of course,
for positive $n$ we have

$\displaystyle U^{(\alpha)}\left(y^{\beta}(1+7x)^{n}\right)$
$\displaystyle=\sum_{k=0}^{n}{{n}\choose{k}}\cdot 7^{k}\cdot
U^{(\alpha)}\left(y^{\beta}x^{k}\right).$

Finally,

$\displaystyle U^{(\alpha)}\left(y^{\beta}x^{m}\right)$
$\displaystyle=-\sum_{j=0}^{13}a_{j}(\tau)\cdot
U^{(\alpha)}\left(y^{\beta}x^{m+j-14}\right).$

Therefore, if we know $U^{(\alpha)}\left(y^{\beta}z^{k}\right)$ for 14
consecutive values of $k$, and of course for $\beta\in\\{0,1\\}$, then we can
utilize Theorems 2.15, 2.8 to construct our more general relations.

We have the full construction in our Mathematica supplement, which may be
found online at https://www3.risc.jku.at/people/nsmoot/d7congsuppG.nb. Each of
our fundamental relations will have the form

$\displaystyle U^{(\alpha)}\left(y^{\beta}z^{k}\right)$
$\displaystyle=f(\alpha,\beta,k,1/z,y)\text{, for }-6\leq k\leq-1,$ (3.52)
$\displaystyle U^{(\alpha)}\left(y^{\beta}z^{k}\right)$
$\displaystyle=f(\alpha,\beta,k,z,y)\text{, for }0\leq k\leq 7,$ (3.53)

for some $f(\alpha,\beta,k,z,y)\in\mathbb{C}[z]\oplus y\mathbb{C}[z]$. We list
these in the opening section of our supplement.

We first consider (3.52). Recalling the definition of $U^{(\alpha)}$, we have

$\displaystyle
U_{7}\left(\mathcal{A}^{1-\alpha}y^{\beta}(\tau)z(\tau)^{k}\right)=f(\alpha,\beta,k,1/z,y).$
(3.54)

There are various different ways to establish this equality, but we will use a
straightforward approach: we will multiply both sides by an eta quotient which
will induce a pole at $[\infty]$ and annihilate all other poles. We then have
only to compare the principal parts and constants of either side. If they
match, then we must have full equality. In our supplement we will justify many
of our formulae using results from [14].

We have
$\mathcal{A}^{1-\alpha}y^{\beta}(\tau)z(\tau)^{k}\in\mathcal{M}\left(\Gamma_{0}(98)\right)$.
Notice that we work with $z$ rather than $x$, because we have theorems which
determine the poles and zeros of eta quotients, e.g., [14, Theorem 23].

In our Mathematica supplement, we show that if

$\displaystyle\mathfrak{m}(\tau):=\frac{1}{q^{4}}\frac{(q^{2};q^{2})_{\infty}^{5}(q^{7};q^{7})_{\infty}^{7}}{(q;q)_{\infty}(q^{14};q^{14})_{\infty}^{11}}\text{,
then}$ (3.55) $\displaystyle\mathfrak{m}(\tau)^{45}$ $\displaystyle\cdot
f(\alpha,\beta,k,1/z,y)\in\mathcal{M}^{\infty}\left(\Gamma_{0}(14)\right),$
(3.56) $\displaystyle\mathfrak{m}(7\tau)^{45}$
$\displaystyle\cdot\mathcal{A}^{1-\alpha}y^{\beta}(\tau)z(\tau)^{k}\in\mathcal{M}^{\infty}\left(\Gamma_{0}(98)\right).$
(3.57)

We thus want to prove the following:

$\displaystyle
U_{7}\left(\mathfrak{m}(7\tau)^{45}\mathcal{A}^{1-\alpha}y^{\beta}(\tau)z(\tau)^{k}\right)=\mathfrak{m}(\tau)^{45}f(\alpha,\beta,k,1/z,y),\text{
}-6\leq k\leq-1.$ (3.58)

We can directly compute and compare the principal parts and constants of
either side to show that they match, thus confirming the cases. From these
relations, we systematically construct and verify the 196 initial relations we
need for Theorem 3.1.

For (3.53), it is easier to work with $1/z$ than with $z$. The fundamental
relations have a maximum positive $z$-power of 71. We therefore divide both
sides of the relations by $z^{71}$, and we have

$\displaystyle
U_{7}\left(\mathcal{A}^{1-\alpha}y^{\beta}(\tau)z(\tau)^{k}z(7\tau)^{-71}\right)=z^{-71}f(\alpha,\beta,k,z,y)\in\mathbb{C}[1/z]\oplus
y\mathbb{C}[1/z].$ (3.59)

Taking $\mathfrak{m}$ again, we show in our supplement that

$\displaystyle\mathfrak{m}(\tau)^{74}$ $\displaystyle\cdot
z^{-71}f(\alpha,\beta,k,z,y)\in\mathcal{M}^{\infty}\left(\Gamma_{0}(14)\right),$
(3.60) $\displaystyle\mathfrak{m}(7\tau)^{74}$
$\displaystyle\cdot\mathcal{A}^{1-\alpha}y^{\beta}(\tau)z(\tau)^{k}z(7\tau)^{-71}\in\mathcal{M}^{\infty}\left(\Gamma_{0}(98)\right).$
(3.61)

We thus want to prove the following:

$\displaystyle
U_{7}\left(\mathfrak{m}(7\tau)^{74}\cdot\mathcal{A}^{1-\alpha}y^{\beta}(\tau)z(\tau)^{k}z(7\tau)^{-71}\right)=\mathfrak{m}(\tau)^{74}\cdot
z^{-71}f(\alpha,\beta,k,z,y),\text{ }0\leq k\leq 7.$ (3.62)

Proving (3.58), (3.62) therefore confirms our fundamental relations.

We use a similar approach to prove Theorem 2.8, also in the supplement.
Because of the size of each relation, we do not include them here. Every
relation, and every step in our construction, is included in our supplement.

##  4\. Partial Stability:
$\mathcal{V}^{(0)}_{n}\rightarrow\mathcal{V}^{(1)}_{7n+3}$

We can now begin to prove Theorem 1.3. We have the following important result:

######  Theorem 4.1.

Let $f\in\mathcal{V}^{(0)}_{n}$. Then $U^{(0)}(f)\in\mathcal{V}^{(1)}_{7n+3}.$

Thus, applying $U^{(0)}$ to a function like $L_{2\alpha}$ will result in a
function which, at least superficially, resembles the predicted form of
$L_{2\alpha+1}$.

###### Proof.

We let $f\in\mathcal{V}^{(0)}_{n}$ by hypothesis. Let us denote

$\displaystyle f=\frac{1}{(1+7x)^{n}}\sum_{\begin{subarray}{c}0\leq\beta\leq
1\\\ m\geq
1-\beta\end{subarray}}s_{\beta}(m)7^{\theta^{(0)}_{\beta}(m)}y^{\beta}x^{m}.$

Applying $U^{(0)}$ and remembering that the operator is linear, we have

$\displaystyle U^{(0)}\left(f\right)=$
$\displaystyle\frac{1}{(1+7x)^{7n+3}}\sum_{\begin{subarray}{c}0\leq\beta\leq
1\\\ m\geq 1-\beta\\\ 0\leq\gamma\leq 1\\\ r\geq
1-\gamma\end{subarray}}s_{\beta}(m)h^{(0)}_{\beta\gamma}(m,n,r)7^{\pi^{(0)}_{\beta\gamma}(m,r)+\theta^{(0)}_{\beta}(m)}y^{\gamma}x^{r}$
(4.1) $\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{7n+3}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1\\\ r\geq
1-\gamma\end{subarray}}t_{\gamma}(r)7^{\theta^{(1)}_{\gamma}(r)}y^{\gamma}x^{r},$
(4.2)

for

$\displaystyle t_{\gamma}(r)=$
$\displaystyle\sum_{\begin{subarray}{c}0\leq\beta\leq 1\\\ m\geq
1-\beta\end{subarray}}s_{\beta}(m)h^{(0)}_{\beta\gamma}(m,n,r)7^{\pi^{(0)}_{\beta\gamma}(m,r)+\theta^{(0)}_{\beta}(m)-\theta^{(1)}_{\gamma}(r)}.$
(4.3)

It remains for us to check that

$\displaystyle\pi^{(0)}_{\beta\gamma}(m,r)+\theta^{(0)}_{\beta}(m)\geq\theta^{(1)}_{\gamma}(r),$
(4.4)

which confirms membership in $\mathcal{V}^{(1)}_{7n+3}$. This is verified in
the subsections below.

∎

###  4.1.
$\displaystyle{\pi^{(0)}_{00}(m,r)+\theta^{(0)}_{0}(m)\geq\theta^{(1)}_{0}(r)}$

Here our bounds are $m\geq 1$, $r\geq 1$.

  1. (1)

$r\geq 4$, $m\geq 4$:

$\displaystyle\left\lfloor\frac{7r-m-23}{9}\right\rfloor+\left\lfloor\frac{7m-16}{9}\right\rfloor\geq\left\lfloor\frac{7r+6m-47}{9}\right\rfloor$
(4.5)
$\displaystyle\geq\left\lfloor\frac{7r-28}{9}\right\rfloor+\left\lfloor\frac{6m-19}{9}\right\rfloor$
(4.6) $\displaystyle\geq\theta^{(1)}_{0}(r)$ (4.7)

  2. (2)

$r\geq 4$, $1\leq m\leq 3$:

$\displaystyle\left\lfloor\frac{7r-m-23}{9}\right\rfloor+0\geq\left\lfloor\frac{7r-26}{9}\right\rfloor$
(4.8) $\displaystyle\geq\theta^{(1)}_{0}(r)$ (4.9)

  3. (3)

$r=3$, $m\geq 3$:

$\displaystyle-1+\left\lfloor\frac{7m-16}{9}\right\rfloor\geq-1$ (4.10)
$\displaystyle=\theta^{(1)}_{0}(r)$ (4.11)

  4. (4)

$1\leq r\leq 3$, $1\leq m\leq 2$:

$\displaystyle-1+\theta^{(0)}_{0}(m)\geq-1$ (4.12)
$\displaystyle=\theta^{(1)}_{0}(r)$ (4.13)

###  4.2.
$\displaystyle{\pi^{(0)}_{01}(m,r)+\theta^{(0)}_{0}(m)\geq\theta^{(1)}_{1}(r)}$

Here our bounds are $m\geq 1$, $r\geq 0$.

  1. (1)

$r\geq 4$, $m\geq 4$:

$\displaystyle\left\lfloor\frac{7r-m-9}{9}\right\rfloor+\left\lfloor\frac{7m-16}{9}\right\rfloor\geq\left\lfloor\frac{7r+6m-33}{9}\right\rfloor$
(4.14)
$\displaystyle\geq\left\lfloor\frac{7r-14}{9}\right\rfloor+\left\lfloor\frac{6m-19}{9}\right\rfloor$
(4.15) $\displaystyle\geq\theta^{(1)}_{1}(r)$ (4.16)

  2. (2)

$r\geq 4$, $1\leq m\leq 3$:

$\displaystyle\left\lfloor\frac{7r-m-9}{9}\right\rfloor+0\geq\left\lfloor\frac{7r-12}{9}\right\rfloor$
(4.17) $\displaystyle\geq\theta^{(1)}_{1}(r)$ (4.18)

  3. (3)

$2\leq r\leq 3$, $m\geq 4$:

$\displaystyle\left\lfloor\frac{7r-m-9}{9}\right\rfloor+\left\lfloor\frac{7m-16}{9}\right\rfloor\geq\left\lfloor\frac{7r+6m-33}{9}\right\rfloor$
(4.19)
$\displaystyle\geq\left\lfloor\frac{7r-14}{9}\right\rfloor+\left\lfloor\frac{6m-19}{9}\right\rfloor$
(4.20) $\displaystyle\geq\theta^{(1)}_{1}(r)$ (4.21)

  4. (4)

$2\leq r\leq 3$, $1\leq m\leq 3$:

$\displaystyle\left\lfloor\frac{7r-m-9}{9}\right\rfloor+0\geq\left\lfloor\frac{7r-12}{9}\right\rfloor$
(4.22) $\displaystyle\geq\theta^{(1)}_{1}(r)$ (4.23)

  5. (5)

$0\leq r\leq 1$, $m\geq 1$:

$\displaystyle-1+\theta^{(0)}_{0}(m)\geq-1=\theta^{(1)}_{1}(r)$ (4.24)

###  4.3.
$\displaystyle{\pi^{(0)}_{10}(m,r)+\theta^{(0)}_{1}(m)\geq\theta^{(1)}_{0}(r)}$

Here our bounds are $m\geq 0$, $r\geq 1$.

  1. (1)

$r\geq 4$, $m\geq 2$:

$\displaystyle\left\lfloor\frac{7r-m-27}{9}\right\rfloor+\left\lfloor\frac{7m-5}{9}\right\rfloor\geq\left\lfloor\frac{7r+6m-40}{9}\right\rfloor$
(4.25)
$\displaystyle\geq\left\lfloor\frac{7r-28}{9}\right\rfloor+\left\lfloor\frac{6m-12}{9}\right\rfloor$
(4.26) $\displaystyle\geq\theta^{(1)}_{0}(r)$ (4.27)

  2. (2)

$r\geq 4$, $m=1$:

$\displaystyle\left\lfloor\frac{7r-28}{9}\right\rfloor+0\geq\theta^{(1)}_{0}(r)$
(4.28)

  3. (3)

$r\geq 4$, $m=0$:

$\displaystyle\left\lfloor\frac{7r-28}{9}\right\rfloor+0\geq\theta^{(1)}_{0}(r)$
(4.29)

  4. (4)

$1\leq r\leq 3$, $m\geq 0$:

$\displaystyle-1+\theta^{(0)}_{1}(m)\geq-1=\theta^{(1)}_{0}(r).$ (4.30)

###  4.4.
$\displaystyle{\pi^{(0)}_{11}(m,r)+\theta^{(0)}_{1}(m)\geq\theta^{(1)}_{1}(r)}$

Here our bounds are $m\geq 0$, $r\geq 0$.

  1. (1)

$r\geq 2$, $m\geq 3$:

$\displaystyle\left\lfloor\frac{7r-m-13}{9}\right\rfloor+\left\lfloor\frac{7m-5}{9}\right\rfloor\geq\left\lfloor\frac{7r-6m-26}{9}\right\rfloor$
(4.31)
$\displaystyle\geq\left\lfloor\frac{7r-14}{9}\right\rfloor+\left\lfloor\frac{6m-12}{9}\right\rfloor$
(4.32) $\displaystyle\geq\theta^{(1)}_{1}(r)$ (4.33)

  2. (2)

$r\geq 2$, $m=2$:

$\displaystyle\left\lfloor\frac{7r-15}{9}\right\rfloor+1\geq\left\lfloor\frac{7r-6}{9}\right\rfloor$
(4.34) $\displaystyle\geq\theta^{(1)}_{1}(r)$ (4.35)

  3. (3)

$r\geq 2$, $0\leq m\leq 1$:

$\displaystyle\left\lfloor\frac{7r-m+5}{9}\right\rfloor+0\geq\left\lfloor\frac{7r+4}{9}\right\rfloor$
(4.36) $\displaystyle\geq\theta^{(1)}_{1}(r)$ (4.37)

  4. (4)

$0\leq r\leq 1$, $m\geq 0$:

$\displaystyle-1+\theta^{(0)}_{1}(m)\geq-1=\theta^{(1)}_{1}(r)$ (4.38)

##  5\. Partial Stability:
$\mathcal{V}^{(1)}_{n}\rightarrow\mathcal{V}^{(0)}_{7n}$

We now come to the more difficult step. We would like to prove a result
analogous to that of Theorem 4.1. Suppose we have some

$\displaystyle f\in\mathcal{V}^{(1)}_{n}.$

We want to prove that $U^{(1)}(f)\in\mathcal{V}^{(1)}_{7n}$. Again, we can
denote

$\displaystyle f=\frac{1}{(1+7x)^{n}}\sum_{\begin{subarray}{c}0\leq\beta\leq
1\\\ m\geq
1-\beta\end{subarray}}s_{\beta}(m)7^{\theta^{(1)}_{\beta}(m)}y^{\beta}x^{m},$

and upon applying $U^{(1)}$,

$\displaystyle U^{(1)}\left(f\right)=$
$\displaystyle\frac{1}{(1+7x)^{7n}}\sum_{\begin{subarray}{c}0\leq\beta\leq
1\\\ m\geq 1-\beta\\\ 0\leq\gamma\leq 1\\\ r\geq
1-\gamma\end{subarray}}s_{\beta}(m)h^{(1)}_{\beta\gamma}(m,n,r)7^{\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)}y^{\gamma}x^{r}$
(5.1) $\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{7n}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1\\\ r\geq
1-\gamma\end{subarray}}t_{\gamma}(r)7^{\theta^{(0)}_{\gamma}(r)}y^{\gamma}x^{r},$
(5.2)

for

$\displaystyle t_{\gamma}(r)=$
$\displaystyle\sum_{\begin{subarray}{c}0\leq\beta\leq 1\\\ m\geq
1-\beta\end{subarray}}s_{\beta}(m)h^{(1)}_{\beta\gamma}(m,n,r)7^{\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)-\theta^{(0)}_{\gamma}(r)}.$
(5.3)

Ideally, we would want to confirm that

$\displaystyle\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)\geq\theta^{(0)}_{\gamma}(r)+1.$
(5.4)

This will not be true for all cases. However, we can bound the counterexamples
to a finite set of $(m,r)$.

###  5.1.
$\displaystyle{\pi^{(1)}_{00}(m,r)+\theta^{(1)}_{0}(m)\geq\theta^{(0)}_{0}(r)+1}$

Here our bounds are $m\geq 1$, $r\geq 1$.

  1. (1)

$r\geq 3$, $m\geq 5$:

$\displaystyle\left\lfloor\frac{7r-m}{9}\right\rfloor+\left\lfloor\frac{7m-28}{9}\right\rfloor\geq\left\lfloor\frac{7r-6m-36}{9}\right\rfloor$
(5.5)
$\displaystyle\geq\left\lfloor\frac{7r-16}{9}\right\rfloor+\left\lfloor\frac{6m-20}{9}\right\rfloor$
(5.6) $\displaystyle\geq\theta^{(0)}_{0}(r)+1$ (5.7)

  2. (2)

$r\geq 3$, $m=4$:

$\displaystyle\left\lfloor\frac{7r-4}{9}\right\rfloor+0\geq\left\lfloor\frac{7r-13}{9}\right\rfloor+1$
(5.8) $\displaystyle\geq\theta^{(0)}_{0}(r)+1$ (5.9)

  3. (3)

$r\geq 9$, $1\leq m\leq 3$:

$\displaystyle\left\lfloor\frac{7r+2}{9}\right\rfloor-1\geq\left\lfloor\frac{7r-7}{9}\right\rfloor$
(5.10) $\displaystyle\geq\theta^{(0)}_{0}(r)+1.$ (5.11)

  4. (4)

$r=2$, $m\geq 6$:

$\displaystyle\left\lfloor\frac{14-m}{9}\right\rfloor+\left\lfloor\frac{7m-28}{9}\right\rfloor\geq\left\lfloor\frac{6m-22}{9}\right\rfloor$
(5.12) $\displaystyle\geq\theta^{(0)}_{0}(2)+1.$ (5.13)

  5. (5)

$r=2$, $4\leq m\leq 5$:

$\displaystyle\left\lfloor\frac{14-m}{9}\right\rfloor+\left\lfloor\frac{7m-28}{9}\right\rfloor\geq
0+1$ (5.14) $\displaystyle\geq\theta^{(0)}_{0}(2)+1.$ (5.15)

  6. (6)

$r=1$, $m\geq 6$:

$\displaystyle\left\lfloor\frac{7-m}{9}\right\rfloor+\left\lfloor\frac{7m-28}{9}\right\rfloor\geq
0+1$ (5.16) $\displaystyle\geq\theta^{(0)}_{0}(1)+1.$ (5.17)

Our deviant cases are:

  * •

$2\leq r\leq 8$, $1\leq m\leq 3$,

  * •

$r=1$, $1\leq m\leq 5$.

###  5.2.
$\displaystyle{\pi^{(1)}_{01}(m,r)+\theta^{(1)}_{0}(m)\geq\theta^{(0)}_{1}(r)+1}$

Here our bounds are $m\geq 1$, $r\geq 0$.

  1. (1)

$r\geq 2$, $m\geq 4$:

$\displaystyle\left\lfloor\frac{7r-m+16}{9}\right\rfloor+\left\lfloor\frac{7m-28}{9}\right\rfloor\geq\left\lfloor\frac{7r+6m-20}{9}\right\rfloor$
(5.18)
$\displaystyle\geq\left\lfloor\frac{7r-5}{9}\right\rfloor+\left\lfloor\frac{6m-15}{9}\right\rfloor$
(5.19) $\displaystyle\geq\theta^{(0)}_{1}(r)+1$ (5.20)

  2. (2)

$r\geq 2$, $1\leq m\leq 3$:

$\displaystyle\left\lfloor\frac{7r-m+16}{9}\right\rfloor-1$ (5.21)
$\displaystyle\geq\left\lfloor\frac{7r-3+7}{9}\right\rfloor=\left\lfloor\frac{7r+4}{9}\right\rfloor$
(5.22) $\displaystyle\geq\left\lfloor\frac{7r-5}{9}\right\rfloor+1$ (5.23)
$\displaystyle\geq\theta^{(0)}_{1}(r)+1$ (5.24)

  3. (3)

$r=1$, $m\geq 4$:

$\displaystyle\left\lfloor\frac{23-m}{9}\right\rfloor+\left\lfloor\frac{7m-28}{9}\right\rfloor\geq
0+\left\lfloor\frac{6m-13}{9}\right\rfloor$ (5.25)
$\displaystyle\geq\theta^{(0)}_{1}(1)+1$ (5.26)

  4. (4)

$r=1$, $1\leq m\leq 3$:

$\displaystyle\left\lfloor\frac{23-m}{9}\right\rfloor-1\geq\left\lfloor\frac{20}{9}\right\rfloor-1$
(5.27) $\displaystyle=2-1$ (5.28) $\displaystyle\geq\theta^{(0)}_{1}(1)+1$
(5.29)

  5. (5)

$r=0$, $m\geq 5$:

$\displaystyle\left\lfloor\frac{16-m}{9}\right\rfloor+\left\lfloor\frac{7m-28}{9}\right\rfloor\geq
0+\left\lfloor\frac{6m-20}{9}\right\rfloor$ (5.30)
$\displaystyle\geq\theta^{(0)}_{1}(0)+1$ (5.31)

  6. (6)

$r=0$, $m=4$:

$\displaystyle\left\lfloor\frac{12}{9}\right\rfloor+0=1$ (5.32)
$\displaystyle=\theta^{(0)}_{1}(0)+1$ (5.33)

The deviant cases are

  * •

$r=0$, $1\leq m\leq 3$.

###  5.3.
$\displaystyle{\pi^{(1)}_{10}(m,r)+\theta^{(1)}_{1}(m)\geq\theta^{(0)}_{0}(r)+1}$

Here our bounds are $m\geq 0$, $r\geq 1$.

  1. (1)

$r\geq 3$, $m\geq 3$:

$\displaystyle\left\lfloor\frac{7r-m-2}{9}\right\rfloor+\left\lfloor\frac{7m-14}{9}\right\rfloor\geq\left\lfloor\frac{7r+6m-24}{9}\right\rfloor$
(5.34)
$\displaystyle\left\lfloor\frac{7r-16}{9}\right\rfloor+\left\lfloor\frac{6m-8}{9}\right\rfloor$
(5.35) $\displaystyle\geq\theta^{(0)}_{0}(r)+1$ (5.36)

  2. (2)

$r\geq 3$, $m=2$:

$\displaystyle\left\lfloor\frac{7r-4}{9}\right\rfloor+0\geq\left\lfloor\frac{7r-7}{9}\right\rfloor$
(5.37) $\displaystyle=\theta^{(0)}_{0}(r)+1$ (5.38)

  3. (3)

$r\geq 11$, $0\leq m\leq 1$:

$\displaystyle\left\lfloor\frac{7r+2}{9}\right\rfloor-1=\left\lfloor\frac{7r-7}{9}\right\rfloor$
(5.39) $\displaystyle=\theta^{(0)}_{0}(r)+1$ (5.40)

  4. (4)

$r=2$, $m\geq 4$:

$\displaystyle\left\lfloor\frac{12-m}{9}\right\rfloor+\left\lfloor\frac{7m-14}{9}\right\rfloor\geq\left\lfloor\frac{6m-10}{9}\right\rfloor$
(5.41) $\displaystyle\geq 1$ (5.42) $\displaystyle=\theta^{(0)}_{0}(2)+1$
(5.43)

  5. (5)

$r=2$, $m=2$:

$\displaystyle\left\lfloor\frac{10}{9}\right\rfloor+0=1$ (5.44)
$\displaystyle=\theta^{(0)}_{0}(2)+1$ (5.45)

  6. (6)

$r=1$, $m\geq 4$:

$\displaystyle\left\lfloor\frac{5-m}{9}\right\rfloor+\left\lfloor\frac{7m-13}{9}\right\rfloor=0+\left\lfloor\frac{7m-13}{9}\right\rfloor$
(5.46) $\displaystyle\geq 1$ (5.47) $\displaystyle=\theta^{(0)}_{0}(1)+1.$
(5.48)

The deviant cases are

  * •

$r=1$, $0\leq m\leq 3$,

  * •

$r=2$, $m=3$,

  * •

$2\leq r\leq 10$, $0\leq m\leq 1$.

###  5.4.
$\displaystyle{\pi^{(1)}_{11}(m,r)+\theta^{(1)}_{1}(m)\geq\theta^{(0)}_{1}(r)+1}$

Here our bounds are $m\geq 0$, $r\geq 0$.

  1. (1)

$r\geq 2$, $m\geq 3$:

$\displaystyle\left\lfloor\frac{7r-m+3}{9}\right\rfloor+1+\left\lfloor\frac{7m-14}{9}\right\rfloor\geq\left\lfloor\frac{7r+6m-10}{9}\right\rfloor$
(5.49)
$\displaystyle\geq\left\lfloor\frac{7r-5}{9}\right\rfloor+\left\lfloor\frac{6m-5}{9}\right\rfloor$
(5.50) $\displaystyle\geq\theta^{(0)}_{1}(r)+1$ (5.51)

  2. (2)

$r\geq 2$, $m=2$:

$\displaystyle\left\lfloor\frac{7r+1}{9}\right\rfloor+1+0=\left\lfloor\frac{7r+10}{9}\right\rfloor$
(5.52) $\displaystyle\geq\theta^{(0)}_{1}(r)+1$ (5.53)

  3. (3)

$r\geq 7$, $0\leq m\leq 1$:

$\displaystyle\left\lfloor\frac{7r+4}{9}\right\rfloor+1-1$ (5.54)
$\displaystyle=\left\lfloor\frac{7r+4}{9}\right\rfloor$ (5.55)
$\displaystyle=\theta^{(0)}_{1}(r)+1$ (5.56)

  4. (4)

$r=1$, $m\geq 2$:

$\displaystyle\left\lfloor\frac{7r-m+3}{9}\right\rfloor+1+\left\lfloor\frac{7m-14}{9}\right\rfloor\geq
1$ (5.57) $\displaystyle=\theta^{(0)}_{1}(1)+1$ (5.58)

The deviant cases are

  * •

$1\leq r\leq 6$, $0\leq m\leq 1$.

##  6\. Congruence Ideal Stability

Let us review our current situation. We have a sequence
$\left(L_{\alpha}\right)_{\alpha\geq 1}$ of modular functions, together with
an alternating sequence of operators $U^{(1)}$, $U^{(0)}$ which take each
$L_{\alpha}$ to $L_{\alpha+1}$. The odd-indexed elements are apparently
members of a set $\mathcal{V}^{(1)}$, and the even-indexed elements members of
$\mathcal{V}^{(0)}$. Generally, an element of $\mathcal{V}^{(\alpha)}$ should
have the form

$\displaystyle f=\frac{1}{(1+7x)^{n}}\sum_{\begin{subarray}{c}0\leq\beta\leq
1\\\ m\geq
1-\beta\end{subarray}}s_{\beta}(m)7^{\theta^{(\alpha)}_{\beta}(m)}y^{\beta}x^{m}.$

We understand that applying $U^{(0)}$ to an element of $\mathcal{V}^{(0)}$
(e.g., $L_{2\alpha}$) produces an element of $\mathcal{V}^{(1)}$. Moreover, we
want the application of $U^{(1)}$ to an element of $\mathcal{V}^{(1)}$ (e.g.,
$L_{2\alpha-1}$) to produce an element of $\mathcal{V}^{(0)}$ with an
additional factor of 7\. However, we know that this is not generally true:
$U^{(1)}(f)$ will produce a rational polynomial in which most monomials will
gain the necessary power of 7, but not all. We are thus forced to conclude
that these deviant monomials must somehow cancel out. We must therefore
examine the various exceptional cases and find a way to keep track of the
complex behavior between the coefficients $s_{\beta}(m)$ which ensures this
cancelation.

To illustrate how this can be done, we will recall our notation. We let

$\displaystyle U^{(1)}\left(f\right)=$
$\displaystyle\frac{1}{(1+7x)^{7n}}\sum_{\begin{subarray}{c}0\leq\beta\leq
1\\\ m\geq 1-\beta\\\ 0\leq\gamma\leq 1\\\ r\geq
1-\gamma\end{subarray}}s_{\beta}(m)h^{(1)}_{\beta\gamma}(m,n,r)7^{\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)}y^{\gamma}x^{r}$
(6.1) $\displaystyle=$
$\displaystyle\frac{1}{(1+7x)^{7n}}\sum_{\begin{subarray}{c}0\leq\gamma\leq
1\\\ r\geq
1-\gamma\end{subarray}}t_{\gamma}(r)7^{\theta^{(0)}_{\gamma}(r)}y^{\gamma}x^{r},$
(6.2)

and

$\displaystyle t_{\gamma}(r)=$
$\displaystyle\sum_{\begin{subarray}{c}0\leq\beta\leq 1\\\ m\geq
1-\beta\end{subarray}}s_{\beta}(m)h^{(1)}_{\beta\gamma}(m,n,r)7^{\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)-\theta^{(0)}_{\gamma}(r)}.$
(6.3)

We first consider $t_{0}(1)$. By Sections 5.1 and 5.3, we have the following
expression:

$\displaystyle t_{0}(1)=$
$\displaystyle\frac{h^{(1)}_{00}(1,n,1)}{7}s_{0}(1)+\frac{h^{(1)}_{00}(2,n,1)}{7}s_{0}(2)+\frac{h^{(1)}_{00}(3,n,1)}{7}s_{0}(3)+h^{(1)}_{00}(4,n,1)s_{0}(4)+h^{(1)}_{00}(5,n,1)s_{0}(5)$
(6.4)
$\displaystyle+\frac{h^{(1)}_{10}(0,n,1)}{7}s_{1}(0)+\frac{h^{(1)}_{10}(1,n,1)}{7}s_{1}(1)+h^{(1)}_{10}(2,n,1)s_{1}(2)+h^{(1)}_{10}(3,n,1)s_{1}(3)+7M,$

for some $M\in\mathbb{Z}$. Not only do we need $t_{0}(1)$ to be an integer,
but we need it to be a multiple of 7. As such, we multiply through by 7 to
kill the denominators, and we have the following integer relation:

$\displaystyle
h^{(1)}_{00}(1,n,1)s_{0}(1)+h^{(1)}_{00}(2,n,1)s_{0}(2)+h^{(1)}_{00}(3,n,1)s_{0}(3)+7h^{(1)}_{00}(4,n,1)s_{0}(4)+7h^{(1)}_{00}(5,n,1)s_{0}(5)$
$\displaystyle+h^{(1)}_{10}(0,n,1)s_{1}(0)+h^{(1)}_{10}(1,n,1)s_{1}(1)+7h^{(1)}_{10}(2,n,1)s_{1}(2)+7h^{(1)}_{10}(3,n,1)s_{1}(3)\equiv
0\pmod{49}.$

Recall that $n\equiv 3\pmod{7}$, and all of the coefficients are subject to
Theorem 3.4. We can therefore restrict ourselves to working with
$h^{(1)}_{\beta\gamma}(m,3,1)$, which we can readily compute for the finite
number of necessary values of $m$:

$\displaystyle
43s_{0}(1)+48s_{0}(2)+15s_{0}(3)+42s_{0}(4)+7s_{0}(5)+8s_{1}(0)+27s_{1}(1)+7s_{1}(2)+42s_{1}(3)\equiv
0\pmod{49}.$ (6.5)

If this relation is satisfied, then $t_{0}(1)$ will be an integer divisible by
7, as we need.

Examining (1.13), we find that

$\displaystyle s_{0}(1)$ $\displaystyle=320013737\equiv 29\pmod{49},$
$\displaystyle s_{0}(2)$ $\displaystyle=29164229489\equiv 45\pmod{49},$
$\displaystyle s_{0}(3)$ $\displaystyle=1226655768017\equiv 1\pmod{49},$
$\displaystyle s_{0}(4)$ $\displaystyle=4505536916704\equiv 47\pmod{49},$
$\displaystyle s_{0}(5)$ $\displaystyle=79044206825472\equiv 42\pmod{49},$
$\displaystyle s_{1}(0)$ $\displaystyle=-320013688\equiv 20\pmod{49},$
$\displaystyle s_{1}(1)$ $\displaystyle=-28844055074\equiv 40\pmod{49},$
$\displaystyle s_{1}(2)$ $\displaystyle=-171156188528\equiv 32\pmod{49},$
$\displaystyle s_{1}(3)$ $\displaystyle=-4337927987008\equiv 0\pmod{49}.$

With these substitutions, we find that

$\displaystyle
43(29)+48(45)+15(1)+42(47)+7(42)+8(20)+27(40)+7(32)+42(0)=7154=146(49).$ (6.6)

This of course makes sense, because $L_{2}$ is indeed divisible by 7, and is a
member of $\mathcal{V}^{(0)}_{21}$.

We can similarly examine $t_{\gamma}(r)$ for the remaining deviant cases. As
it happens, this is the only relation that we will need to take modulo 49. The
remaining necessary relations may be taken modulo powers of 7, as we note in
Tables 1, 2.

$r$ | $t_{0}(r)\bmod 7$
---|---
$2$ | $h^{(1)}_{00}(1,n,2)s_{0}(1)+h^{(1)}_{00}(2,n,2)s_{0}(2)+h^{(1)}_{00}(3,n,2)s_{0}(3)+h^{(1)}_{10}(0,n,2)s_{1}(0)+h^{(1)}_{10}(1,n,2)s_{1}(1)$
$3$ | $0$
$4$ | $h^{(1)}_{10}(0,n,4)s_{1}(0)+h^{(1)}_{10}(1,n,4)s_{1}(1)$
$5$ | $h^{(1)}_{00}(3,n,5)s_{0}(3)+h^{(1)}_{10}(0,n,5)s_{1}(0)$
$6$ | $0$
$7$ | $0$
$8$ | $h^{(1)}_{10}(0,n,8)s_{1}(0)+h^{(1)}_{10}(1,n,8)s_{1}(1)$
$9$ | $h^{(1)}_{10}(0,n,9)s_{1}(0)+h^{(1)}_{10}(1,n,8)s_{1}(1)$
$10$ | $h^{(1)}_{10}(0,n,10)s_{1}(0)+h^{(1)}_{10}(1,n,10)s_{1}(1)$
Table 1. Nonvanishing terms of $t_{0}(r)\bmod 7$ $r$ | $t_{1}(r)\bmod 7$
---|---
$0$ | $h^{(1)}_{01}(3,n,0)s_{0}(3)+h^{(1)}_{11}(0,n,0)s_{1}(0)+h^{(1)}_{11}(1,n,0)s_{1}(1)$
$1$ | $0$
$2$ | $h^{(1)}_{11}(0,n,2)s_{1}(0)+h^{(1)}_{11}(1,n,2)s_{1}(1)$
$3$ | $0$
$4$ | $0$
$5$ | $0$
$6$ | $h^{(1)}_{11}(0,n,6)s_{1}(0)+h^{(1)}_{11}(1,n,6)s_{1}(1)$
Table 2. Nonvanishing terms of $t_{1}(r)\bmod 7$

These coefficients are again subject to Theorem 3.4. We can simplify to the
relations in Tables 3, 4.

$r$ | $t_{0}(r)\bmod 7$
---|---
$2$ | $3s_{0}(1)+2s_{0}(2)+2s_{0}(3)+3s_{1}(0)+4s_{1}(1)$
$3$ | $0$
$4$ | $4s_{1}(0)+5s_{1}(1)$
$5$ | $5s_{0}(3)+5s_{1}(0)$
$6$ | $0$
$7$ | $0$
$8$ | $3s_{1}(0)+2s_{1}(1)$
$9$ | $3s_{1}(0)+2s_{1}(1)$
$10$ | $s_{1}(0)+3s_{1}(1)$
Table 3. Value of $t_{0}(r)\bmod 7$ $r$ | $t_{1}(r)\bmod 7$
---|---
$0$ | $s_{0}(3)+5s_{1}(0)+5s_{1}(1)$
$1$ | $0$
$2$ | $6s_{1}(0)+4s_{1}(1)$
$3$ | $0$
$4$ | $0$
$5$ | $0$
$6$ | $s_{1}(0)+3s_{1}(1)$
Table 4. Value of $t_{1}(r)\bmod 7$

We can quickly verify by computer (or even by hand, were we so inclined) that
the coefficients $s_{\beta}(m)$ of $L_{1}$ do indeed satisfy these relations.
For want of space, we verify this in our Mathematica supplement.

Notice, however, that we are not done. It is not sufficient to verify that
$L_{1}$ satisfies the relations that we have just given. After all, there is
no way of knowing whether $L_{3}$, $L_{5}$, or $L_{44847}$ are also so
composed such that applying $U^{(1)}$ will cause all deviant monomials to
cancel out. To determine this, we construct what we call the congruence ideal
sequence associated with the congruence family. From there, we determine ideal
stability.

Let us consider a function

$\displaystyle
f_{\alpha}=\frac{1}{(1+7x)^{n}}\sum_{\begin{subarray}{c}0\leq\beta\leq 1\\\
m\geq
1-\beta\end{subarray}}s_{\alpha,\beta}(m)7^{\theta^{(1)}_{\beta}(m)}y^{\beta}x^{m},$
(6.7)

in which $\beta\in\\{0,1\\}$ as always, but in which $\alpha$ can be any
positive integer index. For our purposes, $f_{\alpha}$ will be associated with
$L_{2\alpha-1}/7^{\alpha-1}$.

It will be useful for us to define the associated vector

$\displaystyle\mathbf{s}_{\alpha}$
$\displaystyle:=\left(s_{\alpha,0}(1),s_{\alpha,0}(2),s_{\alpha,0}(3),s_{\alpha,0}(4),s_{\alpha,0}(5),s_{\alpha,1}(0),s_{\alpha,1}(1),s_{\alpha,1}(2),s_{\alpha,1}(3)\right).$
(6.8)

Now we wish to encapsulate the relations in (6.5) and Tables 3 and 4. Notice
that the relations in Tables 3, 4 are all mod 7 relations, i.e., we can
express them as polynomials in a finite field. As such, we can define the
associated generated set of functions which summarizes our relations in Tables
3 and 4. This gives us the following:

$\displaystyle 4X_{2}+6X_{3}+6X_{6}+6X_{7},$ $\displaystyle
4X_{3}+6X_{6}+6X_{7},$ $\displaystyle 4X_{6}+5X_{7},$

in which we use the indeterminate vector
$\mathbf{X}=\left(X_{1},X_{2},X_{3},X_{4},X_{5},X_{6},X_{7},X_{8},X_{9}\right)$
in place of $\mathbf{s}_{\alpha}$. We cannot do as much with (6.5), since it
is a mod $7^{2}$ relation. Instead, we simply multiply our function elements
through by 7 and work over a coefficient ring $\mathbb{Z}/49\mathbb{Z}$. We
then have the ideal

$\displaystyle I(\mathbf{X}):=$
$\displaystyle\big{(}p_{1}(\mathbf{X}),p_{2}(\mathbf{X}),p_{3}(\mathbf{X}),p_{4}(\mathbf{X})\big{)}\leq\left(\mathbb{Z}/49\mathbb{Z}\right)[\mathbf{X}],$

in which the polynomials $p_{k}(\mathbf{X})$ are defined:

$\displaystyle p_{1}(\mathbf{X})$
$\displaystyle=43X_{1}+48X_{2}+15X_{3}+42X_{4}+7X_{5}+8X_{6}+27X_{7}+7X_{8}+42X_{9},$
$\displaystyle p_{2}(\mathbf{X})$
$\displaystyle=28X_{2}+42X_{3}+42X_{6}+42X_{7},$ $\displaystyle
p_{3}(\mathbf{X})$ $\displaystyle=28X_{3}+42X_{6}+42X_{7},$ $\displaystyle
p_{4}(\mathbf{X})$ $\displaystyle=28X_{6}+35X_{7},$

By a slight abuse of notation, we let

$\displaystyle I(\alpha):=I\left(\mathbf{s}_{\alpha}\right)=$
$\displaystyle\big{(}p_{1}(\mathbf{s}_{\alpha}),p_{2}(\mathbf{s}_{\alpha}),p_{3}(\mathbf{s}_{\alpha}),p_{4}(\mathbf{s}_{\alpha})\big{)}\leq\left(\mathbb{Z}/49\mathbb{Z}\right)[\mathbf{s}_{\alpha}].$
(6.9)

######  Definition 6.1.

Let $f_{\alpha}\in\mathcal{V}^{(1)}_{n}$ as in (6.7) and $n\equiv 3\pmod{7}$,
and $\mathbf{s}_{\alpha}$ defined as in (6.8). The ideal $I(\alpha)$ is the
congruence ideal associated with $f_{\alpha}$.

######  Lemma 6.2.

Let $n\equiv 3\pmod{7}$, $f_{\alpha}\in\mathcal{V}^{(1)}_{n}$ as in (6.7), and
let $I(\alpha)$ be the associated congruence ideal. If $I(\alpha)=(0)$, then

$\displaystyle\frac{1}{7}U^{(1)}\left(f_{\alpha}\right)\in\mathcal{V}^{(0)}_{7n}.$

###### Proof.

Of course, (6.5) is the very first relation of $I(\alpha)$. The relations in
Tables 3, 4 are taken mod 7. We therefore multiply each of these relations by
7, and show that the results are members of $I(\alpha)$. If we consider the
first relation of Table 3, for example, we get

$\displaystyle
86(7)\left(3s_{0}(1)+2s_{0}(2)+2s_{0}(3)+3s_{1}(0)+4s_{1}(1)\right)=$
$\displaystyle
42p_{1}(\mathbf{s}_{\alpha})-29p_{2}(\mathbf{s}_{\alpha})+64p_{3}(\mathbf{s}_{\alpha})-1764s_{0}(4)$
$\displaystyle-294s_{0}(5)-196s_{1}(1)-294s_{1}(2)-1764s_{1}(3)$
$\displaystyle\equiv$ $\displaystyle
42p_{1}(\mathbf{s}_{\alpha})-29p_{2}(\mathbf{s}_{\alpha})+64p_{3}(\mathbf{s}_{\alpha})\pmod{49}.$

Notice that 86 is coprime with 7. Therefore,

$\displaystyle
7\left(3s_{0}(1)+2s_{0}(2)+2s_{0}(3)+3s_{1}(0)+4s_{1}(1)\right)\in I(\alpha),$

and will vanish if $I(\alpha)=(0).$ We verify membership of each 7-multiple of
the elements of Tables 3, 4 in our Mathematica supplement. ∎

We see, then, that the congruence ideal contains the information related to
the complex interactions between the basis functions that make up
$f_{\alpha}$. On applying $U^{(1)}$ to a typical element of
$\mathcal{V}^{(1)}_{n}$, as we have seen, we do not necessarily gain a
7-multiple of an element in $\mathcal{V}^{(0)}_{7n}$. However, if the
coefficients of $f_{\alpha}$ are so chosen that $I(\alpha)=(0)$, then we will
gain such a function. In particular, as we checked in (6.6) and our
Mathematica supplement, we have

$\displaystyle I(1)=(0).$ (6.10)

But, as we already noted, this is not enough. What we need to determine is
whether the interactions encoded in $I(\alpha)$ are also present in the
successor $f_{\alpha+1}$ to $f_{\alpha}$. We can of course construct the ideal
$I(\alpha+1)$ analogous to $I(\alpha)$, and we want to determine the
relationship that the former bears with respect to the latter. If we can prove
that $I(\alpha)=(0)$ implies $I(\alpha+1)=(0)$, then the structure of the
coefficients of $f_{\alpha}$ which ensures that a power of 7 will be gained
after applying $U^{(0)}\circ U^{(1)}$ will also be present with
$f_{\alpha+1}$.

######  Lemma 6.3.

Let $n\equiv 3\pmod{7}$, $f_{\alpha}\in\mathcal{V}^{(1)}_{n}$ as in (6.7) with
congruence ideal $I(\alpha)$, and let

$\displaystyle f_{\alpha+1}=$ $\displaystyle\frac{1}{7}U^{(0)}\circ
U^{(1)}\left(f_{\alpha}\right)=\frac{1}{(1+7x)^{49n+3}}\sum_{\begin{subarray}{c}0\leq\delta\leq
1\\\ w\geq
1-\delta\end{subarray}}s_{\alpha+1,\delta}(m)7^{\theta^{(1)}_{\delta}(w)}y^{\delta}x^{w}.$

Define $\mathbf{s}_{\alpha+1}$ in a manner analogous to that of
$\mathbf{s}_{\alpha}$, with congruence ideal $I(\alpha+1)$. Then we have

$\displaystyle I(\alpha+1)\subseteq I(\alpha).$

This critical lemma allows us to complete the proof of Theorem 1.3, and with
it Theorem 1.2.

###### Proof.

Recalling (5.3), adjusted with the notation of (6.7), we have

$\displaystyle t_{\gamma}(r)=\sum_{\begin{subarray}{c}0\leq\beta\leq 1\\\
m\geq
1-\beta\end{subarray}}s_{\alpha,\beta}(m)h^{(1)}_{\beta\gamma}(m,n,r)7^{\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)-\theta^{(0)}_{\gamma}(r)-1}.$
(6.11)

We take (4.1), replacing $f$ with $U^{(1)}(f_{\alpha})$. By Theorem 4.1, we
have

$\displaystyle s_{\alpha+1,\delta}(w)=$
$\displaystyle\sum_{\begin{subarray}{c}0\leq\gamma\leq 1\\\ r\geq
1-\gamma\end{subarray}}\left(\sum_{\begin{subarray}{c}0\leq\beta\leq 1\\\
m\geq
1-\beta\end{subarray}}s_{\alpha,\beta}(m)h^{(1)}_{\beta\gamma}(m,n,r)7^{\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)-\theta^{(0)}_{\gamma}(r)-1}\right)$
(6.12) $\displaystyle\times
h^{(0)}_{\gamma\delta}(r,7n,w)7^{\pi^{(0)}_{\gamma\delta}(r,w)+\theta^{(0)}_{\gamma}(r)-\theta^{(1)}_{\delta}(w)}$
(6.13) $\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}0\leq\gamma\leq
1\\\ r\geq
1-\gamma\end{subarray}}t_{\gamma}(r)h^{(0)}_{\gamma\delta}(r,7n,w)7^{\pi^{(0)}_{\gamma\delta}(r,w)+\theta^{(0)}_{\gamma}(r)-\theta^{(1)}_{\delta}(w)}.$
(6.14)

Notice that for $t_{\gamma}(r)\in\mathbb{Z}$ we must have
$s_{\alpha+1,\delta}(w)\in\mathbb{Z}$.

We want to examine, say, $p_{1}(\mathbf{s}_{\alpha+1})$ modulo 49. Notice that
$0\leq m\leq 5$ for all of our deviant cases; for $m\geq 6$, we have

$\displaystyle\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)-\theta^{(0)}_{\gamma}(r)\geq
1.$

Moreover, for $m\geq 7$, we have

$\displaystyle\pi^{(1)}_{\beta\gamma}(m,r)+\theta^{(1)}_{\beta}(m)-\theta^{(0)}_{\gamma}(r)\geq
2.$

Similarly, for $r\geq 7$, we have

$\displaystyle\pi^{(0)}_{\gamma\delta}(r,w)+\theta^{(0)}_{\gamma}(r)-\theta^{(1)}_{\delta}(w)\geq
2.$

As such, to properly study $p_{1}(\mathbf{s}_{\alpha+1})$ modulo 49, we need
only take $0\leq m\leq 6$, $0\leq r\leq 6$.

We compute this explicitly in our Mathematica supplement. Notably, by using
Theorem 3.4, we can reduce our auxiliary function
$h^{(0)}_{\beta\gamma}(r,7n,w)$ to $h^{(0)}_{\beta\gamma}(r,7,w)$. Doing so
gives us the following congruence:

$\displaystyle p_{1}(\mathbf{s}_{\alpha+1})\equiv$ $\displaystyle
25h^{(1)}_{00}(1,n,1)s_{\alpha,0}(1)+7h^{(1)}_{00}(1,n,2)s_{\alpha,0}(1)+35h^{(1)}_{01}(1,n,0)s_{\alpha,0}(1)+25h^{(1)}_{00}(2,n,1)s_{\alpha,0}(2)$
$\displaystyle+7h^{(1)}_{00}(2,n,2)s_{\alpha,0}(2)+35h^{(1)}_{01}(2,n,0)s_{\alpha,0}(2)+25h^{(1)}_{00}(3,n,1)s_{\alpha,0}(3)+7h^{(1)}_{00}(3,n,2)s_{\alpha,0}(3)$
$\displaystyle+35h^{(1)}_{01}(3,n,0)s_{\alpha,0}(3)+28h^{(1)}_{00}(4,n,1)s_{\alpha,0}(4)+28h^{(1)}_{00}(5,n,1)s_{\alpha,0}(5)+25h^{(1)}_{10}(0,n,1)s_{\alpha,1}(0)$
$\displaystyle+7h^{(1)}_{10}(0,n,2)s_{\alpha,1}(0)+35h^{(1)}_{11}(0,n,0)s_{\alpha,1}(0)+25h^{(1)}_{10}(1,n,1)s_{\alpha,1}(1)+7h^{(1)}_{10}(1,n,2)s_{\alpha,1}(1)$
$\displaystyle+35h^{(1)}_{11}(1,n,0)s_{\alpha,1}(1)+28h^{(1)}_{10}(2,n,1)s_{\alpha,1}(2)+28h^{(1)}_{10}(3,n,1)s_{\alpha,1}(3)\pmod{49}.$

Notice that the remaining non-vanishing coefficients are also subject to
Theorem 3.4. We therefore reduce $h^{(1)}_{\beta\gamma}(m,n,r)$ to
$h^{(1)}_{\beta\gamma}(m,3,r)$ mod 49, and we have

$\displaystyle p_{1}(\mathbf{s}_{\alpha+1})\equiv$ $\displaystyle
18s_{\alpha,0}(1)+38s_{\alpha,0}(2)+32s_{\alpha,0}(3)+21s_{\alpha,0}(4)+28s_{\alpha,0}(5)+4s_{\alpha,1}(0)+45s_{\alpha,1}(1)+28s_{\alpha,1}(2)+21s_{\alpha,1}(3)$
$\displaystyle\equiv$ $\displaystyle
46\left(43s_{\alpha,0}(1)+20s_{\alpha,0}(2)+22s_{\alpha,0}(3)+42s_{\alpha,0}(4)+7s_{\alpha,0}(5)+15s_{\alpha,1}(0)+34s_{\alpha,1}(1)+7s_{\alpha,1}(2)+42s_{\alpha,1}(3)\right)$
$\displaystyle\equiv$ $\displaystyle
46\cdot\left(p_{1}(\mathbf{s}_{\alpha})-p_{2}(\mathbf{s}_{\alpha})\right)\pmod{49}.$

That is,

$\displaystyle p_{1}(\mathbf{s}_{\alpha+1})$ $\displaystyle\in I(\alpha).$

We similarly determine in our Mathematica supplement that

$\displaystyle
p_{2}(\mathbf{s}_{\alpha+1}),p_{3}(\mathbf{s}_{\alpha+1}),p_{4}(\mathbf{s}_{\alpha+1})$
$\displaystyle\in I(\alpha).$

We therefore have

$\displaystyle(0)\subseteq I(\alpha+1)\subseteq I(\alpha).$ (6.15)

∎

This gives us enough to prove Theorem 1.3.

###  6.1. Proof of Theorem 1.3 (I): Demonstrating Ideal Stability

Suppose that for some $\alpha\geq 1$ we have

$\displaystyle\frac{1}{7^{\alpha-1}}L_{2\alpha-1}=f_{\alpha}\in\mathcal{V}^{(1)}_{n},$
(6.16)

with $n\equiv 3\pmod{7}$ and $f_{\alpha}$ defined as in (6.7). In particular,
by hypothesis,

$\displaystyle L_{2\alpha-1}\equiv 0\pmod{7^{\alpha-1}}.$

We define the associated congruence ideal $I(\alpha)$ as in (6.9). Suppose
that

$\displaystyle I(\alpha)=(0).$ (6.17)

Then by Lemma 6.2 we have

$\displaystyle\frac{1}{7^{\alpha}}L_{2\alpha}=\frac{1}{7^{\alpha}}U^{(1)}\left(L_{2\alpha-1}\right)=\frac{1}{7}U^{(1)}\left(f_{\alpha}\right)\in\mathcal{V}^{(0)}_{7n}.$
(6.18)

Thus, $L_{2\alpha}$ is divisible by $7^{\alpha}$. Moreover, by Lemma 6.3,

$\displaystyle\frac{1}{7^{\alpha}}L_{2\alpha+1}=\frac{1}{7^{\alpha}}U^{(0)}\circ
U^{(1)}\left(L_{2\alpha-1}\right)=\frac{1}{7}U^{(0)}\circ
U^{(1)}\left(f_{\alpha}\right)=f_{\alpha+1}\in\mathcal{V}^{(1)}_{49n+3},$
(6.19)

and by (6.17)

$\displaystyle(0)\subseteq I(\alpha+1)\subseteq I(\alpha)=(0),$

whence $I(\alpha+1)=(0)$. To complete the induction, we note from (1.13) that
$L_{1}\in\mathcal{V}^{(1)}_{3}$, and by (6.10) we have $I(1)=(0)$.

###  6.2. Proof of Theorem 1.3 (II): Computing the Localizing Factor

Checking the values of $\psi(\alpha)$ is a straightforward exercise in
elementary number theory. If for some $\alpha\geq 1$ we have

$\displaystyle\psi(2\alpha-1)=\left\lfloor\frac{7^{2\alpha}}{16}\right\rfloor,$

as in (1.15), then we know from Theorem 3.1 that on applying $U^{(1)}$ to
$L_{2\alpha-1}$, our resultant denominator will be $1+7x$ raised to the power
of

$\displaystyle\psi(2\alpha)=7\psi(2\alpha-1)$
$\displaystyle=7\left\lfloor\frac{7^{2\alpha}}{16}\right\rfloor=7\left(\frac{7^{2\alpha}-1}{16}\right)=\frac{7^{2\alpha+1}-7}{16}=\left\lfloor\frac{7^{2\alpha+1}}{16}\right\rfloor.$

We may similarly verify that

$\displaystyle\psi(2\alpha+1)=7\psi(2\alpha)+3=\left\lfloor\frac{7^{2\alpha+2}}{16}\right\rfloor.$

We note that

$\displaystyle\psi(1)=\left\lfloor\frac{7^{2}}{16}\right\rfloor=3,$

which matches the localizing factor we have for $L_{1}$ in (1.13).

###  6.3. Proof of Theorem 1.3 (III): Accounting for the Anomalous Term
$r_{L}$

Finally, we note the occurence of the term $r_{L}$. Notice that for a given
element of $\mathcal{V}^{(1)}$, the power of 7 attached to $x^{m}$ for $1\leq
m\leq 3$ is $-1$. This is also true for $y,yx$. If we take

$\displaystyle s_{0}(1)x+s_{0}(2)x^{2}+s_{0}(3)x^{3}+s_{1}(0)y+s_{1}(1)xy$
$\displaystyle\equiv
s_{0}(1)\left(x+s_{0}(2)s_{0}(1)^{-1}x^{2}+s_{0}(3)s_{0}(1)^{-1}x^{3}+s_{1}(0)s_{0}(1)^{-1}y+s_{1}(1)s_{0}(1)^{-1}xy\right)\pmod{7},$

we should expect

$\displaystyle s_{0}(2)$ $\displaystyle\equiv 3s_{0}(1)\pmod{7},$
$\displaystyle s_{0}(3)$ $\displaystyle\equiv s_{0}(1)\pmod{7},$
$\displaystyle s_{1}(0)$ $\displaystyle\equiv 6s_{0}(1)\pmod{7},$
$\displaystyle s_{1}(1)$ $\displaystyle\equiv 5s_{0}(1)\pmod{7};$

or, equivalently,

$\displaystyle 7(s_{0}(2)-3s_{0}(1))$ $\displaystyle\equiv 0\pmod{49},$ (6.20)
$\displaystyle 7(s_{0}(3)-s_{0}(1))$ $\displaystyle\equiv 0\pmod{49},$ (6.21)
$\displaystyle 7(s_{1}(0)-6s_{0}(1))$ $\displaystyle\equiv 0\pmod{49},$ (6.22)
$\displaystyle 7(s_{1}(1)-5s_{0}(1))$ $\displaystyle\equiv 0\pmod{49}.$ (6.23)

Indeed, as we verify in our Mathematica supplement, we have

$\displaystyle
7(s_{0}(2)-3s_{0}(1)),7(s_{0}(3)-s_{0}(1)),7(s_{1}(0)-6s_{0}(1)),7(s_{1}(1)-5s_{1}(1))\in
I(\alpha).$

Therefore, for $I(\alpha)=(0)$, we have

$\displaystyle\frac{1}{7}(s_{0}(1)x+s_{0}(2)x^{2}+s_{0}(3)x^{3}+s_{1}(0)y+s_{1}(1)xy)=s_{0}(1)r_{L}+M_{0}(x)+yM_{1}(x),$
(6.24)

for some $M_{0}(x),M_{1}(x)\in\mathbb{Z}[x].$

∎

##  7\. Further Work

Despite the tediousness of the proof, the underlying elegance, especially of
the last step, is astonishing. We see that each generating function
$L_{\alpha}$ is composed of a combination of basis functions $y^{\beta}x^{m}$
in just the right way so that any deviant terms will disappear in passing to
its successor… and that this composition is itself very precisely inherited by
its successor. This in a sense explains why the congruence family exists at
all. We could certainly imagine some more arbitrary sequence of functions over
$\mathrm{X}_{0}(14)$, each related to its successor by a modified Atkin
$U_{7}$ operator. However, a given function in such a sequence is unlikely to
contain such a precise genetic disposition to cancel deviant terms; and even
if it did, it would be unlikely to pass that precise disposition on to the
next function in so stabilizing a manner.

A congruence family, then, is the manifestation of the occasional (rare)
function sequence which does possess this precise cancelation and inheritance
property.

We finish now by considering the following additional questions which are
raised by this proof:

###  7.1. Congruence Families of Higher Genus and Cusp Count

The natural question that arises from our work is whether our methods can be
used to prove a given congruence family when either the genus or the cusp
count of the associated curve $\mathrm{X}$ is increased.

It is important to note that in the case of the main result of this paper, the
genus is a complicating factor, but it does not appear to fundamentally alter
the accessibility of the proof. Our methods are in large measure identical to
those used in previous applications of localization to prove congruences,
except for a greater degree of necessary calculations, together with a bit of
“index gymnastics.” As such, any congruence families in which $\mathrm{X}$ has
higher genus ought to be accessible to proof, provided that the cusp count of
$\mathrm{X}$ remains 4 or less. We are currently investigating possible
congruence families whose curves carry such a topology.

A more ambitious problem is for us to adapt our methods to encompass
congruence families in which $\mathrm{X}$ has a cusp count greater than 4.
These are among the most difficult and the least-understood congruence
families. The proofs of these families—when we have them at all—are not
generally easy to understand, and often involve algebraic structures or
functions which are not clearly understood.

As an example, we consider the Andrews–Sellers congruence family. The family
was proposed in 1994, and concerned the generalized 2-color Frobenius function
$c\phi_{2}(n)$. It resisted proof until Paule and Radu’s work in 2012. The
techniques developed by Paule and Radu have been extremely important and
applicable. On the other hand, their proof involves certain algebraic
structures, e.g., a pair of rank 2 $\mathbb{Z}[X]$ modules, which do not arise
from a clear theoretical understanding. The proof techniques certainly work,
but we do not fully know why.

The situation is analogous to the proofs of the irrationality of $\sqrt{2}$.
The classical “Pythagorean” proof is especially elegant, but it does not
highlight why $\sqrt{2}$ is irrational in the same way as a proof which
utilizes the property of unique factorization in $\mathbb{Z}$. The latter
proof relates irrationality to a much deeper and more general property of
$\mathbb{Z}$, and immediately allows us to generalize to a much broader class
of irrational numbers.

The appeal of localization is that our proof not only provides an algebraic
framework which arises naturally from the topology of the underlying Riemann
surface (and which easily reduces to the classical case for simple
topologies), but it also allows us to highlight what makes some congruences
more difficult to prove than others. The ideal structure that plays such a
prominent role in our proof in this article emphasizes the complex interaction
and inheritance between basis functions which might not be noticeable
otherwise.

###  7.2. Ideal Vs. Kernel Approaches

This article constitutes the second collaboration between Banerjee and Smoot
to prove a congruence family using the localization method. Our first result
[5] concerned $d_{5}(n)$ modulo powers of 5. In that paper we used an approach
which is equivalent but which relies on the properties of certain vector
spaces. Instead of working with a sequence of ideals, we build a vector space,
and work with a linear operator $\Omega$ on that vector space. For every odd-
indexed $L_{2\alpha-1}$ we have a certain vector $v_{\alpha}$. Proving the
congruence family amounts to proving that

$\displaystyle v_{\alpha}\in\mathrm{ker}\left(\Omega\right)\text{ implies
}v_{\alpha+1}\in\mathrm{ker}\left(\Omega\right).$

The approach is equivalent to our approach in this article, except that in our
current case we would be forced to work over a
$\mathbb{Z}/49\mathbb{Z}$-module, rather than a vector space.

Part of the reason that we used a different formulation is that we recognize
limitations to the current localization method. It cannot currently be used to
prove congruences associated with a modular curve of cusp count 6. We hope to
modify the method to properly account for the most difficult congruence
families.

With this goal in mind, our two papers show that one can consider
modifications to the method either from the perspective of linear operators on
certain modules and vector spaces, or from the perspective of ideal chains
associated with a given polynomial ring. One of these approaches may
ultimately prove more useful to the task than the other.

##  8\. Appendix I

$\displaystyle\theta^{(1)}_{0}(m)$ $\displaystyle=\begin{cases}-1,&1\leq m\leq
3,\\\ \left\lfloor\frac{7m-28}{9}\right\rfloor&m\geq 4\end{cases}$
$\displaystyle\theta^{(1)}_{1}(m)$ $\displaystyle=\begin{cases}-1,&0\leq m\leq
1,\\\ \left\lfloor\frac{7m-14}{9}\right\rfloor&m\geq 2\end{cases}$
$\displaystyle\theta^{(0)}_{0}(m)$ $\displaystyle=\begin{cases}0,&1\leq m\leq
2,\\\ \left\lfloor\frac{7m-16}{9}\right\rfloor&m\geq 3\end{cases}$
$\displaystyle\theta^{(0)}_{1}(m)$ $\displaystyle=\begin{cases}0&0\leq m\leq
1,\\\ \left\lfloor\frac{7m-5}{9}\right\rfloor&m\geq 2\end{cases}$

$\displaystyle\pi^{(1)}_{00}(m,r)$
$\displaystyle=\begin{cases}\left\lfloor\frac{7r+2}{9}\right\rfloor&1\leq
m\leq 3\text{ and }r\geq 9,\\\
\max\left(0,\left\lfloor\frac{7r-m}{9}\right\rfloor\right)&\text{
otherwise}\end{cases}$ $\displaystyle\pi^{(1)}_{01}(m,r)$
$\displaystyle=\max\left(0,\left\lfloor\frac{7r-m+16}{9}\right\rfloor\right)$
$\displaystyle\pi^{(1)}_{10}(m,r)$
$\displaystyle=\begin{cases}\left\lfloor\frac{7r+2}{9}\right\rfloor&0\leq
m\leq 1\text{ and }r\geq 11,\\\
\left\lfloor\frac{7r-3}{9}\right\rfloor&m=0\text{ and }1\leq r\leq 10,\\\
\max\left(0,\left\lfloor\frac{7r-m-2}{9}\right\rfloor\right)&\text{
otherwise}\end{cases}$ $\displaystyle\pi^{(1)}_{11}(m,r)$
$\displaystyle=\begin{cases}\left\lfloor\frac{7r+4}{9}\right\rfloor+1&0\leq
m\leq 1\text{ and }r\geq 7,\\\
\left\lfloor\frac{7r+2}{9}\right\rfloor+1&m=0,\\\
\max\left(0,\left\lfloor\frac{7r-m+3}{9}\right\rfloor+1\right)&\text{
otherwise}\end{cases}$ $\displaystyle\pi^{(0)}_{00}(m,r)$
$\displaystyle=\begin{cases}-1&1\leq r\leq 3,\\\
\max\left(0,\left\lfloor\frac{7r-m+4}{9}\right\rfloor-3\right)&r\geq
4\end{cases}$ $\displaystyle\pi^{(0)}_{01}(m,r)$
$\displaystyle=\begin{cases}-1&0\leq r\leq 1,\\\
\max\left(0,\left\lfloor\frac{7r-m}{9}\right\rfloor-1\right)&r\geq
4\end{cases}$ $\displaystyle\pi^{(0)}_{10}(m,r)$
$\displaystyle=\begin{cases}-1&1\leq r\leq 3,\\\
\max\left(0,\left\lfloor\frac{7r-1}{9}\right\rfloor-3\right)&m=0,r\geq 4\\\
\max\left(0,\left\lfloor\frac{7r-m}{9}\right\rfloor-3\right)&r\geq
4\end{cases}$ $\displaystyle\pi^{(0)}_{11}(m,r)$
$\displaystyle=\begin{cases}-1&0\leq r\leq 1,\\\
\max\left(0,\left\lfloor\frac{7r-m+4}{9}\right\rfloor-2\right)&m=0,r\geq 2\\\
\max\left(0,\left\lfloor\frac{7r-m+5}{9}\right\rfloor-2\right)&r\geq
2\end{cases}$

##  9\. Appendix II

$\displaystyle b_{0}(z)$ $\displaystyle=z^{14}$ $\displaystyle b_{1}(z)$
$\displaystyle=-2z^{7}-35z^{8}-84z^{9}-133z^{10}+161z^{11}+112z^{12}+63z^{13}-96z^{14}$
$\displaystyle b_{2}(z)$
$\displaystyle=1-14z+35z^{2}+84z^{3}+476z^{4}-1876z^{5}-4228z^{6}-8859z^{7}+9940z^{8}$
$\displaystyle+9254z^{9}+5138z^{10}+1022z^{11}-5152z^{12}-5152z^{13}+4224z^{14}$
$\displaystyle b_{3}(z)$
$\displaystyle=-12+203z-826z^{2}-826z^{3}+25242z^{4}-111958z^{5}-308154z^{6}$
$\displaystyle-1027606z^{7}+944559z^{8}+345632z^{9}+203924z^{10}-218015z^{11}+67312z^{12}$
$\displaystyle+190400z^{13}-112640z^{14}$ $\displaystyle b_{4}(z)$
$\displaystyle=66-1260z+7364z^{2}-6650z^{3}+136430z^{4}-345436z^{5}-4566688z^{6}$
$\displaystyle-38513350z^{7}+41646514z^{8}+4955069z^{9}-5994226z^{10}+4396224z^{11}$
$\displaystyle+683200z^{12}-4193280z^{13}+2027520z^{14}$ $\displaystyle
b_{5}(z)$
$\displaystyle=-220+4515z-32480z^{2}+99967z^{3}-24493z^{4}+7498575z^{5}-5736031z^{6}$
$\displaystyle-565652279z^{7}+610835141z^{8}-53989607z^{9}+34022968z^{10}-30421440z^{11}$
$\displaystyle-33660928z^{12}+61071360z^{13}-25952256z^{14}$ $\displaystyle
b_{6}(z)$
$\displaystyle=495-10500z+82306z^{2}-354480z^{3}+669914z^{4}+23325603z^{5}+256881359z^{6}$
$\displaystyle-3911740166z^{7}+4057736697z^{8}-673766856z^{9}+182358400z^{10}-54240256z^{11}$
$\displaystyle+501760000z^{12}-616562688z^{13}+242221056z^{14}$ $\displaystyle
b_{7}(z)$
$\displaystyle=-792+16758z-127988z^{2}+493185z^{3}-3807692z^{4}+41690572z^{5}+1462044654z^{6}$
$\displaystyle-14498962786z^{7}+11696357232z^{8}+2668196608z^{9}-1949538304z^{10}$
$\displaystyle+2020085760z^{11}-4193910784z^{12}+4393009152z^{13}-1660944384z^{14}$
$\displaystyle b_{8}(z)$
$\displaystyle=924-18816z+122500z^{2}-105938z^{3}+2849350z^{4}-84220857z^{5}+4057736697z^{6}$
$\displaystyle-31293921328z^{7}+16440406976z^{8}+11942708736z^{9}+2743967744z^{10}$
$\displaystyle-11615600640z^{11}+21576024064z^{12}-22020096000z^{13}+8304721920z^{14}$
$\displaystyle b_{9}(z)$
$\displaystyle=-792+14910z-65744z^{2}-475335z^{3}+4252871z^{4}-53989607z^{5}+4886681128z^{6}$
$\displaystyle-36201745856z^{7}-2936847872z^{8}+30714163200z^{9}-802586624z^{10}+26205749248z^{11}$
$\displaystyle-68115496960z^{12}+75749130240z^{13}-29527900160z^{14}$
$\displaystyle b_{10}(z)$
$\displaystyle=495-8190z+10675z^{2}+549528z^{3}-5994226z^{4}+39640552z^{5}+2665376896z^{6}$
$\displaystyle-19718835200z^{7}-18705154048z^{8}-11319246848z^{9}+35764305920z^{10}$
$\displaystyle-13946060800z^{11}+123547418624z^{12}-169114337280z^{13}+70866960384z^{14}$
$\displaystyle b_{11}(z)$
$\displaystyle=-220+2975z+8414z^{2}-218015z^{3}+1631392z^{4}+22120448z^{5}+483614208z^{6}$
$\displaystyle-4209074176z^{7}-10097590272z^{8}-29349117952z^{9}+52936310784z^{10}-13857980416z^{11}$
$\displaystyle-110863843328z^{12}+217969590272z^{13}-103079215104z^{14}$
$\displaystyle b_{12}(z)$
$\displaystyle=66-644z-5152z^{2}+8176z^{3}+328832z^{4}+4738048z^{5}+40714240z^{6}-290291712z^{7}$
$\displaystyle-1108344832z^{8}-3934257152z^{9}+7985954816z^{10}+11274289152z^{11}+37580963840z^{12}$
$\displaystyle-120259084288z^{13}+68719476736z^{14}$ $\displaystyle b_{13}(z)$
$\displaystyle=-12+63z+896z^{2}+10304z^{3}-68096z^{4}-344064z^{5}-1146880z^{6}-524288z^{7}$
$\displaystyle b_{14}(z)$ $\displaystyle=1.$

$\displaystyle a_{0}(z)$
$\displaystyle=3x+175x^{2}+4606x^{3}+73059x^{4}+794731x^{5}+6487502x^{6}+42824236x^{7}+238945119x^{8}$
$\displaystyle+1117547851x^{9}+4162186322x^{10}+11541131602x^{11}+22033069422x^{12}+25705247659x^{13}$
$\displaystyle+13841287201x^{14}$ $\displaystyle a_{1}(z)$
$\displaystyle=203x+12005x^{2}+322077x^{3}+5251673x^{4}+59434354x^{5}+511403396x^{6}+3581353209x^{7}$
$\displaystyle+21042935438x^{8}+101911799164x^{9}+387181329563x^{10}+1085915564370x^{11}$
$\displaystyle+2089469416853x^{12}+2453862488063x^{13}+1328763571296x^{14}$
$\displaystyle a_{2}(z)$
$\displaystyle=6265x+376201x^{2}+10314990x^{3}+173583039x^{4}+2053551290x^{5}+18689384000x^{6}$
$\displaystyle+138788289969x^{7}+855550469291x^{8}+4274768767815x^{9}+16532786315885x^{10}$
$\displaystyle+46861231432855x^{11}+90848559682384x^{12}+107376751451872x^{13}+58465597137024x^{14}$
$\displaystyle a_{3}(z)$
$\displaystyle=116459x+7115969x^{2}+200101839x^{3}+3492479508x^{4}+43419223008x^{5}+419167823718x^{6}$
$\displaystyle+3295778614744x^{7}+21226671646068x^{8}+109033294196141x^{9}+428467687129601x^{10}$
$\displaystyle+1226396760506785x^{11}+2394725729734352x^{12}+2847983254477760x^{13}+1559082590320640x^{14}$
$\displaystyle a_{4}(z)$
$\displaystyle=1449833x+90415745x^{2}+2619811999x^{3}+47714242940x^{4}+626857433013x^{5}$
$\displaystyle+6431855317139x^{6}+53404125506473x^{7}+357773867642002x^{8}+1882980021363602x^{9}$
$\displaystyle+7505614687422502x^{10}+21678572346541824x^{11}+42622306155271360x^{12}$
$\displaystyle+50994149398993920x^{13}+28063486625771520x^{14}$ $\displaystyle
a_{5}(z)$
$\displaystyle=12707135x+812295414x^{2}+24406700227x^{3}+467289947852x^{4}+6523210723638x^{5}$
$\displaystyle+71180537383858x^{6}+621830080144679x^{7}+4313683223069388x^{8}+23190084180310873x^{9}$
$\displaystyle+93616781749765064x^{10}+272672550787560000x^{11}+539628856248745984x^{12}$
$\displaystyle+649372072346640384x^{13}+359212628809875456x^{14}$
$\displaystyle a_{6}(z)$
$\displaystyle=80168088x+5286545768x^{2}+166136541193x^{3}+3372698052990x^{4}+50267896083671x^{5}$
$\displaystyle+582755302614765x^{6}+5332050237612588x^{7}+38130306660737921x^{8}+208794282026692488x^{9}$
$\displaystyle+852473027477812608x^{10}+2502347102814884864x^{11}+4983330104745803776x^{12}$
$\displaystyle+6030418071535484928x^{13}+3352651202225504256x^{14}$
$\displaystyle a_{7}(z)$
$\displaystyle=-3+365390731x+25095939659x^{2}+834433504065x^{3}+18132408865284x^{4}$
$\displaystyle+289513052752098x^{5}+3555914358207064x^{6}+33901256261028254x^{7}+248847657182595856x^{8}$
$\displaystyle+1384458034838074112x^{9}+5709778330183280640x^{10}+16881992402381180928x^{11}$
$\displaystyle+33821264829712826368x^{12}+41149283911545126912x^{13}+22989608243832029184x^{14}$
$\displaystyle a_{8}(z)$
$\displaystyle=-119+1187495288x+86220830934x^{2}+3079413154580x^{3}+72315867551242x^{4}$
$\displaystyle+1238343447960010x^{5}+16042203009007441x^{6}+158492327221426800x^{7}$
$\displaystyle+1189441617391337408x^{8}+6708389085147929088x^{9}+27916706629722251264x^{10}$
$\displaystyle+83096997943085629440x^{11}+167427147653571936256x^{12}+204766748751880519680x^{13}$
$\displaystyle+114948041219160145920x^{14}$ $\displaystyle a_{9}(z)$
$\displaystyle=-1988+2661685929x+209205847900x^{2}+8181772833318x^{3}+209365036375208x^{4}$
$\displaystyle+3838510233679961x^{5}+52139774281593432x^{6}+530898816397465152x^{7}$
$\displaystyle+4058869323511299584x^{8}+23161327540107079680x^{9}+97163280575812763648x^{10}$
$\displaystyle+291030344013883113472x^{11}+589573740171907563520x^{12}+724686857267692175360x^{13}$
$\displaystyle+408704146557013852160x^{14}$ $\displaystyle a_{10}(z)$
$\displaystyle=-17913+3854368588x+341084691626x^{2}+14946334588157x^{3}+418884673555446x^{4}$
$\displaystyle+8183050172059224x^{5}+115757691604548224x^{6}+1208423052153490432x^{7}$
$\displaystyle+9382000579851124736x^{8}+54076398124284641280x^{9}+228495346771705724928x^{10}$
$\displaystyle+688405122921636298752x^{11}+1401821100823034200064x^{12}+1731419384505167773696x^{13}$
$\displaystyle+980889951736833245184x^{14}$ $\displaystyle a_{11}(z)$
$\displaystyle=-91840+3161053504x+337306540599x^{2}+16974499008929x^{3}+520991685008480x^{4}$
$\displaystyle+10762401101720576x^{5}+157420717177934336x^{6}+1676798937863856128x^{7}$
$\displaystyle+13184178950079578112x^{8}+76647275568092938240x^{9}+325973004513076838400x^{10}$
$\displaystyle+987439454642002460672x^{11}+2020696994730586669056x^{12}+2507426478357575892992x^{13}$
$\displaystyle+1426749020708121083904x^{14}$ $\displaystyle a_{12}(z)$
$\displaystyle=-254016+1050345289x+154541941232x^{2}+9126241254480x^{3}+304769343501952x^{4}$
$\displaystyle+6593965302027264x^{5}+99034088979775488x^{6}+1071829647175024640x^{7}$
$\displaystyle+8515062058317774848x^{8}+49869621987073064960x^{9}+213334790308255236096x^{10}$
$\displaystyle+649528704899328507904x^{11}+1335454068490190716928x^{12}+1664540524159474597888x^{13}$
$\displaystyle+951166013805414055936x^{14}$ $\displaystyle a_{13}(z)$
$\displaystyle=-296009-12511233x-224214592x^{2}-2204529600x^{3}-12808201728x^{4}-43783176192x^{5}$
$\displaystyle-80957571072x^{6}-61681958912x^{7}$ $\displaystyle a_{14}(z)$
$\displaystyle=1.$

##  10\. Acknowledgments

The first author was funded by the Austrian Science Fund (FWF): W1214-N15,
Project DK6. The second author was funded in whole by the Austrian Science
Fund (FWF): Einzelprojekte P 33933, “Partition Congruences by the Localization
Method”. Our most profound thanks to the Austrian Government and People for
their generous support.

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