A single-variable proof of the omega SPT congruence family over powers of 5

In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (q)$$\end{document}ω(q). Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}[X]$$\end{document}Z[X]-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang’s family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}[X]$$\end{document}Z[X]. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.


Introduction
In 2018, Liuquan Wang and Yifan Yang proved [19] the existence of an infinite family of congruences for the smallest parts function for the mock theta function ω(q). Wang and Yang's proof employed methods developed by Paule and Radu. However, the author has discovered that this family of congruences lends itself to a singlevariable proof similar in theory to the classical methods of Ramanujan and Watson. In such a proof certain very interesting algebraic intricacies emerge, which-to our knowledge-are different from all known proofs of partition congruence families. The purpose of this paper is to elaborate this alternative proof.
This family of congruences has a form which is common to the field; in addition to Ramanujan's iconic congruences [3,8,16,20], an enormous number of similar results have been found; recent results include [6,11,17]. The proof of each of these results generally involves an induction argument which takes advantage of some algebraic structure on the associated space of modular functions.
In the case of Ramanujan's classical result for powers of 5, one works over the congruence subgroup Γ 0 (5) and an associated space of modular functions under specific meromorphic conditions. Because the algebraic structure of this space of functions is isomorphic to Z[X ], the full family of congruences may be proved by a straightforward process. This is similarly true for Ramanujan's modified congruence family for powers of 7. The general technique was first published by Watson [20], although Ramanujan appears to have been the first to understand it [4].
However, Ramanujan's congruence family for powers of 11 is more difficult to prove. This is owed in large measure to the fact that the underlying modular curve X 0 (11) associated with Γ 0 (11) has genus 1. To compare, X 0 (5) and X 0 (7) each have genus 0. This fact ensures that, as a consequence of the Weierstrass gap theorem [12], the necessary space of functions on Γ 0 (11) is isomorphic not to Z[X ], but rather to a rank 2 Z[X ]-module.
The necessary modification of Ramanujan and Watson's approach was first developed by Atkin [3]. Later families of congruences associated with a genus 1 modular curve include the Andrews-Sellers conjecture, proved by Paule and Radu [11], and the Choi-Kim-Lovejoy conjecture, proved by the author [17]. The method used for both of these proofs was developed by Paule and Radu as an important modification of Atkin's approach.
Wang and Yang also employ this method. In particular, they develop a sequence of weakly holomorphic modular forms (L α ) α≥1 such that with Φ α an integer power series with constant term 1, and λ α the minimum positive solution to 12x ≡ 1 (mod 5 α ) (see Sect. 2). They show that L α 5 α · F = f 0,α (t) + ρ · f 1,α (t), (1.3) in which f i,α ∈ Z[X ], is a weight 2 holomorphic modular form, and t, ρ are modular functions which take the form of eta quotients with integer expansions in the Fourier variable q. The relevant congruence subgroup is Γ 0 (10). This is standard to Paule and Radu's approach. Note the free rank 2 Z[X ]-module structure of (1.3), which is characteristic of the method.
However, Paule and Radu developed their method in order to overcome the complications which arise from a congruence family in which the associated modular curve has nonzero genus. The genus of X 0 (10) is 0.
It is this extremely important and telling fact that drove us to attempt a more classical proof of Wang and Yang's theorem.
An interesting complication emerges in the factor (1 + 5y) −3 . One might correctly guess that our relevant space of modular functions for all α ≥ 1 is isomorphic to a localization of Z[X ], rather than to Z[X ] itself. Indeed, we have the following remarkable result, the principal theorem of this paper: Then for all α ≥ 1, We are not aware of any other congruence families in which the proof necessitates such a ring structure, and it would be interesting to know whether any additional examples exist.
What makes this theorem especially interesting is that the algebraic structure of the representation of L α reveals not only important arithmetic information (i.e., divisibility by 5 α ), but also topological information relating to the underlying modular curve-in this case, X 0 (10). Certainly, the single-variable representation suggests that the curve has genus 0, since any modular function with a pole only at a single cusp must have a polynomial expression in a Hauptmodul (see Sect. 6).
More interesting still is the fact that L α requires a rational polynomial expression in the Hauptmodul y. This indicates that L α has poles at more than one cusp, which in turn suggests that X 0 (10) has more than two cusps; in fact, the curve has four. Thus, the arithmetic properties of c(n) and the topology of X 0 (10) are given together in the algebraic structure of Theorem 1.3.
Silviu Radu and I have developed some algorithmic machinery [15], using the theory of modular functions, which might easily be modified to examine identities with a form similar to (1.6). On a more ambitious note, the steps in this proof are straightforward, despite their complexity. Because so much of the key algebraic structure derives from the modular curve, which itself can be quickly identified from the initial cases of the congruence family, this technique may prove to be completely general, especially when the associated modular curve has a cusp count greater than 2.
The author strongly believes that localized rings may yet prove to be an enormously productive environment in which to examine new arithmetic properties in partition theory, especially for situations in which more traditional methods fail.
Another interesting difficulty arises in the somewhat irregular 5-adic convergence of each term of L α under repeated application of the corresponding U 5 operators. In particular, in mapping L 2α−1 to L 2α , the individual components of the linear coefficient do not increase piecewise with respect to their 5-adic value-rather, the components must be shown to sum to the necessary multiple of 5 (see Definition 4.3 and Theorem 4.4).
Such a strange complication necessitates a very precise manipulation on the 5adic convergence of our critical functions, together with a careful examination of the coefficients in our auxiliary functions modulo 5. As in the matter of localization, we are unaware of any other examples of congruence families which demand such a constructive method of verifying divisibility by 5.
A final complication emerges in the base cases of our key lemmas. We require the verification of 50 initial relations. However, we can show that these 50 are algebraically dependent, and that a total of only 10 initial relations need be directly established. This stands in contrast to the 20 that Wang and Yang require for their proof. From these 10 relations, the 50 relations necessary for our induction may be assembled and verified with relative ease through a computer algebra system. The computational complexity is striking; nevertheless, the reliance (in principle, at least) upon so few relations, together with the single-variable approach, compels us to call our proof classical, or perhaps "semiclassical." In total, these complications seem overwhelming, and it is understandable that a single-variable proof has not been found before now. It seems that the genus of the underlying modular curve alone is sufficient to compel a single-variable proof, in spite of the many considerable difficulties.
The remainder of our paper is outlined as follows: in Sect. 2 we define the necessary functions L α , along with certain auxiliary functions. We also define the modular function y, from which we will develop certain modular equations. In Sect. 3 we will develop some important module structures, including our polynomial localization. Following this, we develop our modified U 5 operators, and prove certain key properties of U 5 on our localized ring, including some important arithmetical information.
In Sect. 4, we carefully demonstrate 5-adic convergence of our relevant space of functions. In particular, we impose certain additional arithmetic constraints on our functions in order to control the somewhat irregular behavior of our 5-adic convergence, especially over the coefficients of the smaller powers of y. This culminates in the proof of our Main Theorem. As a reference, we provide certain useful 5-adic valuation tables in "Appendix I." In Sect. 5, we outline the proof of our fundamental relations, and the algebraic method by which we may verify the 50 initial relations needed for the induction of our Main Lemma. The proof of the ten fundamental relations is based on the cusp analysis established from the theory of modular functions. We give these ten fundamental relations in "Appendix II." However, for want of space, we will post the computations of our 50 initial relations (along with our cusp analysis calculations, and some miscellaneous calculations) in an Online Mathematica Supplement, which can be found at https://www3.risc.jku.at/people/nsmoot/online3.nb.
The importance of a computational approach to this problem cannot be overstated. Not only was there a need to calculate certain relations to complete our induction (most of which would prove too tedious to demonstrate by hand); but many of our results which can in principle be proved without any reliance on computation (e.g., the sufficiency of the additional conditions imposed in Definition 4.3) were only found through many weeks of exhaustive experimentation.
In our final section, we attempt to provide some insight as to why our technique may be more difficult to apply to a family of congruences in which the associated modular curve has genus 1, e.g., the Andrews-Sellers family.
We will now define our key generating functions, and their behavior under the standard U 5 operator.
Therefore, one could write L α in the form We now define the standard Hecke U 5 operator: We list some of the important properties of U 5 . The proofs are straightforward, and can be found in [2,Chap. 10] and [8,Chap. 8].

The modular equations
Our most important functions are the following: . (2.11) Notice that, by the Freshman's Dream,

This yields
It is not difficult to verify that from which x = 1 + 5y follows.
Then we have a j (5τ )y j = 0. (2.14) Proof Because y(5τ ) is a modular function with only one pole, we may prove this equation using cusp analysis. See the end of Sect. 5.
For convenience of notation, in later sections we will define b 5 (τ ) := 1.

Localized ring
We will begin to construct the algebra structure needed for our proof, beginning with the peculiar localization property. Define the multiplicatively closed set We will prove that for every α ≥ 1, L α is a member of the localization of Z[y] at S, which we will denote by Z[y] S . Notice that because 1/x n = 1/(1 + 5y) n is an eta quotient with an integer power series expansion in q = e 2πiτ for every n ≥ 1, we can expand every element of the localization into an integer power series in q, i.e., We need to define two general classes of subsets of Z[y] S . But first, we need a key definition: We now need to construct suitable sets to contain our L α . Due to a somewhat irregular pattern of 5-adic convergence, we must define our 5-adic valuation function very carefully.
Now we take an arbitrary n ≥ 1, and define the following:

Recurrence relation
We now define the following maps: for i = 0, 1. Now we are ready to utilize our modular equations, together with our U (i) operators to build certain helpful recurrence relations.

Lemma 3.2
For all m, n ∈ Z, and i ∈ {0, 1}, we have for n ≥ 1. Writing x in terms of y, we have If we multiply both sides by y m for some m ≥ 1, then (3.8) Now we expand each power of y with its modular equation and rearrange Now multiply both sides by F · Z 1−i : (3.10) We are now ready to take the U 5 operator. Recall that by Line 2 of Lemma 2.2, for any functions f (τ ), g(τ ),

This gives us
(3.11) Dividing both sides by F, we achieve our formula.

Main lemma
We need to provide certain general relations for U (i) y m (1+5y) n . For this, we will define the following: , m = 3, and r = 2, , m = 2, and r = 3, 4, 5, We will, therefore, begin by proving the following: Lemma 3.4 Let κ, δ ∈ Z ≥0 and μ ∈ Z be fixed, and fix i to either 0 or 1. If there exists a discrete array h i such that for 1 ≤ m ≤ 5, 1 ≤ n ≤ 5, then such a relation can be made to hold for all m ≥ 1, n ≥ 1.

Lemma 3.5
Let κ, δ ∈ Z ≥0 and m 0 ∈ Z ≥1 be fixed, and fix i to either 0 or 1. If there exists a discrete array h i such that for 1 ≤ n ≤ 5, then such a relation can be made to hold for all n ≥ 1.
Proof of Lemma 3. 4 We will use induction. Suppose that the relation holds for all positive integers m, n strictly less than some m 0 , n 0 ∈ Z ≥6 , respectively. We want to show that the relation can be made to hold for m 0 and n 0 . We have This can be demonstrated by a simple expansion of a j (τ )b k (τ )(1+5y) 5(k−1) . Expanding w( j, k), we have Notice that for any M, N ∈ Z, Because of this, We now have where and Notice that After relabeling our powers of y, we have Proof of Lemma 3. 5 We will again use induction, though only over n, while holding m 0 fixed. Suppose that the relation holds for all positive integers n strictly less than some n 0 ∈ Z ≥6 . We want to show that the relation can be made to hold for n 0 .
This can be demonstrated with a simple expansion ofŵ(k). Expanding, we have With a change of index, we have This ensures that the smallest powers of 5 dividing the coefficients of U (i) y m 0 (1+5y) n 0 derive precisely from the sum Therefore, if our relation is established for 1 ≤ n ≤ 5, then it must be true for all n ≥ 6 as well.
We may now rearrange our sum and define a new discrete array in a manner similar to (3.29) to finish the proof. As an additional consequence of Lemmas 3.4, 3.5, we have the following important result on the behavior of the coefficients in these expansions: Corollary 3.6 For all n ∈ Z ≥1 we have: For all n ∈ Z ≥1 and 1 ≤ m ≤ 3 we have: . We, therefore, need only establish that (3.49) is true in all relevant cases. Thereafter, we can simply compute the relevant coefficients for five consecutive values of n. We note that for (3.37), r ≥ 1 and l ≥ 1. Because of this, r + l ≥ 2, and (3.37) will contribute nothing to the linear coefficient. On the other hand, for (3.36), the only possibility is for l = 0 and r = 1. Because 5(0)+10 6 > 1, we easily get (3.49).
To prove (3.44), (3.45), (3.46), we note that we may directly compute π 0 (m, r ). Notice that the only way for r + l = 2 to be true is for r = l = 1 or r = 2 and l = 0. For the first case, we have Here, (3.39) is strict; for the second case, i.e., for r = 2 and l = 0, (3.39) follows immediately. Once again, we need only examine each case for five consecutive values of n.
To prove (3.47), we take into account that there are three different ways for r +l = 3 to be true. Either r = 1 and l = 2, or r = 2 and l = 1, or r = 3, l = 0. We, therefore, have (1+5y) n can be made. As only five values of m will contribute to the coefficient that we want, we, therefore, only need to check (3.48) for 1 ≤ m ≤ 5, and for 1 ≤ n ≤ 5.

Main theorem
With the necessary relations established for U (i) y m (1+5y) n , we can now work towards the main theorem. We begin with the following theorem: n . Then we can express f as Using Theorem 3.3, we write We examine φ(m) + π 0 (m, r ) for r ≥ (m + 2)/5 and m ≥ 1: For m = 1: For m = 2: In both cases, φ(2) + π 0 (2, r ) ≥ θ(r ) + 1. For m = 3: For m = 4: For m ≥ 5: Notice that, in all cases, Theorem 4.2 Let f ∈ V (1) n and denote (4.5) n . Then we can express f as We write (1) y m (1 + 5y) n (4.7) Let us denote the coefficient of  With m = 3, , r = 3 or r ≥ 4. (4.11) Notice that for 1 ≤ m ≤ 3, θ(m) + π 1 (m, r ) ≥ φ(r ) + 1 except when r = 1. Finally, for m ≥ 4, Our last two theorems are very nearly sufficient to give us the 5-adic increase we need, with the notable exception of the components which contribute to the coefficient of y (1 + 5y) 5n−4 . Indeed, the individual components need not be divisible by 5 at all. We, therefore, need one additional condition for our purposes.  This small additional condition is at last sufficient for our purposes. Not only do we need to prove that this allows 5-adic convergence upon application of U (1) , but that the condition is stable, i.e., that
Notice that this implies that giving us the first case of Theorem 1.3. More specifically, by examining the coefficients of (1+5y) ψ(1)

5·F
· L 1 , we know that Suppose that for some α ∈ Z ≥1 , we have for some n ∈ Z ≥1 . Then By (4.18) of Theorem 4.4, we know that there exists some f 2α ∈ V (0) 5n−4 such that Therefore, Moreover, Therefore, This shows that for some integer sequence n(α). We now verify that n(α) = ψ(α). It is a fact of elementary number theory that for all α ≥ 1, Let us suppose, then, that On applying U (1) , Theorem 3.3 shows that our localizing factor (1 + 5y) ψ(2α−1) increases to (1 + 5y) 5ψ(2α−1)−4 . We, therefore, have Thus, In similar fashion, it can be proved that if then by Theorem 3.3, applying U (0) will increase the power of our localizing factor from (1 + 5y) ψ(2α) to (1 + 5y) 5ψ(2α)−2 , and it can be quickly proved that This is compatible with the increase in the localizing powers in Theorem 3.3. Finally, ψ(1) = 3 is the localizing power for L 1 .

Initial relations
For i fixed, our theorem for expanding U (i) y m (1+5y) n requires 25 initial relations to be justified. However, these relations are ultimately dependent on far fewer relations, since one can very quickly verify that We can very quickly compute any value of U (i) y m (1+5y) n , provided we have exact expressions for U (i) ((1 + 5y) To compute U (i) y m (1+5y) n for 1 ≤ m, n ≤ 5, we need to have expressions for U (i) ((1 + 5y) r ) for −5 ≤ r ≤ 4. However, we have the degree 5 modular equation for x = 1 + 5y, which yields Moreover, for n ≥ 0 we obviously have and y follows a degree 5 modular equation. Therefore, in order to determine U (i) y m (1+5y) n for any m, n ∈ Z and i fixed at 0 or 1, we only need to determine relations for U (i) y k for five consecutive values of k.
For both values of i, that makes 10 relations. Once these relations are established, verification of the 50 initial relations follows as a relatively simple, if somewhat tedious, computational exercise. The calculation is straightforward, but we detail it in the Mathematica Supplement to this paper, which can be found online at https://www3.risc.jku.at/people/nsmoot/ online3.nb.
We choose to justify these ten relations using finiteness conditions from the theory of modular functions.
This approach is useful, given that the right-hand sides of eight of the ten relations below are already modular functions with a pole only at a single cusp of X 0 (10) (the remaining two can become such functions with a slight adjustment). The remaining task is merely to verify that the left-hand side of each relation is also a modular function with a pole at the same cusp, and then to compare the principal parts from either side. Because it is slightly easier to expand both sides of each relation with respect to the pole at ∞, we will divide by a sufficient power of y to induce a pole at ∞ rather than at 0.
We will provide some, though not all, details to our cusp analysis computations here. The full computations are detailed in our Online Mathematica Supplement.
We preface our results with a brief overview of the theory of modular forms. For a classic review of the theory, see [8]. For a more modern treatment, see [5].
We denote SL(2, Z) as the set of all 2 × 2 integer matrices with determinant 1. For any given N ∈ Z ≥1 , let We define a group action If γ = a b c d and τ ∈Ĥ, then we write The orbits of this action are defined as The detailed properties of X 0 (N ), including its Riemann surface structure, are given in [5,Chaps. 2,3]. For want of space, we will only add that X 0 (N ) possesses a unique nonnegative integer g (X 0 (N )) called its genus. This number is referenced in the Introduction, and may be computed using Theorem 3.1.1 of [5,Chap. 3]. For an understanding of the connection of the genus to module rank, e.g., via the Paule-Radu method, see Sect. 6. Due to its precise symmetry on Γ 0 (N ), any modular function f ∈ M (Γ 0 (N )) induces a well-defined function The notions of pole order and cusps of f used in Definition 5.4 have been constructed so as to coincide with these notions applied tof on X 0 (N ). In particular, (5.6) represents the principal part off in local coordinates near the cusp [a/c] N . Notice that as τ → i∞, we must have γ τ → a/c, and q → 0. Because f is holomorphic on H by definition, any possible poles forf must be found for [τ ] N ⊆Q. The number and order of these poles is of paramount importance to us.
We now give an extremely important result [9, Theorem 1.37] in the general theory of Riemann surfaces: Theorem 5.6 Let X be a compact Riemann surface, and letf : X −→ C be analytic on all of X . Thenf must be a constant function.
The importance of this theorem cannot be overstated. As an immediate consequence we have This is immensely useful for verifying that two modular functions over the same space are equivalent. For example, let us take f , g ∈ M ∞ (Γ 0 (N )). Then this means that f , g both have principal parts only at a single cusp.
If these principal parts of each function match, then f − g ∈ M (Γ 0 (N )) can have no poles at any cusp. This implies thatf −ĝ is analytic on the whole of X 0 (N ), which forcesf −ĝ, and therefore, f − g, to be a constant. If the constants of f and g also match, then f − g = 0, i.e., f = g.
We now give two key theorems that will prove useful in checking the modularity of certain functions. We will use Dedekind's eta function [8,Chap. 3]: The first is a theorem by Newman [10, Theorem 1]: Theorem 5.8 Let f = δ|N η(δτ ) r δ , withr = (r δ ) δ|N an integer-valued vector, for some N ∈ Z ≥1 . Then f ∈ M (Γ 0 (N )) if and only if the following apply: To study the order of an eta quotient at a given cusp, we make use of a theorem that can be found in [14,Theorem 23], generally attributed to Ligozat: N )) , then the order of f at the cusp [a/c] N is given by the following:

Computing the initial cases
Despite its apparent recondite nature, the theory above imposes various finiteness conditions which can be used to a computational advantage. Our initial relations consist of for 1 ≤ l ≤ 4, and Recalling the definition of U (i) , and multiplying through by F, the relations have the form Using Newman's theorem, we can prove that y ∈ M(Γ 0 (10)) Z ∈ M(Γ 0 (50)).
More difficult is the function F, which is a modular form of weight 2 [6, Corollary 2.3]. We want to convert F to modular functions. We have are Jacobi's theta functions. Moreover, these functions can be expanded [13,Sect. 78] so that From here, we can show that by applying Newman's theorem to Similarly, by Newman's theorem again, 20)).

This shows that
Therefore, we can multiply either side of (5.9), (5.10) by φ(τ ). Our left-hand sides now have the form The interior functions are modular on Γ 0 (100). On the other hand, the right-hand sides of (5.9), (5.10) are modular functions on Γ 0 (20).
We, therefore, need to take μ(5τ ) 40+5+2 = μ(5τ ) 47 to account for the possible poles of φ(5τ ) · F(τ ) together with possible powers of y and Z . If we multiply both sides of our prospective relations, we have This is the complete form for our verification process. On the left-hand side, we take 0 ≤ i ≤ 1 and 1 ≤ l ≤ 4. On the right-hand side, we take the polynomials p i,l (y) to be the right-hand side of the relation for U (i) (x l ) for 1 ≤ l ≤ 4 [and the numerator of the right-hand side in the case of U (i) (1)]. Both sides are modular functions on Γ 0 (20) with a pole only at the cusp [∞] 20 . We simply compare the principal parts on either side.
The explicit constructions above, including order computation and principal part comparison, are given in our Mathematica Supplement.
Both of these applications are also given in our Mathematica supplement.

Necessity of the Paule-Radu technique For genus 1 surfaces
The proof strategy used above was able to give a single-variable proof of Theorem 1.2. We certainly believe that such a strategy can be extended to other families of congruences over any genus 0 curve. It is tempting to try a similar proof for any family of congruences over a genus 1 curve in which the standing proof employs the Paule-Radu technique, e.g., the proof of the Andrews-Sellers theorem. This is a much more difficult problem, and we will try to briefly explain why. For a more in-depth discussion of the relationship between genus and the rank of the Z[X ] modules of interest to us, see [12]. Wang and Yang define [19] the functions They go on to demonstrate that for all α ≥ 1, This is indicative of the machinery of the Paule-Radu technique, which has been successfully applied in cases such as proving the Andrews-Sellers conjecture [11] and the Choi-Kim-Lovejoy conjecture [17]. These latter problems could not be solved by the classical techniques developed by Watson in 1938 (and indeed, known by Ramanujan himself some decades prior), in which the relevant functions L α are understood as elements of Z[t], for a given modular function t.
In the case of the Andrews-Sellers conjecture, the necessary spaces of functions are free rank 2 Z[X ]-modules. Why does the Andrews-Sellers conjecture necessitate this more complex module structure while the functions associated with spt ω (n)) can be described with the simpler localized ring Z[X ] S ?
Notice that h is a Hauptmodul, i.e., M ∞ (Γ 0 (10)) = C[h]. In the case of the Andrews-Sellers congruences, the corresponding functions are modular over X 0 (20), which has genus 1. By the Weierstrass gap theorem, we cannot reduce M ∞ (Γ 0 (20)) to a 1-variable polynomial ring; indeed, M ∞ (Γ 0 (20)) must be isomorphic to a rank 2 module < 1, h 3 > C[h 2 ] , for ord (10) ∞ (h 2 ) = −2, ord Because of this, we are forced to take a rank 2 module, and therefore, a two-variable system. From this, the necessity of the Paule-Radu apparatus must follow.
As a final important note on the potential of our technique to produce new results, we give a very brief description of how we discovered that a localized ring structure was more useful than the standard polynomial ring. Our initial attempt to describe L α in terms of the function x failed almost immediately: one can verify that with the appropriate substitutions of the function x into (1.5), we get This clearly does not work. However, at the advice of Silviu Radu, we attempted to adjust our function x. We discovered the appropriate substitution in the form of x = 1 + 5y. Substituting into (6.1) and simplifying, we derive (1.6).
The critical point is that we would prefer the function needed to annihilate the poles of L α to be equal to (or a power of) the function used to describe the right-hand side of the witness identity. This is the situation that the author has studied, together with Radu [15]. It is of course far more probable that these functions are not equal. Nevertheless, if the function used on the right-hand side, e.g., y, is a Hauptmodul, then we could still use this function to describe our prefactor function (e.g., x = 1 + 5y). This necessarily induces a localized ring.
Such ring structure may appear more complicated, and more daunting, than that of the traditional methods. Nevertheless, as we hope to have shown, the resulting complications and fears can be overcome.
Acknowledgements A huge thanks to the anonymous referee, whose advice substantially improved the form of the paper. I am extremely grateful to Liuquan Wang and Yifan Yang for their work on this family of congruences, without which this project would never have developed. I am enormously grateful to Drs. Cristian-Silviu Radu and Johannes Middeke for their advice on technical matters, and to Koustav Banerjee for calming my doubts and insisting that my insights were worth investigating. Finally, an enormous thanks to Professor Peter Paule for his patience, as the prospects of this project waxed and waned time and again.
Funding Open access funding provided by Austrian Science Fund (FWF).