Construction of modular function bases for Γ0(121) related to p(11n+6)

Motivated by arithmetic properties of partition numbers , our goal is to find algorithmically a Ramanujan type identity of the form , where is a polynomial in products of the form with . To this end we multiply the left side by an appropriate factor such the result is a modular function for having only poles at infinity. It turns out that polynomials in the do not generate the full space of such functions, so we were led to modify our goal. More concretely, we give three different ways to construct the space of modular functions for having only poles at infinity. This in turn leads to three different representations of not solely in terms of the but, for example, by using as generators also other functions like the modular invariant .


k=1
(1 − q k ) 4 (1 − q 7k ) 4 . (1.2) For the 11 case Ramanujan did not present any such identity. Only recently, with the help of his Ramanujan-Kolberg algorithm, Radu was able to derive such kind of a witness identity of Ramanujan type; see [2, (58)]. Radu's work triggered further algorithmic developments on this theme. We mention a few examples. First, another witness identity was derived by Hemmecke in a generalized algebraic setting [3, (9)]; this identity reveals the 11 divisibility in explicit manner.
Despite that fact that f and t being again modular functions, in M ∞ (11), from an algebraic point of view the structure of (1.5) is more involved -and also more interesting. Namely, it expresses the algebraic fact that (1. 6) This means that the C-algebra of polynomials in f and t with complex coefficients can be represented as a module freely generated by 1, f , . . . , f 4 over the ring C[t] of polynomials in t with complex coefficients. The free generation is obvious, as in all cases we will consider, owing to the fact that the pole order of the generators f j are pairwise different. In general, for non-constant modular functions t, b 1 , . . . , b n−1 in M ∞ (N), our notation for such modules is: In various applications one needs a C[t]-module representation of the whole space M ∞ (N). For example, when using modular functions to prove Ramanujan's congruences for powers of 11 (i.e. 11 2 | p(11 2 n + 116), 11 3 | p(11 3 n + 721), etc.) one needs to work with a C[t]-module representation of M ∞ (11). According to the Weierstraß gap theorem, see [5,Thm. 12.2] for a version in the context of modular functions, there is a representation Atkin [6] was the first to construct such F j explicitly. In [7] Atkin's construction was revisited and a simpler representation of the F j was found by using a trace operator; more precisely, a special instance of [8, (1)]. An explicit discussion of the representation (1.7) can also be found in [5]. Summarizing, despite the usefulness of the module representation (1.6), it does not give the full space, for instance, F 2 ∈ C[t, f ] in view of the pole orders 4 and 5 of f and t, respectively. It is the main objective of this note to show how a basis of the full space M ∞ (N) can be obtained algorithmically. To be as concrete as possible, we will do this in the form of a case study where we fix N := 121. Despite being a special case, we feel this specialization will allow to illustrate general features and, on the other hand, will be sufficiently general to lead also to interesting non-trivial number theoretic applications.
To construct module bases for the full space such that two concepts turn out to be fundamental: order-completeness and the notion of an integral basis; see Definition 4.1. The connection to the classical notion of integral elements is made by Moreover, if f ∈ M ∞ (N), then there exists an algebraic relation with n = .
The main reason to aim at the computation of an integral basis is the following. Many of the modular functions arising in q-series identities can be modified in a straight-forward manner to turn them into members of M ∞ (N), for example, by multiplying with eta quotients. In such cases the q-expansions at the pole ∞ often are available in a 'natural fashion'. The knowledge of an integral basis B for M ∞ (N) that is computed from known functions such as eta-quotients or the Klein j function then allows to algorithmically express f as a C[t]-linear combination of the elements of B.
The content of our note is structured as follows. Section 2 recalls the most important modular function notions needed. Section 3 gives a brief summary of the main problem of this note which is solved by three different methods in Sections 6-8. To describe these solutions we need some preparations. Section 4 discusses the problem of using eta-quotients for module representations. Section 5 returns to the theme of representing the generating function n≥1 p(11n + 6)q n and prepares the ground for the computation of integral bases. Section 6 solves the problem of computing an integral basis for M ∞ (121) by using the modular invariant; i.e. Klein's j function. Section 7 solves the integral basis problem using the trace operator already mentioned in connection with (1.7). Finally, Section 8 explains how the Maple package algcurves can be invoked to derive the desired integral basis.

Notation
Let H = {τ ∈ C | Im(τ ) > 0 } denote the complex upper half-plane. In the following N denotes a positive integer. We define the groups This action induces an action on meromorphic functions f : (2.1) Because of γ ∞ := lim Im(τ )→∞ γ τ = a/c, we say that (2.1) is a q-expansion of f at a/c. Understanding a/0 = ∞, this extends to defining q-expansions at ∞. Note that if γ ∞ = γ ∞ = a/c then γ = γ ±1 h 0 ±1 for some h ∈ Z and, thus, i.e. we can (uniquely) extend the definition of f to points on H : Let M ∞ (N) be the set of modular functions for 0 (N) that only have a pole (if any) at infinity. An element f ∈ M ∞ (N) has a representation as a Laurent series in q.
We denote by pord(f ) = −ord q f the pole order (at infinity) of f ; here ord q f is defined as the index of the least non-zero coefficient in the expansion (2.1) of f in powers of q. In view of (2.2) with c = 0 and thus w N (c) = 1, we note that q-expansions at infinity are unique in integer powers of q.
Denote by M ∞ Q (N) the elements of M ∞ (N) whose q-series expansion have rational coefficients. From Theorem 3.52 of [9] it follows that M ∞ (N) is generated as a C-vectorspace by elements of M ∞ Q (N). The action of SL 2 (Z) on H extends in an obvious way to an action on H. The orbits of the action of the subgroup 0 (N) ⊂ SL 2 (Z) are denoted by The set of all such orbits is denoted by There are only finitely many cosets with respect to 0 (N); more precisely, for N ≥ 2, Owing to this fact together with the observation Q ∪ {∞} = {γ ∞ | γ ∈ SL 2 (Z) }, there are only finitely many orbits [τ ] N with τ ∈ Q ∪ {∞}. These orbits are called cusps of X 0 (N).
As usual, denotes the Dedekind eta function. Let 1 = δ 1 < δ 2 < · · · < δ n = N be the positive divisors of N. For convenience, we allow to index n-dimensional vectors by the divisors of N, instead of the usual index set {1, . . . , n}.
We define R(N) to be the set of integer tuples r = (r δ 1 , . . . , r δ n ) ∈ Z n . With R * (N) we denote the subset of all tuples r = (r δ ) δ|N of R(N) that fulfill the following conditions: By [10,Theorem 1], the elements of are modular functions for 0 (N). Moreover, we define If L is a ring and S is a subset of an L-module, we denote by S L the set of L-linear combinations of elements of S. If L is a field, then S L is a vector space. If S ⊂ L, then S L is an ideal of L.

The problem
As pointed out in the introduction, this case study was inspired by recent algorithmic progress made in connection with classical number theoretic observations made by Ramanujan. For N = 121, the problem is to find an integral basis of the space M ∞ (121) of modular functions having a pole (if any) only at infinity. We present a solution to this problem by following three different approaches, namely, by using Klein's j-invariant, by using series that are obtained by the trace operator, applied to some eta-quotients living in M ∞ (242), and by employing the integral_basis command from Maple.
Essentially, in each of the three approaches we construct a basis for M ∞ (121). These bases cannot be shown explicitly in this paper because of size. For this reason we have put the explicit expressions of the bases at https://risc.jku.at/people/hemmecke/papers/ integralbasis/. Each basis at the above URL has been computed in the computer algebra system FriCAS 1 by the package QEta. 2 In this article, we explicitly mark the references to these bases by giving the name of the file that contains the respective expression(s). This filename has to be appended to the above URL in order to retrieve the data from the internet.

A basis for the eta-quotients in M ∞
Q (121) In [2], Radu shows that E ∞ (N) is a finitely generated (multiplicative) monoid; i.e. there exist m 1 , . . . , m k ∈ E ∞ (N) such that any element of f ∈ E ∞ (N) can be written as Radu also describes an algorithm to compute such monoid generators. Furthermore, Radu gives an algorithm to compute elements , and {z 0 , . . . , z −1 } is ordercomplete. This latter notion is defined in For N = 121, Radu's algorithm [2] delivers two monoid generators in E ∞ (121), namely and Next, by application of the algorithm samba to t and u one obtains the element such that i.e. B = 1, z, z 2 , z 3 , z 4 forms an order-complete basis of Q[t, u] = E ∞ (121) Q . We remark that in this simple example the reduction expressed by the left equality in (4.4) can be 'seen' immediately. Also note that, by using (1 − x) 11 ≡ 1 − x 11 (mod 11), one can easily show that the q-series of z has integer coefficients.

The generating function for p(11n + 6)
In this section, we use a method described in the proof of Proposition 4.3 in [5] to find a new relation for the generating function of p(11n + 6) that shows 11 | p(11n + 6) for all n ∈ N. We aim at computing the cofactor d and the coefficients c i as described in the following Lemma. and, therefore, f / ∈ M = Q[t, u]. Since the maximal pole order of an element of the basis from (4.5) is pord(z 4 ) = 196, it is possible by (4.1) to (algorithmically) reduce any element f ∈ M ∞ (121) to an element r of pole order ≤ 191.
Let us consider the 192 coefficients of the principal part of the (reduced) elements for t i f , for i = 0, . . . , 192 and put them into a matrix (one row for any element), i.e. the (i, j)th entry of the matrix is [q −j ](t i f ) (i.e. the coefficient of q −j in the q-series expansion of t i f ). Since there are more rows than columns, it is clear that there must be a Q-linear relation among the rows of this matrix. We can thus find a polynomial d ∈ Q[T] with the property that d(t)f can be reduced to a modular function with vanishing principal part, i.e.
It turns out that the polynomial d that we have computed can be factored as d =  = c(t, z), in other words, we have found another identity for the generating function for p(11n + 6) in term of eta-quotients. The polynomial c can be factored to reveal a factor of 11 and the degrees of the c k are 75, 66, 56, 46, and 36, respectively, see dc.input.
As mentioned above the q-series expansion of z has integer coefficients. Thus the identity reveals and proves divisibility by 11 of p(11n + 6) for all n ∈ N.
With (4.5) we have found an order-complete basis of M = Q[t, u] with g M = 96. The computations described above not only gave us an identity for f, but they also showed that Adding f to the generators, we can determine an order-complete basis But this does not lead to a better basis, because f 2 ∈ Q[t, u, f ] which is seen by the relation displayed at f2.input.

An integral basis by using the Klein j function
Let us come back to the basis B (f ) computed in Section 5, see bf.input. This basis is not an integral basis, so we must consider to include other elements of M ∞ (121).
Klein's j-invariant (also called modular invariant or absolute invariant) is a modular function for 0 (1) = SL 2 (Z), In the theory of modular functions the j-invariant is fundamental because every modular function can be expressed as a rational function in j. For a definition and further properties of j see, for example, Chapter VII of [12].
We attempt to add more modular functions for 0 (N) and call samba in the hope to get an integral basis. We know, for example, that j(τ ) is a modular function for SL 2 (Z) and consequently also for 0 (N). Since for a b c d ∈ 0 (N) we have j(N aτ +b cτ +d ) = j( a(Nτ )+Nb (c/N)(Nτ )+d ) = j(Nτ ), also j(Nτ ) is a modular function for 0 (N).
We multiply j 0 (τ ) := j(τ ) and j 2 (τ ) := j(11 2 τ ) by suitable eta-quotients with the goal to arrive at elements in M ∞ (121). Let us define where t and u are the eta-quotients defined in (4.2) and (4.3), then j ∞ 0 , j ∞ 2 ∈ M ∞ (121). Calling samba from our QEta package with input t, u, j ∞ 0 , j ∞ 2 leads to an order-complete basis i.e. Q[t, u, j ∞ 0 , j ∞ 2 ] has gap number 6. In other words, . We were also able to compute the representation of the elements of B (j) in terms of the original functions t, u, j 0 and j 2 , but these polynomials are too big to be presented in this article; see bj.input on our website.
Since B (j) is an integral basis for M ∞ (121), we can construct a polynomial p (j) ∈ Q[T, U, J 0 , J 2 ] such that f = p (j) (t, u, j 0 , j 2 ) by reducing f with respect to B (j) and keeping track of the cofactors of this reduction, see fj.input. Unfortunately, this relation cannot be used to demonstrate 11 | p(11n + 6).

An integral basis obtained with the trace map
We can generate a new modular function by applying the trace operator to a modular function from M ∞ (242).
In our case we have is the matrix corresponding to the Atkin-Lehner involution, and U 2 is the operator on functions φ : H → C so that For concrete computations with such trace maps the reader is referred to [7]. Here we only remark that if φ(τ ) = n≥m c(n)q n , q = exp(2πiτ ), is the q-series expansion of φ, then the effect of the action of U 2 is Also, notice that the action of W 242 2 is defined via the slightly more general action of the general linear group GL 2 (Z): for all γ = a b c d ∈ GL 2 (Z), Similar to the computation of (4. By exploiting the modular transformation properties it is straight-forward to compute the See ds.input on our website. That the additional polynomial d 6 appears is, in fact, not a surprise, but can be explained by the following Theorem 7.1. It is a factor of the discriminant (wrt. y) of a polynomial p(x, y) such that p(t, z) = 0, see dz.input.
Theorem 7.1 is an extension of Lemma 5.1 in the sense that we do not just claim the existence of a polynomial d(x), but rather state that choosing the discriminant (see exact formulation in the theorem) will work for any f ∈ M ∞ (N). Theorem 7.1: Let f , t, z ∈ M ∞ (N), with pord(t) ≥ 1 and gcd(pord(t), pord(z)) = 1. Let p(x, y) = y n + p n−1 (x)y n−1 + · · · + p 1 (x)y + p 0 (x) ∈ C[x, y] be the minimal polynomial between t and z, that is p(t, z) = 0. Let D(x) = Disc y (p(x, y)) be the usual discriminant of p(x, y) with respect to y. Then D(t)f = c 0 (t) + c 1 (t)z + · · · + c n−1 (t)z n−1 for some polynomials c i (x) ∈ C[x], i = 0, . . . , n − 1.
Proof: Note that the existence of such a monic polynomial p(x, y) is given by Lemma 1.1. Definition 7.1 of [5] defines the discriminant polynomial D t (z)(x) for an order-complete basis 1, z, . . . , z n−1 of the C[t]-module C [t, z]. In that definition we see that the discriminant is the square of the determinant of a certain Vandermonde matrix: |V(z * 1 , . . . , z * n )| 2 . By comparing the definition of the entries z * i in [5] with the definition of F • (G|U i ) −1 in the proof of Theorem 7.1 of [7], we observe that they coincide. However, in that paper G corresponds to t * and F to z * where t * and z * denote the functions X 0 (N) → C corresponding to t and z, see Remark 5.1 in [5]. Furthermore, in the proof of Theorem 7.1 of [7] it is also shown that the symmetric functions are exactly the coefficients of the polynomial p(t * , y), i.e. p n−i (t * ) = (−1) i e i (z * 1 , . . . , z * n ), i = 1, . . . , n − 1. By considering modular functions instead of functions X 0 (N) → C, we can remove the star and, therefore, p(t, y) = (y − z 1 )(y − z 2 ) · · · (y − z n ). We see that by definition Disc y (p(t, y)) = i<j (z i − z j ) 2 which coincides with |V(z 1 , .., z n )| 2 and also with D(t).
The discriminant polynomials D t (z)(x) = D t (z, z 2 , . . . , z n−1 )(x) is a special case of the concept of an order-reduction polynomial D t (b 1 , . . . , b n−1 )(x) ∈ C[x], see [5, t, u, u 2 , u 3 , v(t, u). This computation takes about 30 minutes and yields a basis 4 . Note that if also the non-leading terms of the q-expansion of the basis elements are reduced by the other basis elements, then B (v) agrees with the basis B (j) computed in Section 6.
Unfortunately, we cannot give the explicit form of the p i,j (T) ∈ Q[T], because they are too big and therefore only listed on our website in the file bv.input. However, we give their degree for the reader to get an idea. We define the matrix A = (a i,j )

Conclusion
There exist already several identities for expressing p(11n + 6)q n in terms of etaquotients. In [3] we gave a relation in terms of eta-quotients for 0 (2 · 11). Initially, our goal was to get rid of the factor 2, which in this context seems unnatural, and try to work with eta-quotients for 0 (11 · 11). As shown in Section 5, f cannot be expressed in that way. However, it is possible to find a polynomial d such that d(t)f indeed is a sum of eta-quotients from M ∞ (121).
The wish to avoid such a polynomial prefactor d in turn led us to attempts to express f by other functions, like the Klein j invariant, an eta-quotient from M ∞ (242), and a modular function v(t, u) coming from the output of van Hoeij's Maple package. Any of these additions not only gave us a way to express f, but, more generally, a way way to compute an integral basis B such that any function of φ ∈ M ∞ (121) can be expressed as a C[t]-linear combination of elements of B. Moreover, in contrast to the basis returned by van Hoeij's Maple package, our basis can be used to algorithmically find polynomials p b ∈ Q[t] with φ = b∈B p p (t)b.