Kronecker’s Limit Formula, Holomorphic Modular Functions, and q-Expansions on Certain Arithmetic Groups

ABSTRACT

For any square-free integer N such that the “moonshine group” Γ₀(N)⁺ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmodul of Γ₀(N)⁺ to certain McKay–Thompson series associated to the representation theory of the Fischer–Griess monster group. In particular, the Hauptmoduli admits a q-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus groups Γ₀(N)⁺. For all such arithmetic groups of genus up to and including three, we prove that the corresponding function field admits two generators whose q-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity. As corollary, we derive a polynomial relation which defines the underlying projective curve, and we deduce whether i∞ is a Weierstrass point. Our method of proof is based on modular forms and includes extensive computer assistance, which, at times, applied Gauss elimination to matrices with thousands of entries, each one of which was a rational number whose numerator and denominator were thousands of digits in length.

Keywords:

• Hauptmodul
• Kronecker limit formula
• Eisenstein series
• q-expansions

2010 AMS Subject Classification:

• 11F11
• 11F06
• 11F20
• 20H05

1. Introduction

1.1. Classical aspects of the j-Invariant

The action of the discrete group on the hyperbolic upper half plane yields a quotient space which has genus zero and one cusp. By identifying with a well-known fundamental domain for the action of on , we will, as is conventional, identify the cusp of with the point denoted by i∞. From the uniformization theorem, we have the existence of a single-valued meromorphic function on which has a simple pole at i∞ and which maps the one-point compactification of onto the one-dimensional projective space . Let z denote the global coordinate on , and set q = e^2πiz, which is a local coordinate in a neighborhood of i∞ in the compactification of . The bi-holomorphic map f from the compactification of onto is uniquely determined by specifying constants c₋₁ ≠ 0 and c₀ such that the local expansion of f near i∞ is of the form f(q) = c₋₁q⁻¹ + c₀ + O(q) as q → 0. For reasons coming from the theory of automorphic forms, one chooses c₋₁ = 1 and c₀ = 744. The unique function obtained by setting of c₋₁ = 1 and c₀ = 744 is known as the j-invariant, which we denote by j(z).

Let One defines the holomorphic Eisenstein series E_k of even weight k ≥ 4 by the series: (1-1) Classically, it is known that (1-2) which yields an important explicit expression for the j-invariant. As noted on page 90 of [Serre 73], which is a point we will emphasize in the next subsection, the j-invariant admits the series expansion: (1-3) An explicit evaluation of the coefficients in the expansion (1–3(1-3) ) was established by Rademacher in [Rademacher 38], and we refer the reader to the fascinating article [Knopp 90] for an excellent exposition on the history of the j-invariant in the setting of automorphic forms. More recently, there has been some attention on the computational aspects of the Fourier coefficients in (1–3(1-3) ); see, for example, [Baier and Köhler 03] and, for the sake of completeness, we mention the monumental work in [Edixhoven and Couveignes 11] with its far reaching vision.

For quite some time it has been known that the j-invariant has importance far beyond the setting of automorphic forms and the uniformization theorem as applied to . For example, in 1937 T. Schneider initiated the study of the transcendence properties of j(z). Specifically, Schneider proved that if z is a quadratic irrational number in the upper half plane then j(z) is an algebraic integer, and if z is an algebraic number but not imaginary quadratic then j(z) is transcendental. More specifically, if z is any element of an imaginary quadratic extension of , then j(z) is an algebraic integer, and the field extension is abelian. In other words, special values of the j-invariant provide the beginning of explicit class field theory. The seminal work of Gross–Zagier [Gross and Zagier 85] studies the factorization of the j-function evaluated at imaginary quadratic integers, yielding numbers known as singular moduli, from which we have an abundance of current research which reaches in various directions of algebraic and arithmetic number theory.

There is so much richness in the arithmetic properties of the j-invariant that we are unable to provide an exhaustive list, but rather ask the reader to accept the above-mentioned examples as indicating the important role played by the j-invariant in number theory and algebraic geometry.

1.2. Monstrous moonshine

The j-invariant attained another realm of importance with the discovery of “Monstrous Moonshine.” We refer to the article [Gannon 06a] for a fascinating survey of the history of monstrous moonshine, as well as the monograph [Gannon 06b] for a thorough account of the underlying mathematics and physics surrounding the moonshine conjectures. At this point, we offer a cursory overview in order to provide motivation for this article.

In the mid-1900s, there was considerable research focused toward the completion of the list of sporadic finite simple groups. In 1973, B. Fischer and R. L. Griess discovered independently, but simultaneously, certain evidence suggesting the existence of the largest of all sporadic simple groups; the group itself was first constructed by R. L. Griess in [Griess 82] and is now known as “the monster,” or “the friendly giant,” or “the Fischer–Griess monster,” and we denote the group by . Prior to [Griess 82], certain properties of were deduced assuming its existence, such as its order and various aspects of its character table. At that time, there were two very striking observations. On one hand, A. Ogg observed that the set of primes which appear in the factorization of the order of is the same set of primes such that the discrete group Γ₀(p)⁺ has genus zero. On the other hand, J. McKay observed that the linear-term coefficient in (1–3(1-3) ) is the sum of the two smallest irreducible character degrees of . J. Thompson further investigated McKay’s observation in [Thompson 79a] and [Thompson 79b], which led to the conjectures asserting all coefficients in the expansion (1–3(1-3) ) are related to the dimensions of the components of a graded module admitting action by . Building on this work, J. Conway and S. Norton established the “monstrous moonshine” conjectures in [Conway and Norton 79] which more precisely formulated relations between and the j-invariant (1–3(1-3) ). A graded representation for was explicitly constructed by I. Frenkel, J. Lepowsky, and A. Meurman in [Frenkel et al. 88] thus proving aspects of the McKay–Thompson conjectures from [Thompson 79a] and [Thompson 79b]. Building on this work, R. Borcherds proved a significant portion of the Conway–Norton “monstrous moonshine” conjectures in his celebrated work [Borcherds 92]. More recently, additional work by many authors (too numerous to list here) has extended “moonshine” to other simple groups and other j-invariants associated to certain genus zero Fuchsian groups.

Still, there is a considerable amount yet to be understood within the framework of “monstrous moonshine.” Specifically, we call attention to the following statement by T. Gannon from [Gannon 06a]:

In genus >0, two functions are needed to generate the function field. A complication facing the development of a higher-genus Moonshine is that, unlike the situation in genus 0 considered here, there is no canonical choice for these generators.

In other words, one does not know the analogue of the j-invariant for moonshine groups of genus greater than zero from which one can begin the quest for “higher genus moonshine.”

1.3. Our main result

The motivation behind this article is to address the above statement by T. Gannon. The methodology we developed yields the following result, which is the main theorem of the present paper.

Theorem 1.

Let N be a square-free integer such that the genus g of Γ₀(N)⁺ satisfies the bounds 1 ≤ g ≤ 3. Then the function field associated to Γ₀(N)⁺ admits two generators whose q-expansions have integer coefficients after the lead coefficient has been normalized to equal one. Moreover, the orders of poles of the generators at i∞ are at most g + 2.

In all cases, the generators we compute have the minimal poles possible, as can be shown by the Weierstrass gap theorem. Finally, as an indication of the explicit nature of our results, we compute a polynomial relation associated to the underlying projective curve. For all groups Γ₀(N)⁺ of genus two and genus three, we deduce whether i∞ is a Weierstrass point.

In brief, our analysis involves four steps. First, we establish an integrality theorem which proves that if a holomorphic modular function f on Γ₀(N)⁺ admits a q-expansion of the form f(z) = q^−a + ∑_{k > −a}c_kq^k, then there is an explicitly computable κ such that if for k ≤ κ then for all k. Second, we prove the analogue of Kronecker’s limit formula, resulting in the construction of a non-vanishing holomorphic modular form Δ_N on Γ₀(N)⁺; we refer to Δ_N as the Kronecker limit function. Third, we construct spaces of holomorphic modular functions on Γ₀(N)⁺ by taking ratios of holomorphic modular forms whose numerators are holomorphic Eisenstein series on Γ₀(N)⁺ and whose denominator is a power of the Kronecker limit function. Finally, we employ considerable computer assistance in order to implement an algorithm, based on the Weierstrass gap theorem and Gauss elimination, to derive generators of the function fields from our spaces of holomorphic functions. By computing the q-expansions of the generators out to order q^κ, the integrality theorem from the first step completes our main theorem.

1.4. Additional aspects of the main theorem

There are ways in which one can construct generators of the function fields associated to the groups Γ₀(N)⁺, two of which we now describe. However, the methodologies do not provide all the information which we developed in the proof of our main theorem.

Classically, one can use Galois theory and elementary aspects of Hecke congruence groups Γ₀(N) in order to form modular functions using the j-invariant. In particular, the functions generate all holomorphic functions which are Γ₀(N)⁺ invariant and, from (1–3(1-3) ), admit q-expansions with lead coefficients equal to one and all other coefficients are integers. However, the order of poles at i∞ are much larger than g + 2.

From modern arithmetic algebraic geometry, we have another approach toward our main result. We shall first present the argument, which was first given to us by an anonymous reader of a previous draft of the present article, and then discuss how our main theorem goes beyond the given argument.

The modular curve X₀(N) exists over , is nodal, and its cusps are disjoint -valued points. For all primes p dividing N, the Atkin–Lehner involutions w_p of X₀(N) commute with each other and generate a group G_N of order 2^r. Let X₀(N)⁺ = X₀(N)/G_N, and let c denote the unique cusp of X₀(N)⁺. The cusp can be shown to be smooth over ; moreover, all the fibers of X₀(N)⁺ over are geometrically irreducible. Therefore, the complement U of in X₀(N)⁺ is affine. The coordinate ring is a finitely generated -algebra, with an increasing filtration by sub--modules consisting of the functions with a pole of order at most k along the cusp c. The successive quotients are free -modules, each with rank either zero or one. The k for which the rank is zero form the gap-sequence of , the generic fiber of X₀(N)⁺. If k is not a gap, there is a non-zero element , unique up to sign and addition of elements from . The q-parameter is obtained by considering the formal parameter at c, which is the projective limit of the sequence of quotients where I is the ideal of c. The projective limit is naturally isomorphic to the formal power series ring . With all this, if f is any element of , then there is an integer n such that nf has q-expansion in . In other words, the function field of can be generated by two elements whose q-expansions are in .

Returning to our main result, we can discuss the information contained in our main theorem which goes beyond the above general argument. The above approach from arithmetic algebraic geometry yields generators of the function field with poles of small order with integral q-expansion; however, it was necessary to “clear denominators” by multiplying through by the integer n, thus allowing for the possibility that the lead coefficient is not equal to one. In the case that X₀(N)⁺ has genus one, then results which are well-known to experts in the field (or so we have been told) may be used to show that the generators satisfy an integral Weierstrass equation. However, such arguments do not apply to the genus two and genus three cases which we exhaustively analyzed. Furthermore, we found that our result holds in all cases when N is square-free, independent of the structure of the gap sequence at i∞; see, [Kohnen 03]. As we found, for genus two i∞ was not a Weierstrass point for any N; however, for genus three, there were various types of gap sequences for different levels N.

Finally, given all of the above discussion, one naturally is led to the following conjecture.

Conjecture 2.

For any square-free N, the function field associated to any positive genus g group Γ₀(N)⁺ admits two generators whose q-expansions have integer coefficients after the lead coefficient has been normalized to equal one. Moreover, the orders of poles of the generators at i∞ are at most g + 2.

1.5. Our method of proof

Let us now describe our theoretical results and computational investigations in greater detail.

For any square-free integer N, the subset of defined by is an arithmetic subgroup of . As shown in [Cummins 04], there are precisely 44 such groups which have genus zero and 38, 39, and 31 which have genus one, two, and three, respectively. These groups of genus up to three will form a considerable portion of the focus in this article. There are two reasons for focusing on the groups Γ₀(N)⁺ for square-free N. First, for any positive integer N, the groups Γ₀(N)⁺ are of moonshine-type (see, e.g., Definition 1 from [Gannon 06a]). Second, should N not be square-free, then there exist genus zero groups Γ₀(N)⁺, namely when N = 25, 49, and 50, but those groups correspond to “ghost” classes of the monster. In summary, we are letting known results from “monstrous moonshine” serve as a guide that, perhaps, the information derived in this paper may someday find meaning elsewhere.

For arbitrary square-free N, the discrete group Γ₀(N)⁺ has one-cusp, which we denote by i∞. Associated to the cusp of Γ₀(N)⁺ one has, from spectral theory and harmonic analysis, a well-defined non-holomorphic Eisenstein series denoted by . The real analytic Eisenstein series is defined for and by (1-4) where is the parabolic subgroup of Γ₀(N)⁺. Our first result is to determine the Fourier coefficients of in terms of elementary arithmetic functions, from which one obtains the meromorphic continuation of the real analytic Eisenstein series to all .

As a corollary of these computations, it is immediate that has no pole in the interval (1/2, 1). Consequently, we prove that groups Γ₀(N)⁺ for all square-free N have no residual spectrum besides the obvious one at s = 1.

Using our explicit formulas for the Fourier coefficients of , we are able to study the special values at s = 1 and s = 0, which, of course, are related by the functional equation for . As a result, we arrive at the following generalization of Kronecker’s limit formula. For any square-free N which has r prime factors, the real analytic Eisenstein series admits a Taylor series expansion of the form where η(z) is Dedekind’s eta function associated to . From the modularity of , one concludes that is invariant with respect to the action by Γ0(N)+. Consequently, there exists a multiplicative character ϵN(γ) on Γ0(N)+ such that one has the identity (1-5) We study the order of the character, and we define ℓN to be the minimal positive integer so that ϵℓNN = 1. A priori, it is not immediate that ℓN is finite for general N. Classically, when N = 1, the study of the character ϵ1 on is the beginning of the theory of Dedekind sums; see [Lang 76]. For general N, we prove that ℓN is finite and, furthermore, can be evaluated by the expression where denotes the least common multiple function and σ(N) stands for the sum of divisors of N.

With the above notation, we define the Kronecker limit function Δ_N(z) associated to Γ₀(N)⁺ to be (1-6) Δ_N(z) is a cusp form and we let k_N denote its weight. By combining (1–5(1-5) ) and (1–6(1-6) ), we get that k_N = 2^{r − 1}ℓ_N. In summary, Δ_N(z) is a non-vanishing, weight k_N holomorphic modular form with respect to Γ₀(N)⁺.

Analogous to the setting of , one defines the holomorphic Eisenstein series of even weight k ≥ 4 associated to Γ₀(N)⁺ by the series (1-7) We show that one can express E^(N)_2m in terms of E_2m, the holomorphic Eisenstein series associated to ; namely, for m ≥ 2 one has the relation (1-8)

With the above analysis, we are now able to construct holomorphic modular functions on the space . For any non-negative integer M, the function (1-9) is a holomorphic modular function on , meaning a weight zero modular form with exponential growth in z as z → i∞. The vector b = (b_ν) can be viewed as a weighted partition of the integer k_NM with weights m = (m_ν). For considerations to be described below, we let denote the set of all possible rational functions defined in (1–9(1-9) ) by varying the vectors b and m yet keeping M fixed. Trivially, consists of the constant function {1}, which is convenient to include in our computations.

Dedekind’s eta function can be expressed by the well-known product formula, namely and the holomorphic Eisenstein series associated to admits the q-expansion: (1-10) where B_k denotes the kth Bernoulli number and σ is the generalized divisor function:

It is immediate that each function F_b in (1–9(1-9) ) admits a q-expansion with rational coefficients. However, it is clear that such coefficients are certainly not integers and, actually, can have very large numerators and denominators. Indeed, when combining (1–8(1-8) ) and (1–10(1-10) ), one gets that which, evidentially, can have a large denominator when m and N are large. In addition, note that the function in (1–9(1-9) ) is defined with a product of holomorphic Eisenstein series in the numerators, so the rational coefficients in the q-expansion of (1–9(1-9) ) are even farther removed from being easily described.

With the above theoretical background material, we implement the algorithm described in Section 7.3 to find a set of generators for the function field associated to the genus g group Γ₀(N)⁺. In addition to the theoretical results outlined above, we obtain the following results which are based on our computational investigations:

(1)	For all genus zero groups Γ0(N)+, the algorithm concludes successfully, thus yielding the q-expansion of the Hauptmoduli. In all such cases, the q-expansions have positive integer coefficients as far as computed. The computations were completed for all 44 different genus zero groups Γ0(N)+.
(2)	For all genus one groups Γ0(N)+, the algorithm concludes successfully, thus yielding the q-expansions for two generators of the associated function fields. In all such cases, each q-expansion has integer coefficients as far as computed. The computations were completed for all 38 different genus one groups Γ0(N)+. The function whose q-expansion begins with q−2 is the Weierstrass ℘-function in the coordinate for the underlying elliptic curve.
(3)	For all 38 different genus one groups Γ0(N)+, we computed a cubic relation satisfied by the two generators of the function fields.
(4)	In addition, we consider all groups Γ0(N)+ of genus two and three. In every instance, the algorithm concludes successfully, yielding generators for the function fields whose q-expansions admit integer coefficients as far as computed. Only for the generators of the function field associated to the group Γ0(510)+, the coefficients are half-integers. For the latter case, we present in Section 10 an additional base change such that the coefficients get integers.
(5)	We extend all q-expansions out to order qκ, where κ is given in Tables 1 and 2 or evaluated according to Remark 6, thus showing that the field generators have integer q-expansions. Table 1. Number of expansion coefficients that need to be integer in order that all coefficients in the q-expansion of the Hauptmodul are integer. Listed is the level N and the value of κ according to Remark 6 for the genus zero groups Γ0(N)+. N 1 2 3 5 6 7 10 11 13 14 15 17 19 21 22 23 26 29 30 31 33 34 κ 19 47 48 19 60 19 75 19 19 47 96 19 19 53 47 19 47 19 127 19 48 47 N 35 38 39 41 42 46 47 51 55 59 62 66 69 70 71 78 87 94 95 105 110 119 κ 19 47 48 19 108 47 19 48 19 19 47 60 48 181 19 81 48 47 19 181 89 19 Table 2. Number of expansion coefficients that need to be integer in order that all coefficients in the q-expansions of the field generators are integer. Listed is the level N and the value of κ according to Remark 6 for the genus one groups Γ0(N)+. N 37 43 53 57 58 61 65 74 77 79 82 83 86 89 91 101 102 111 114 κ 222 194 194 285 194 194 634 222 597 194 194 194 194 194 643 194 194 222 285 N 118 123 130 131 138 141 142 143 145 155 159 174 182 190 195 210 222 231 238 κ 194 194 634 194 194 194 194 271 382 382 194 194 643 382 634 599 222 597 599

The fact that all of the q-expansions which we uncovered have integer coefficients is not at all obvious and leads us to believe there is deeper, so-far hidden, arithmetic structure which perhaps can be described as “higher genus moonshine.”

In some instances, the computations from our algorithm were elementary and could have been completed without computer assistance. For instance, when N = 5 or N = 6, the first iteration of the algorithm used a set with only two functions to conclude successfully. However, as N grew, the complexity of the computations became quite large. As an example, for N = 71, which is genus zero and appears in “monstrous moonshine,” the smallest non-zero weight for a denominator in (1–9(1-9) ) was 4, but we needed to consider all functions whose numerators had weight up to 40, resulting in 362 functions whose largest pole had order 120. The most computationally extensive genus one example was N = 79 where the smallest non-zero weight denominator in (1–9(1-9) ) was 12, but we needed to consider all functions whose numerator had weight up to 84, resulting in 13158 functions whose largest pole had order 280.

As one can imagine, the data associated to the q-expansions we considered is massive. In some instances, we encountered rational numbers whose numerators and denominators each occupied a whole printed page. In addition, in the cases where the algorithm required several iterations, the input data of q-expansions of all functions were stored in computer files which if printed would occupy hundreds of thousands of pages. As an example of the size of the problem we considered, it was necessary to write computer programs to search the output from the Gauss elimination to determine if all coefficients of all q-expansions were integers since the output itself, if printed, would occupy thousands of pages.

The computational results are summarized in Sections 8–10. In Tables 5^{Table 6}, 7, and 9, we list the q-expansions of the two generators of function fields associated to each genus one, genus two, and genus three group Γ₀(N)⁺. As T. Gannon’s comment suggests, the information summarized in those tables does not exist elsewhere. In Tables 6, 8, and 10, we list the polynomial relations satisfied by the generators of Tables 5, 7, and 9, respectively. The stated results, in particular the q-expansions, were limited solely by space considerations; a thorough documentation of our findings will be given in forthcoming articles.

All input and output information associated to the computational investigations undertaken in the present article is made available at http://www.efsa.unsa.ba/∼lejla.smajlovic/.

1.6. Further studies

As stated above, the j-invariant can be written in terms of holomorphic Eisenstein series associated to . By storing all information from the computations from Gauss elimination, we obtain similar expressions for the Hauptmoduli for all genus zero groups Γ₀(N)⁺. As one would imagine based upon the above discussion, some of the expressions will be rather large. In [Jorgenson et al. preprint-a], we will report of this investigation, which in many instances yields new relations for the j-invariants and for holomorphic Eisenstein series themselves. For the genus one groups, the computations from Gauss elimination produce expressions for the Weierstrass ℘-function, in the coordinate on , in terms of holomorphic Eisenstein series; these computations will be presented in [Jorgenson et al. in preparation-b]. Finally, in [Jorgenson et al. in preparation-c], we compute values of the Hauptmoduli j_N associated to all genus zero groups Γ₀(N)⁺ at elliptic fixed points using differential equation satisfied by the Schwarzian derivative of j_N and prove that all values are algebraic integers.

1.7. Outline of the paper

In Section 2, we will establish notation and recall various background material. In Section 3, we prove that if a certain number of coefficients in the q-expansions of the generators are integers, then all coefficients are integers. In Section 4, we will compute the Fourier expansion of the non-holomorphic Eisenstein series associated to Γ₀(N)⁺. The generalization of Kronecker’s limit formula for groups Γ₀(N)⁺ is proven in Section 5, including an investigation of its weight and the order of the associated Dedekind sums. In Section 6, we relate the holomorphic Eisenstein series associated to Γ₀(N)⁺ to holomorphic Eisenstein series on , as cited above. Further details regarding the algorithm we develop and implement are given in Section 7. Results of our computational investigations are given in Section 8 for genus zero, Section 9 for genus one, and Section 10 for genus two and genus three. Finally, in Section 11, we offer concluding remarks and discuss directions for future study, most notably our forthcoming articles [Jorgenson et al. preprint-a], [Jorgenson et al. in preparation-b], and [Jorgenson et al. in preparation-c].

1.8. Closing comment

The quote we presented above from [Gannon 06a] indicates that “higher genus moonshine” has yet to have the input from which one can search for the type of mathematical clues that are found in McKay’s observation involving the coefficients of the j-invariant or in Ogg’s computation of the levels of all genus zero moonshine groups Γ₀(N)⁺ and their appearance in the prime factorization of the order of the Fischer–Griess monster . It is our hope that someone will recognize some patterns in the q-expansions we present in this article, as well as in [Jorgenson et al. preprint-a] and [Jorgenson et al. in preparation-b], and then, perhaps, higher genus moonshine will manifest itself.

2. Background material

2.1. Preliminary notation

Throughout we will employ the standard notation for several arithmetic quantities and functions, including: the generalized divisor function σ_a, Bernoulli numbers B_k, the Möbius function μ, and the Euler totient function ϕ, the Jacobi symbol , the greatest common divisor function ( ·, ·), and the least common multiple function . Throughout the paper we denote by {p_i}, i = 1, …, r, a set of distinct primes and by N = p₁⋅⋅⋅p_r a square-free, positive integer.

The convention we employ for the Bernoulli numbers follows [Zagier 08] which is slightly different than [Serre 73] although, of course, numerical evaluations agree when following either set of notation. For a precise discussion, we refer to the footnote on page 90 of [Serre 73].

2.2. Certain arithmetic groups

As stated above, denotes the hyperbolic upper half plane with global variable with z = x + iy and y > 0, and we set q = e^2πiz = e(z).

The subset of , defined by (2-11) is an arithmetic subgroup of . We denote by the corresponding subgroup of .

Basic properties of Γ₀(N)⁺, for square-free N are derived in [Jorgenson et al. 14] and references therein. In particular, we use that the non-compact surface has exactly one cusp, which can be taken to be at i∞.

2.3. Function fields and modular forms

The set of meromorphic functions on X_N is a function field, meaning a degree one transcendental extension of . Let g_N denote the genus of X_N. If g_N = 0, then the function field of X_N is isomorphic to where j_N is a single-valued meromorphic function on X_N. If g_N > 0, then the function field of X_N is generated by two elements which satisfy a polynomial relation.

A meromorphic function f on is a weight 2k meromorphic modular form if we have the relation In other language, the differential is Γ₀(N)⁺ invariant.

The product, resp. quotient, of weight k₁ and weight k₂ meromorphic forms is a weight k₁ + k₂, resp. k₁ − k₂, meromorphic form. Since , each holomorphic modular form admits a Laurent series expansion which, when setting q = e^2πiz, can be written as As is common in the mathematical literature, we consider forms such that c_m = 0 whenever m < m₀ for some . If , the function f frequently will be re-scaled so that when c_m = 0 for m < m₀; the constant m₀ is the order of the pole of f at i∞.

If X_N has genus zero, then by the Hauptmoduli j_N for X_N one means the weight zero holomorphic form which is the generator of the function field on X_N. If g_N > 0, then we will use the notation j_{1; N} and j_{2; N} to denote two generators of the function field.

Unfortunately, the notation of hyperbolic geometry writes the local coordinate on as x + iy, and the notation of the algebraic geometry of curves uses x and y to denote the generators of the function field under consideration; see, for example, page 31 of [Lang 82]. We will follow both conventions and provide ample discussion in order to prevent confusion.

3. Integrality of the coefficients in the q-expansion

In this section we prove that integrality of all coefficients in the q-expansion of the Hauptmodul j_N when g_N = 0 and the generators j_{1; N} and j_{2; N} when g_N > 0 can be deduced from integrality of a certain finite number of coefficients. Also, our proof yields an effective bound on the number of coefficients needed to test for integrality. First, we present a proof in the case when genus is zero, followed by a proof for the higher genus setting which is a generalization of the genus zero case. The proof is based on the property of Hecke operators and Atkin–Lehner involutions. We begin with a simple lemma.

Lemma 3.

For any prime p which is relatively prime to N, let T_p denote the unscaled Hecke operator which acts on Γ₀(N) invariant functions f, If f is a holomorphic modular form on Γ₀(N)⁺, then T_p(f) is also a holomorphic modular form on Γ₀(N)⁺.

Proof.

The form f is Γ₀(N)⁺ invariant, hence f is Γ₀(N) invariant and so is T_p(f). By Lemma 11 of [Atkin and Lehner 70], if W is any coset representative of the quotient group Γ₀(N)\Γ₀(N)⁺, then T_p(f) is also W invariant since f is W invariant. Therefore, T_p(f) is a Γ₀(N)⁺ invariant holomorphic form.

Theorem 4.

Let j_N be the Hauptmodul for a genus zero group Γ₀(N)⁺. Let p₁ and p₂ with p₂ > p₁ be distinct primes which are relatively prime to N. If the q-expansion of j_N contains integer coefficients out to order q^κ for some κ ≥ (p₂/(p₂ − 1)) · (p₁p₂)², then all further coefficients in the q-expansion of j_N are integers.

Proof.

Let c_k denote the coefficient of q^k in the q-expansion of j_N. Let m > κ be arbitrary integer. Assume that c_k is an integer for k < m, and let us now prove that c_m is an integer. To do so, let us consider the Γ₀(N)⁺ invariant functions for specific integers r₁ and r₂ depending on m which are chosen as follows. First, take r₁ to be the unique integer such that We now wish to choose r₂ to be the smallest positive integer such that Let us argue in general the existence of r₂ and establish a bound in terms of p₁ and p₂, noting that for any specific example one may be able to choose r₂ much smaller than determined by the bound below.

Since p₁ and p₂ are primes and 1 ≤ r₁ ≤ p₁ < p₂, we have that (p₂, r₁p₁) = 1. By applying the Euclidean algorithm and Bezout’s identity, there exists an r ≥ 1 such that rp₂ ≡ 1modr₁p₁. Choose A be the smallest positive integer such that and Clearly, and, furthermore In particular, r₁p₁ and r₂p₂ are relatively prime, and we have the bounds r₁p₁ ⩽ p²₁ and r₂p₂ ⩽ p²₁p²₂.

With the above choices of r1 and r2, set f1 = jr1N and f2 = jr2N. The function f1 is a Γ0(N)+ invariant modular function with pole of order r1 at i∞. Therefore, there is a polynomial of degree r1p1 such that (3-12) If we write the q-expansion of f1 as then The coefficients bk are determined by the binomial theorem and the coefficients of jN, namely by By assumption, c1, …, cm − 1 are integers, and m − 1 ≥ (p2/(p2 − 1))(p1p2)2 > r1. From this, we conclude that are integers. In particular, all coefficients of out to order q0 are integers, from which we can compute the coefficients of and conclude that the polynomial has integer coefficients, in particular the lead coefficient is one.

Let us determine the first appearance of the coefficient c_m in . First, the smallest k where c_m appears in the formula for b_k is when k = m + 1 − r₁, and then we have that By our choice of r₁, the index m + 1 − r₁ is positive and divisible by p₁. As a result, the first appearance of c_m in the expansion of is within the coefficient of q^d where d = (m + 1 − r₁)/p₁. Going further, we have that the coefficient of q^d for d = (m + 1 − r₁)/p₁ in the expansion of is of the form p₁r₁c_m plus an integer which can be determined by binomial coefficients and c_k for k < m.

Let us now determine the first appearance of c_m in . Again, by the binomial theorem, we have that c_m first appears as a coefficient of q^e where e = m − r₁p₁ + 1. By the choice of m and r₁, since p₁ < p₂, one has that The above inequality is equivalent to in other words, d < e. As a consequence, we have that the coefficient of q^d in can be written as a polynomial expression involving binomial coefficients and c_k for k < m. By induction, the coefficient of q^d in is an integer.

In summary, by equating the coefficients of q^d where d = (m + 1 − r₁)/p₁ in the formula (3–12(3-12) ), on the left-hand side we get an expression of the formula r₁p₁c_m plus an integer, and on the right-hand side we get an integer. Therefore, c_m is a rational number whose denominator is a divisor of r₁p₁.

Let us now consider . Since m + 1 − r₂ is divisible by p₂, we consider the coefficient of where d′ = (m + 1 − r₂)/p₂. By the choice of m and r₂, recalling that r₂ ⩽ p²₁p₂, we have and this is equivalent to or d′ < e′ = m − r₂p₂ + 1. With all this, we conclude, analogously as in the first case that c_m is a rational number whose denominator is a divisor of r₂p₂. However, r₁p₁ and r₂p₂ are relatively prime, hence c_m is an integer.

By induction on m, the proof of the theorem is complete.

Theorem 5.

Let Γ₀(N)⁺ have genus greater than zero, and assume there exists two holomorphic modular functions j_{1; N} and j_{2; N} which generate the function field associated to Γ₀(N)⁺. Furthermore, assume that the q-expansions of j_{1; N} and j_{2; N} are normalized in the form: Let p₁ and p₂ with p₂ > p₁ be distinct primes which are relatively prime to a₁a₂N. Assume the q-expansions of j_{1; N} and j_{2; N} contain integer coefficients out to order q^κ with κ ≥ a₂(p₂/(p₂ − 1)) · (p₁p₂)². Then all further coefficients in the q-expansions of j_{1; N} and j_{2; N} are integers.

Proof.

The argument is very similar to the proof of Theorem 4. Let c_{l; k} denote the coefficient of q^k in the q-expansion of j_{l; N}, l ∈ {1, 2}. Assume that c_{l; k} is an integer for l ∈ {1, 2} and k < m, and let us prove that c_{1; m} and c_{2; m} are integers.

For i, l ∈ {1, 2}, we study the expression (3-13) for certain polynomials Q_{i, l} of two variables. The left-hand side of the above equation has a pole at i∞ of order a_lp_ir_i. Hence The existence of Q_{i, l}(x, y) follows from the assumption that j_{1; N} and j_{2; N} generate the function field associated to Γ₀(N)⁺ and the observation that is Γ₀(N)⁺ invariant and has a pole of order less than a_lr_ip_i.

Of course, the polynomials Q_{i, l} are not unique since j_{1; N} and j_{2; N} satisfy a polynomial relation. This does not matter. We introduce the following canonical choice in order to uniquely determine the polynomials. Consider the coefficient . If there exist non-negative integers n′₁ and n′₂ such that a₁n′₁ + a₂n′₂ = a₁n₁ + a₂n₂ with n′₁ < n₁ then we set equal to zero.

Integrality of coefficients of j_{1; N} and j_{2; N} out to order q^κ with κ ≥ a₂(p₂/(p₂ − 1)) · (p₁p₂)² implies that all coefficients of out to order q⁰ are integers. The coefficient of the polynomial Q_{i, l}(x, y) first appears on the right-hand side of (3–13(3-13) ) as a coefficient multiplying . The canonical choice of coefficients enables us to deduce, inductively in the degree ranging from −a_lr_ip_i + 1 to zero that all coefficients are integers.

Having established that the coefficients of the polynomials Q_{i, l} are integers, now we wish to determine two values r₁ and r₂ so that the coefficient c_{1; m} first appears in as a factor of q^d for the smallest possible d, which leads to determining r_i from the equation . Such a solution exists provided p₁ and p₂ are relatively prime to a₁. Without loss of generality, we may assume that 1 ≤ r₁ ≤ p₁. As in the proof of Theorem 4, we impose the further condition that (r₁p₁, r₂p₂) = 1. For at least one value of r₂ in the range 1 ≤ r₂ ≤ r₁p₁p₂, we have that and , so, in particular, (r₁p₁, r₂p₂) = 1.

With the above choices of r₁, we determine the first appearance of c_{1; m} in the equation: On the left-hand side, c_{1; m} appears as a coefficient of q^d where d = (m − (r₁ − 1)a₁)/p₁, and, in fact, the coefficient of q^d is equal to an integer plus r₁p₁c_{1; m}. On the right-hand side, c_{1; m} first appears as a coefficient of q^e where e = m − a₁(r₁p₁ − 1). We have that d < e when m > m′ where The resulting expression shows that c_{1; m} can be expressed as a fraction, where the numerator is a finite sum involving integer multiples of positive powers of c_{1; k} and c_{2; k} for k < m and denominator equal to r₁p₁. Similarly, by studying the first appearance of c_{1; m} in the expression: we obtain an expression showing that c_{1; m} can be expressed as a fraction, where the numerator is a finite sum involving integer multiples of positive powers of c_{1; k} and c_{2; k} for k < m and denominator equal to r₂p₂. By induction on m, and that (r₁p₁, r₂p₂) = 1, we conclude that c_{1; m} is an integer.

Analogously, one studies c_{2; m}. In the four different equations studied, the minimum number of coefficients needed to be integers in order to initiate the induction is the smallest integer that is larger than or equal to a₂(p₂/(p₂ − 1))(p₁p₂)², which is assumed to hold in the premise of the theorem.

Remark 6.

Let us now describe, then implement, an algorithm which will reduce the number of coefficients which need to be tested for integrality. For this remark alone, let us set a₁ = a₂ = 1 if g_N = 0.

Let m be the lower bound determined in Theorem 4 and Theorem 5. Let p₁, …, p_k be the set of primes less than m which are relatively prime to a₁a₂N. For each prime, let r_{i, l} be the integer in the range 1 ≤ r_{i, l} ≤ p_i satisfying , l ∈ {1, 2}. Let S_{m, l} denote the set of triples: Note that if r_{i, l} is such that a_l(r_{i, l}(p_i + 1) − 1) ≥ m, then the set S_{m, l} does not contain a triple whose prime is p_i.

Now let Assume that the greatest common divisor of all elements in Rm, 1 is one and that of Rm, 2 is also one. Then, by using all the Hecke operators applied to the functions jri, 1 + jpi1; N and jri, 2 + jpi2; N and the arguments from Theorem 4 and Theorem 5, we can determine c1; m and c2; m from lower indexed coefficients and show that c1; m and c2; m are integers if all lower indexed coefficients are integers.

Although the above observation is too cumbersome to employ theoretically, it does lead to the following algorithm which can be implemented:

(1)	Let m be the lower bound given in Theorem 4 for g = 0 or Theorem 5 for g > 0.
(2)	Construct the sets Sm, l and Rm, l, as described above.
(3)	If the greatest common divisor of all elements in Rm, 1 is one and that of Rm, 2 is one, too, then replace m by m − 1 and repeat with Step 2.
(4)	If the greatest common divisor of all elements of Rm, 1 or Rm, 2 is greater than one, let κ = m.

The outcome of this algorithm yields a reduced number κ of coefficients which need to be tested for integrality in order to conclude that all coefficients are integers.

In Tables 1 and 2 we list the level N and improved bounds on κ which were determined by the above algorithm for all genus zero and genus one groups Γ₀(N)⁺.

4. The Fourier coefficients of the real analytic Eisenstein series

In this section we derive formulas for coefficients in the Fourier expansion of the real analytic Eisenstein series . First, we derive an intermediate result, expressing the Fourier coefficients in terms of Ramanujan sums. Then, we compute coefficients in a closed form.

4.1. The Fourier expansion of real analytic Eisenstein series on Γ₀(N)⁺

Lemma 7.

The real analytic Eisenstein series , defined by (1–4(1-4) ) for z = x + iy and , admits a Fourier series expansion: (4-14) where W_s(mz) is the Whittaker function given by and K_{s − 1/2} is the Bessel function. Furthermore, coefficients of the Fourier expansion (4–14(4-14) ) are given by where is the completed Riemann zeta function and (4-15) where we put

Proof.

The Fourier expansion (4–14(4-14) ) of the real analytic Eisenstein series for is a special case of [Iwaniec 02], Theorem 3.4., when the surface has only one cusp at i∞ with identity scaling matrix. By ϕ_N(s) we denote the scattering matrix, evaluated in [Jorgenson et al. 14] as where

If C_N denotes the set of positive left-lower entries of matrices from Γ₀(N)⁺, then, by [Iwaniec 02], Theorem 3.4. The Fourier coefficients of (4–14(4-14) ) are given by (4-16) where S_N denotes the Kloosterman sum (see [Iwaniec 02], formula (2.23)) defined, for c ∈ C_N, as (4-17) In [Jorgenson et al. 14] we proved that For a fixed , with v∣N and arbitrary, we can take e = v in the definition of Γ₀(N)⁺ to deduce that matrices from Γ₀(N)⁺ with left lower entry c are given by for some integers a, b, and d such that vad − (N/v)bn = 1. Since N is square-free, this equation has a solution if and only if (v, n) = 1 and (d, (N/v)n) = 1.

Therefore, for , and fixed equation (4–17(4-17) ) becomes This is a Ramanujan sum, which can be evaluated using formula (3.3) from [Iwaniec and Kowalski 04], page 44, to get For element belongs to C_N if and only if v∣N and (n, v) = 1, hence equation (4–16(4-16) ) becomes which proves (4–15(4-15) ).

4.2. Computation of Fourier coefficients

In this subsection we will compute coefficients (4–15(4-15) ) in closed form. We will assume that m > 0 and incorporate negative terms via the identity ϕ_N( − m, s) = ϕ_N(m, s).

Let p and q denote prime numbers, and let denote the highest power of p that divides m, i.e., the number α_p is such that and . For any prime p, set and

Theorem 8.

Assume N is square-free and let ϕ_N(m, s) denote the mth coefficient in the Fourier series expansion (4–14(4-14) ) of the Eisenstein series. Then the coefficients ϕ_N(m, s) can be meromorphically continued from the half plane to the whole complex plane by the equation:

Proof.

We employ equation (4–15(4-15) ) and observe that coefficients a_m(n) are multiplicative. Therefore, it is natural to investigate the associated L-series: defined for . The series L_m(s) is defined in [Koyama 09], Lemma 2.2, for , where its analytic continuation to the whole complex plane is given by the formula L_m(s) = σ_{1 − s}(m)/ζ(s).

The multiplicativity of coefficients a_m(n) implies that for one has the Euler product: where E_p(m, s) stands for the pth Euler factor, i.e., (4-18) By the computations presented in [Koyama 09], page 1137, one has hence, E_p(m, s) = F_p(m, s). Therefore, (4-19)

Let v∣N be fixed. In order to compute the sum: using equation (4–19(4-19) ), we use the fact that N is square-free, so (v, N/v) = 1, hence every integer n coprime to v can be represented as (4-20) Since (k, v) = 1 and (k, q) = 1 for every q∣(N/v), we deduce that actually (k, N) = 1. Therefore (4-21) Let q₁, …, q_l denote the set of all prime divisors of N/v. The multiplicativity of coefficients a_m implies that for n given by (4–20(4-20) ) one has hence, by (4–21(4-21) ) one gets that The identity (4–19(4-19) ) is now applied to the inner sum in the above equation, from which we deduce that for .

For all ν = 1, …, l one then has by (4–18(4-18) ). Therefore, (4-22)

Substituting (4–22(4-22) ) into (4–15(4-15) ) completes the proof.

Proposition 9.

For a square-free integer N = ∏^r_{ν = 1}p_ν, D_N(m, s) is multiplicative in N, Furthermore, for a prime p and : where is the highest power of p dividing m.

Proof.

We apply induction on number r of prime factors of N. For r = 1 the statement is immediate.

Assume that the statement is true for all numbers N with r distinct prime factors and let N = ∏^{r + 1}_{ν = 1}p_ν. Since {v: v∣N} = {v: v∣∏^r_{ν = 1}p_ν}∪{v · p_{r + 1}: v∣∏^r_{ν = 1}p_ν}, the statement is easily deduced from the definition of D_N(m, s) and the inductive assumption.

The second statement follows trivially from the properties of the function σ_s(m) and the definition of the Euler factor F_p.

Corollary 10.

For any square-free N, the groups Γ₀(N)⁺ have no residual eigenvalues λ ∈ [0, 1/4) besides the obvious one λ₀ = 0.

Proof.

From Lemma 7 with m = 0, we have a formula for the constant term ϕ_N(s) in the Fourier expansion of the real analytic Eisenstein series, namely see also Lemma 5 of [Jorgenson et al. 14]. It is elementary to see that ϕ_N(s) has no poles in the interval (1/2, 1). From the spectral theory, one has that the residual eigenvalues λ of the hyperbolic Laplacian such that λ ∈ (0, 1/4) correspond to poles s of the real analytic Eisenstein series with s ∈ (1/2, 1) with the relation s(1 − s) = λ. Since poles of real analytic Eisenstein series are exactly the poles of ϕ_N(s), the statement is proved.

5. Kronecker’s limit formula for Γ₀(N)⁺

In this section we derive Kronecker’s limit formula for the Laurent series expansion of the real analytic Eisenstein series. First, we will prove the following, intermediate result.

Proposition 11.

For a square-free integer N = ∏^r_{ν = 1}p_ν, we have the Laurent expansion: (5-23) as s → 1, where and

Proof.

We start with the formula (4–14(4-14) ) for the Fourier expansion of real analytic Eisenstein series at the cusp i∞.

Let ϕ denote the scattering matrix for . Then, one has ϕ_N(s) = ϕ(s)D_N(s). We will use this fact in order to deduce the first two terms in the Laurent series expression of around s = 1.

By the classical Kronecker limit formula for the Laurent series expansion of ϕ(s)y^{1 − s} at s = 1 is given by (see, e.g., [Kühn 01], page 228) where A = 2 − 24ζ′( − 1) − 2log 4π.

The function D_N(s) is holomorphic in any neighborhood of s = 1, hence D_N(s) = D_N(1) + D′_N(1)(s − 1) + O(s − 1)², as s → 1. Therefore, as s → 1.

In order to get the residue and the constant term stemming from ϕ_N it is left to compute values of D_N and D′_N at s = 1, This, together with (4–14(4-14) ) and Theorem 8 completes the proof.

Since the series always has a pole at s = 1 with residue , from the above proposition, we easily deduce a simple formula for the volume of the surface X_N in terms of the level of the group, namely

Theorem 12.

For a square-free integer N = ∏^r_{ν = 1}p_ν, we have the asymptotic expansion as s → 1, where C_{−1, N}and C_{0, N} are defined in Proposition 11 and η is the Dedekind eta function defined for by

Proof.

From the definition of Euler factors Fp(m, s), using the fact that , we get that Therefore, a simple computation implies that hence where we put P(i1, …, ik) = 1 for k = 0, and for k = 1, …, r Every m ≥ 1 can be written as m = pα11⋅⋅⋅pαrr · l, where αi ≥ 0 and (l, N) = 1. Therefore, we may write the sum over m in (5–23(5-23) ) as (5-24) When k = 0 the above sum obviously reduces to (5-25) as deduced from the proof of the classical Kronecker limit formula.

Now, we will take k ∈ {1, …, r} and compute one term in the sum (5–24(5-24) ). Without loss of generality, in order to ease the notation, we may assume that i1 = 1, …, ik = k and compute the term with P(1, …, k). Later, we will take the sum over all indices 1 ≤ i1 < ⋅⋅⋅ < ik ≤ r. First, we observe that P(1, …, k) = 0 if αi = 0, for any i ∈ 1, …, k and if all αi ≥ 1, for i = 1, …, k. Furthermore, for αi ≥ 1, i = 1, …, k one has Therefore, Since every integer m ≥ 1 can be represented as m = pα1 − 11⋅⋅⋅pαk − 1k · pαk + 1k + 1⋅⋅⋅pαrr · l, where α1, …, αk ≥ 1; αk + 1, …, αr ≥ 0 and (l, N) = 1 we immediately deduce that by (5–25(5-25) ).

Let us now sum over all k = 0, …, r and sum over all indices 1 ≤ i₁ < … < i_k ≤ r, which is equivalent to taking the sum over all v∣N. The sum (5–24(5-24) ) then becomes since ∑_v∣Nv = σ(N). Therefore, we get that The proof is complete.

Remark 13.

Since the number of divisors of N = p₁⋅⋅⋅p_r is 2^r, the expression is a geometric mean.

When N = 1, Theorem 12 amounts to the classical Kronecker limit formula.

From Theorem 12 and the functional equation for real analytic Eisenstein series, we can reformulate the Kronecker limit formula as asserting an expansion for as s → 0.

Proposition 14.

For , the real analytic Eisenstein series admits a Taylor series expansion of the form

An immediate consequence of Theorem 12 and the above proposition is the fact that the function (5-26) is invariant under the action of the group Γ₀(N)⁺. Using the fact that the eta function is non-vanishing on , we may deduce the stronger statement.

Proposition 15.

There exists a character ϵ_N(γ) on Γ₀(N)⁺ such that (5-27) for all γ ∈ Γ₀(N)⁺.

Proof.

Let Since the expression (5–26(5-26) ) is a group invariant we have Dividing by and raising the expression to the 2^r/4 power, we get that hence for some function f of absolute value 1.

Since the function η is a holomorphic function, non-vanishing on , for a fixed γ ∈ Γ₀(N)⁺ the function f(γ, z) is a holomorphic function in z on of absolute value 1 on , hence it is constant, as a function of z. Therefore, f(z, γ) = ϵ_N(γ), for all where |ϵ_N(γ)| = 1.

It is left to prove that ϵ_N(γ₁γ₂) = ϵ_N(γ₁)ϵ_N(γ₂), for all γ₁, γ₂ ∈ Γ₀(N)⁺. This is an immediate consequence of formula (5–27(5-27) ).

The character ϵ_N(γ) is the analogue of the classical Dedekind sum.

In the case when the genus of the surface X_N is equal to zero, the group Γ₀(N)⁺ is generated by a finite number of elliptic elements and one parabolic element. The character ϵ_N(γ) is necessarily finite on each elliptic element. The fundamental group of X_N contains a relation which expresses the generator of the parabolic group as a product of finite, cyclic elements; hence ϵ_N(γ) is finite on the parabolic element as well. Therefore, there exists an integer ℓ_N such that for all γ ∈ Γ₀(N)⁺.

Irrespective^{Table 3}of the^{Table 4}genus of the surface X_N, we will prove that the character ϵ_N(γ) is a certain 24th root of unity.

Theorem 16.

Let N = p₁⋅⋅⋅p_r. Let us define the constant ℓ_N by Then, the function is a weight k_N = 2^{r − 1}ℓ_N holomorphic form on Γ₀(N)⁺ vanishing only at the cusp.

Proof.

The vanishing of the function Δ_N(z) at the cusp i∞ only is an immediate consequence of properties of the Dedekind eta function. Therefore, it is left to prove that Δ_N(z) is a weight k_N = 2^{r − 1}ℓ_N holomorphic form on Γ₀(N)⁺.

We begin with the decomposition , where see [Maclachlan 81], page 147, with a slightly different notation; elements and are interchanged.

For each v∣N and v > 1, from [Atkin and Lehner 70], Lemma 8 and Lemma 9 (see also [Maclachlan 81], page 147) it follows that Γ₀(N, v) is the coset under the action of the congruence group Γ₀(N) of a product of at most r non-trivial elements , j ∈ A_v⊆{1, …, r} such that . Therefore, every element of Γ₀(N)⁺ can be written as a finite product of an element from Γ₀(N) and elements from Γ₀(N)⁺ whose square is in Γ₀(N). Since the character ϵ_N defined in Proposition 15 is multiplicative, it is sufficient to prove that for all γ ∈ Γ₀(N) and for all elements τ of Γ₀(N)⁺ such that τ² ∈ Γ₀(N).

Therefore, it is actually sufficient to prove that for for an arbitrary .

For this, we apply results of [Raji 06], chapter 2.2.3, pages 21–23, with f₁ = Δ_N; g = 2^r; δ_l = v, , for all δ_l = v and k = k_N. By the definition of ℓ_N, we see that , therefore, conditions (2.11) and (2.12) of [Raji 06] are fulfilled, so then where denotes the Jacobi symbol. The multiplicativity of the Jacobi symbol implies that hence Therefore, since k_N is divisible by 4. The proof is complete.

In Tables 3 and 4 we list the values of degree ℓ_N and weight k_N for all genus zero and genus one groups Γ₀(N)⁺.

Table 3. The degree ℓ_N and the weight k_N of the modular form Δ_N on Γ₀(N)⁺ for all groups Γ₀(N)⁺ of genus zero. Listed are the level N, and the values of degree ℓ_N and weight k_N.

N	1	2	3	5	6	7	10	11	13	14	15	17	19	21	22	23	26	29	30	31	33	34
ℓN	24	8	12	4	2	12	4	4	12	2	2	4	12	6	2	4	4	4	1	12	2	4
kN	12	8	12	4	4	12	8	4	12	4	4	4	12	12	4	4	8	4	4	12	4	8
N	35	38	39	41	42	46	47	51	55	59	62	66	69	70	71	78	87	94	95	105	110	119
ℓN	2	2	6	4	1	2	4	2	2	4	2	1	2	1	4	1	2	2	2	1	1	2
kN	4	4	12	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4	4

Table 4. The degree ℓ_N and the weight k_N of the modular form Δ_N on Γ₀(N)⁺ for all groups Γ₀(N)⁺ of genus one. Listed are the level N and the values of degree ℓ_N and weight k_N.

N	37	43	53	57	58	61	65	74	77	79	82	83	86	89	91	101	102	111	114
ℓN	12	12	4	6	4	12	2	4	2	12	4	4	2	4	6	4	1	6	1
kN	12	12	4	12	8	12	4	8	4	12	8	4	4	4	12	4	4	12	4
N	118	123	130	131	138	141	142	143	145	155	159	174	182	190	195	210	222	231	238
ℓN	2	2	2	4	1	2	2	2	2	2	2	1	1	1	1	1	1	1	1
kN	4	4	8	4	4	4	4	4	4	4	4	4	4	4	4	8	4	4	4

6. Holomorphic Eisenstein series on Γ₀(N)⁺

In the classical case, holomorphic Eisenstein series of even weight k ≥ 4 are defined by formula (1–1(1-1) ). Modularity is an immediate consequence of the definition. We proceed analogously in the case of the arithmetic groups Γ₀(N)⁺ and define for even k ≥ 4 modular forms by (1–7(1-7) ). The following proposition relates E^(N)_k(z) to E_k(z).

Proposition 17.

Let E^(N)_k(z) be the modular form defined by (1–7(1-7) ). Then, for all even k = 2m ≥ 4 and all one has

 

(6-28) where E_k is defined by (1–1(1-1) ).

Proof.

Taking e ∈ {v: v∣N} in the definition (2–11(2-11) ) of the arithmetic group Γ₀(N)⁺ we see that where

Now, we use the fact that N is square-free to deduce that for a fixed v₁∣N we have the disjoint union decomposition Therefore, for a fixed v₁∣N we have that Multiplying the above equation by v^{− k/2}₁ and taking the sum over all v₁∣N we get Once we fix v₁∣N and put , it is easy to see that v ranges over all divisors of N, as u₁ runs through divisors of v₁ and u₂ runs through divisors of N/v₁.

Furthermore, for fixed v₁, v∣N there exists a unique pair (u₁, u₂) of positive integers such that u₁∣v₁, u₂∣(N/v₁) and . Reasoning in this way, we see that By the definition of the function σ_m(N) the above equation is equivalent to (6–28(6-28) ), which completes the proof.

7. Searching for generators of function fields

With the above analysis, we are now able to define holomorphic modular functions on the space .

7.1. q-expansions

According to Theorem 16, the form Δ_N has weight k_N, and its q-expansion is easily obtained from inserting the product formula of the classical Δ function: in This allows us to get the q-expansion for any power of Δ_N, including negative powers.

Similarly, the q-expansion of the holomorphic Eisenstein series E^(N)_k is obtained from the q-expansion of the classical Eisenstein series E_k. Using the notation of [Zagier 08], we write where B_k stands for the kth Bernoulli number (e.g., −8/B₄ = 240; −12/B₆ = −504, etc.). From (6–28(6-28) ) we deduce

7.2. Subspaces of rational functions

Now it is evident that for any positive integer M, the function: (1–9(1-9) ) is a holomorphic modular function on , meaning a weight zero modular form with exponential growth in z as z → i∞. Its q-expansion follows from substituting E^(N)_k and Δ_N by their q-expansions.

Let denote the set of all possible rational functions defined in (1–9(1-9) ) for all possible vectors b = (b_ν) and m = (m_ν) with fixed M. Our rationale for finding a set of generators for the function field of the smooth, compact algebraic curve associated to X_N is based on the assumption that a finite span of contains the set of generators for the function field. Subject to this assumption, the set of generators follows from a base change. The base change is performed as follows.

7.3. Our algorithm

Choose a non-negative integer κ. Let M = 1 and set . Let g_N denote the genus of X_N.

(1)	Form the matrix of coefficients from the q-expansions of all elements of , where each element in is expanded along a row with each column containing the coefficient of a power, negative, zero, or positive, of q. The expansion is recorded out to order qκ.
(2)	Apply Gauss elimination to , thus producing a matrix which is in row-reduced echelon form.
(3)	Implement the following decision to determine whether the algorithm has completed: If the gN lowest non-trivial rows at the bottom of correspond to q-expansions whose lead terms have precisely gN gaps in the set {q−1, …, q−2gN}, then the algorithm is completed. If the indicator to stop fails, then replace M by M + 1, by and reiterate the algorithm.

We described the algorithm with the choice of an arbitrary κ. For reasons of efficiency, we initially selected κ to be zero, so that all coefficients for q^ν for ν ≤ κ are included in , but finally increased κ to its desired value according to Remark 6.

The rationale for the stopping decision in Step 3 above is based on two ideas, one factual and one hopeful. First, the Weierstrass gap theorem states that for any point P on a compact Riemann surface there are precisely g_N gaps in the set of possible orders from 1 to 2g_N of functions whose only pole is at P. For the main considerations of this paper, we study g_N ≤ 3. For instance, when g_N = 1 there occurs exactly one gap which for topological reasons is always {q⁻¹}. Second, for any genus, the assumption which is hopeful is that the function field is generated by the set of holomorphic modular functions defined in (1–9(1-9) ). The latter assumption is not obvious, but it has turned out to be true for all groups Γ₀(N)⁺ that we have studied so far. This includes in particular all groups Γ₀(N)⁺ of genus zero, genus one, genus two, and genus three.

7.4. Implementation notes

We have implemented C code that generates in integer arithmetic the set of all possible rational functions defined in (1–9(1-9) ) for any given positive integer K and any square-free level N. The set is passed in symbolic notation to PARI/GP [The PARI Group 11] which computes in rational arithmetic the q-expansions of all elements of up to any given positive order κ and forms the matrix . Theoretically, we could have used PARI/GP for the Gauss elimination of . However, for reasons of efficiency, we have written our own C code, linked against the GMP MP library [Granlund and the GMP development team 12], to compute the Gauss elimination in rational arithmetic. The result is a matrix whose rows correspond to q-expansions with rational coefficients. Inspecting the rational coefficients of , we find that the denominators are 1, i.e., after Gauss elimination the coefficients turn out to be integers. All computations are in rational arithmetic, hence are exact.^{Table 5Table 6}

After the fact, we select κ as in Tables 1 and 2 in case when genus is zero or one, or compute κ as explained in Remark 6 when genus is bigger than one, and extend the q-expansions of the field generators out to order q^κ. Inspecting the coefficients we find that they are integer which proves that the field generators have integer q-expansions.

It is important that we compute the Gauss elimination with a small value of κ first, and extend the q-expansions for the field generators out to the larger number κ according to Remark 6, later. There are two reasons for this: (i) It would be a waste of resources to start with a large value of κ. Once the Gauss elimination is done with a small value of κ, it is easy to extend the q-expansions out to a larger value of κ using reverse engineering of the Gauss elimination. (ii) If the genus is larger than one, we do not know the gap sequence for the field generators in advance. We can determine a₁ and a₂, and hence κ according to Remark 6 only if we have completed the Gauss elimination with some other zero or positive value of κ before.^{Table 7Table 8Table 9Table 10}

8. Genus zero

The genus zero case, due to its simplicity, is suitable for demonstration of the algorithm. We do that at the example of holomorphic modular functions on X_N for the level N = 17. After four iterations, the set consists of 15 functions. With the above notation, the functions are as follows, with their associated q-expansions: After Gauss elimination, we derive a set of functions with the following q-expansions: Every function f_k with k = 1, …, 11 can be written as a polynomial in f₁₂. For example, f₁₁ = f²₁₂ − 14. Further relations are easily derived.

As stated, the j-invariant, also called the Hauptmodul, is the unique holomorphic modular function with simple pole at z = i∞ with residue equal to 1 and constant term equal to zero. For Γ₀(17)⁺, the Hauptmodul is given by j₁₇ = f₁₂(z). It is remarkable that the q-expansion for j₁₇ has integer coefficients, especially when considering how we obtained j₁₇.

In the case when the surface X_N has genus equal to zero, q-expansions of Hauptmoduls j_N coincide with McKay–Thompson series for certain classes of the monster . Therefore, we were able to compare results obtained using the algorithm explained above with well known expansions that may be found, e.g., at [Sloane 10]. All expansions were computed out to order q^κ and they match with the corresponding expansions of McKay–Thompson series available at [Sloane 10].

The successful execution of our algorithm in the genus zero case, besides direct check of the correctness of the algorithm provides two additional corollaries. First, the j-invariant is expressed as a linear combination of elements from the set , for some M, thus showing that the Hauptmoduli can be written as a rational function in holomorphic Eisenstein series and the Kronecker limit function, thus generalizing (1–2(1-2) ). Second, since the function field associated to the underlying algebraic curve is generated by the j-invariant, we conclude that all holomorphic modular forms on X_N are generated by a finite set of holomorphic Eisenstein series and a power of the Kronecker limit function, again generalizing a classical result for . A complete analysis of these two results for all genus zero groups Γ₀(N)⁺ will be given in [Jorgenson et al. preprint-a].

9. Genus one

The smallest N such that Γ₀(N)⁺ has genus one is N = 37. In that example, the Kronecker limit function is , which has weight 12. After four iterations, the algorithm completes successfully. However, consists of 434 functions, and this set contains functions whose pole at i∞ has order as large as 76. For instance, the function F₍₄₈₎(z) = E⁽³⁷⁾₄₈(z)/(Δ₃₇(z))⁴ is in , and the first few terms in the q-expansion of F₍₄₈₎ are where a₋₇₅ and a₋₇₄ are the following rational numbers:

In the setting of genus one, there is no holomorphic modular function with a simple pole at i∞. However, there does exist a holomorphic modular function with an order two pole at i∞, which is equal to the Weierstrass ℘-function. We let j_{1; N} denote the ℘-function for Γ₀(N)⁺. Also, let j_{2; N} denote the holomorphic modular function with a third order pole at i∞. In the coordinates of the complex plane, j_{2; N} is the derivative of j_{1; N}. However, in the coordinate of the upper half plane, either z or q, one will not have that j_{2; N} is the derivative of j_{1; N}, as can be seen by the chain rule from one variable calculus. In all cases, the functions j_{1; N} and j_{2; N} generate the function field of the underlying elliptic curve.

Our algorithm yields the q-expansions of the functions j_{1; N} and j_{2; N}. In the case N = 37, we obtain the expansions: and

As we have stated, the coefficients in the q-expansions of j_{1; 37} and j_{2; 37} are integers. Again, noting the size of the denominators of the coefficients in the q-expansions of elements of , it is striking that the generators of the function fields would have integer coefficients.

In Table 5, we list the generators of the function field of the underlying algebraic curve for all 38 genus one groups Γ₀(N)⁺. In some instances, the computations behind Table 5 were quite intensive. For N = 79, the Kronecker limit function has weight 12, and we needed seven iterations of the algorithm. When considering all functions whose numerator had weight up to 84, the set had 13158 functions, and the orders of the pole at i∞ was as large as 280. As N became larger, the size of the denominators of the rational coefficients in the q-expansions for grew very large, in some cases becoming thousands of digits in length. Nonetheless, we have computed q-expansions of generators of the function field in all 38 genus one cases out to order q^κ where κ is listed in Table 2. The exact arithmetic of the program and successful completion of the algorithm, together with Theorem 5, yields the following result.

Table 5. q-expansion of y = j_{2; N}(z), and q-expansion of x = j_{1; N}(z) for the genus one groups Γ₀(N)⁺.

N	y	x
37	q^−3 + 3q^−1 + 19q + 38q^2 + 93q^3 + 176q^4 + 347q^5 + 630q^6 + 1139q^7 + ⋅⋅⋅	q^−2 + 2q^−1 + 9q + 18q^2 + 29q^3 + 51q^4 + 82q^5 + 131q^6 + 199q^7 + ⋅⋅⋅
43	q^−3 + 2q^−1 + 13q + 24q^2 + 55q^3 + 98q^4 + 186q^5 + 318q^6 + 549q^7 + ⋅⋅⋅	q^−2 + 2q^−1 + 7q + 13q^2 + 20q^3 + 33q^4 + 50q^5 + 77q^6 + 112q^7 + ⋅⋅⋅
53	q^−3 + 3q^−1 + 10q + 16q^2 + 33q^3 + 50q^4 + 90q^5 + 140q^6 + 227q^7 + ⋅⋅⋅	q^−2 + q^−1 + 4q + 7q^2 + 10q^3 + 17q^4 + 23q^5 + 35q^6 + 48q^7 + ⋅⋅⋅
57	q^−3 + q^−1 + 6q + 10q^2 + 27q^3 + 36q^4 + 61q^5 + 106q^6 + 156q^7 + ⋅⋅⋅	q^−2 + 2q^−1 + 5q + 8q^2 + 9q^3 + 15q^4 + 23q^5 + 30q^6 + 43q^7 + ⋅⋅⋅
58	q^−3 + 3q^−1 + 10q + 10q^2 + 30q^3 + 34q^4 + 72q^5 + 90q^6 + 172q^7 + ⋅⋅⋅	q^−2 + q^−1 + 3q + 7q^2 + 7q^3 + 14q^4 + 17q^5 + 29q^6 + 32q^7 + ⋅⋅⋅
61	q^−3 + 2q^−1 + 7q + 10q^2 + 22q^3 + 32q^4 + 53q^5 + 80q^6 + 127q^7 + ⋅⋅⋅	q^−2 + q^−1 + 3q + 6q^2 + 7q^3 + 11q^4 + 16q^5 + 23q^6 + 30q^7 + ⋅⋅⋅
65	q^−3 + 2q^−1 + 7q + 10q^2 + 17q^3 + 24q^4 + 43q^5 + 66q^6 + 102q^7 + ⋅⋅⋅	q^−2 + q^−1 + 3q + 4q^2 + 7q^3 + 11q^4 + 13q^5 + 19q^6 + 23q^7 + ⋅⋅⋅
74	q^−3 + 3q^−1 + 7q + 6q^2 + 17q^3 + 16q^4 + 35q^5 + 38q^6 + 71q^7 + ⋅⋅⋅	q^−2  +   q + 4q^2 + 3q^3 + 7q^4 + 6q^5 + 13q^6 + 13q^7 + 22q^8 + ⋅⋅⋅
77	q^−3 + 3q^−1 + 7q + 6q^2 + 14q^3 + 16q^4 + 28q^5 + 38q^6 + 56q^7 + ⋅⋅⋅	q^−2  +   q + 3q^2 + 3q^3 + 7q^4 + 7q^5 + 10q^6 + 13q^7 + 16q^8 + ⋅⋅⋅
79	q^−3 + q^−1 + 4q + 5q^2 + 11q^3 + 14q^4 + 23q^5 + 32q^6 + 47q^7 + ⋅⋅⋅	q^−2 + q^−1 + 2q + 4q^2 + 4q^3 + 6q^4 + 8q^5 + 11q^6 + 14q^7 + 19q^8 + ⋅⋅⋅
82	q^−3 + 2q^−1 + 5q + 4q^2 + 13q^3 + 12q^4 + 24q^5 + 26q^6 + 49q^7 + ⋅⋅⋅	q^−2 + q^−1 + 2q + 4q^2 + 3q^3 + 6q^4 + 7q^5 + 11q^6 + 11q^7 + 18q^8 + ⋅⋅⋅
83	q^−3 + q^−1 + 4q + 5q^2 + 9q^3 + 12q^4 + 20q^5 + 27q^6 + 40q^7 + ⋅⋅⋅	q^−2 + q^−1 + 2q + 3q^2 + 4q^3 + 6q^4 + 7q^5 + 10q^6 + 12q^7 + 16q^8 + ⋅⋅⋅
86	q^−3 + 2q^−1 + 5q + 4q^2 + 11q^3 + 10q^4 + 22q^5 + 22q^6 + 41q^7 + ⋅⋅⋅	q^−2  +   q + 3q^2 + 2q^3 + 5q^4 + 4q^5 + 9q^6 + 8q^7 + 14q^8 + ⋅⋅⋅
89	q^−3 + q^−1 + 3q + 4q^2 + 9q^3 + 10q^4 + 16q^5 + 22q^6 + 32q^7 + ⋅⋅⋅	q^−2 + q^−1 + 2q + 3q^2 + 3q^3 + 5q^4 + 6q^5 + 8q^6 + 10q^7 + 13q^8 + ⋅⋅⋅
91	q^−3  +   2q + 4q^2 + 6q^3 + 10q^4 + 15q^5 + 18q^6 + 28q^7 + ⋅⋅⋅	q^−2 + 2q^−1 + 3q + 3q^2 + 4q^3 + 5q^4 + 7q^5 + 10q^6 + 11q^7 + ⋅⋅⋅
101	q^−3 + 2q^−1 + 3q + 4q^2 + 7q^3 + 8q^4 + 13q^5 + 16q^6 + 23q^7 + ⋅⋅⋅	q^−2  +   q + 2q^2 + 2q^3 + 3q^4 + 3q^5 + 5q^6 + 6q^7 + 8q^8 + ⋅⋅⋅
102	q^−3 + q^−1 + 3q + 2q^2 + 8q^3 + 6q^4 + 12q^5 + 14q^6 + 22q^7 + ⋅⋅⋅	q^−2 + q^−1 + q + 3q^2 + 2q^3 + 4q^4 + 5q^5 + 6q^6 + 6q^7 + 10q^8 + ⋅⋅⋅
111	q^−3  +   q + 2q^2 + 6q^3 + 5q^4 + 8q^5 + 12q^6 + 14q^7 + 18q^8 + ⋅⋅⋅	q^−2 + 2q^−1 + 3q + 3q^2 + 2q^3 + 3q^4 + 4q^5 + 5q^6 + 7q^7 + 9q^8 + ⋅⋅⋅
114	q^−3 + q^−1 + 2q + 2q^2 + 7q^3 + 4q^4 + 9q^5 + 10q^6 + 16q^7 + 16q^8 + ⋅⋅⋅	q^−2  +   q + 2q^2 + q^3 + 3q^4 + q^5 + 4q^6 + 5q^7 + 6q^8 + ⋅⋅⋅
118	q^−3 + q^−1 + 3q + 2q^2 + 5q^3 + 4q^4 + 9q^5 + 8q^6 + 15q^7 + 14q^8 + ⋅⋅⋅	q^−2 + q^−1 + q + 2q^2 + 2q^3 + 3q^4 + 3q^5 + 5q^6 + 4q^7 + 7q^8 + ⋅⋅⋅
123	q^−3 + q^−1 + 2q + 2q^2 + 5q^3 + 4q^4 + 7q^5 + 10q^6 + 11q^7 + 14q^8 + ⋅⋅⋅	q^−2  +   q + q^2 + q^3 + 3q^4 + 2q^5 + 3q^6 + 4q^7 + 4q^8 + ⋅⋅⋅
130	q^−3 + 2q^−1 + 3q + 2q^2 + 5q^3 + 4q^4 + 7q^5 + 6q^6 + 14q^7 + 10q^8 + ⋅⋅⋅	q^−2 − q^−1 − q + 2q^2 + q^3 + q^4 + q^5 + 3q^6 + q^7 + 5q^8 + 2q^9 + ⋅⋅⋅
131	q^−3 + q^−1 + 2q + 2q^2 + 4q^3 + 4q^4 + 6q^5 + 7q^6 + 10q^7 + 12q^8 + ⋅⋅⋅	q^−2  +   q + q^2 + q^3 + 2q^4 + 2q^5 + 3q^6 + 3q^7 + 4q^8 + ⋅⋅⋅
138	q^−3 + q^−1 + 2q + q^2 + 5q^3 + 3q^4 + 5q^5 + 6q^6 + 10q^7 + 8q^8 + ⋅⋅⋅	q^−2 + q^−1 + q + 2q^2 + q^3 + 2q^4 + 3q^5 + 3q^6 + 2q^7 + 5q^8 + ⋅⋅⋅
141	q^−3 + q^−1 + q + 2q^2 + 4q^3 + 3q^4 + 5q^5 + 7q^6 + 8q^7 + 9q^8 + ⋅⋅⋅	q^−2  +   q + q^2 + q^3 + 2q^4 + q^5 + 2q^6 + 3q^7 + 3q^8 + ⋅⋅⋅
142	q^−3 + q^−1 + 2q + q^2 + 4q^3 + 3q^4 + 6q^5 + 5q^6 + 9q^7 + 8q^8 + ⋅⋅⋅	q^−2 + q^−1 + q + 2q^2 + q^3 + 2q^4 + 2q^5 + 3q^6 + 3q^7 + 4q^8 + ⋅⋅⋅
143	q^−3 + q^−1 + q + 2q^2 + 3q^3 + 3q^4 + 6q^5 + 7q^6 + 8q^7 + 9q^8 + ⋅⋅⋅	q^−2  +   q + q^2 + q^3 + 2q^4 + q^5 + 2q^6 + 2q^7 + 3q^8 + ⋅⋅⋅
145	q^−3  +   q + q^2 + 4q^3 + 2q^4 + 4q^5 + 6q^6 + 7q^7 + 9q^8 + ⋅⋅⋅	q^−2 + q^−1 + q + 2q^2 + q^3 + 2q^4 + 2q^5 + 2q^6 + 3q^7 + 4q^8 + ⋅⋅⋅
155	q^−3 + q^−1 + q + 2q^2 + 3q^3 + 3q^4 + 4q^5 + 4q^6 + 7q^7 + 7q^8 + ⋅⋅⋅	q^−2  +   q + q^2 + q^3 + q^4 + q^5 + 2q^6 + 2q^7 + 3q^8 + 2q^9 + ⋅⋅⋅
159	q^−3  +   q + q^2 + 3q^3 + 2q^4 + 3q^5 + 5q^6 + 5q^7 + 6q^8 + ⋅⋅⋅	q^−2 + q^−1 + q + q^2 + q^3 + 2q^4 + 2q^5 + 2q^6 + 3q^7 + 3q^8 + ⋅⋅⋅
174	q^−3  +   q + q^2 + 3q^3 + q^4 + 3q^5 + 3q^6 + 4q^7 + 4q^8 + ⋅⋅⋅	q^−2 + q^−1 + q^2 + q^3 + 2q^4 + 2q^5 + 2q^6 + 2q^7 + 3q^8 + 2q^9 + ⋅⋅⋅
182	q^−3  +   2q + 2q^3 + 2q^4 + 3q^5 + 2q^6 + 4q^7 + 4q^8 + 7q^9 + ⋅⋅⋅	q^−2  +   q + q^2 + q^4 + q^5 + 2q^6 + q^7 + 2q^8 + q^9 + 3q^10 + ⋅⋅⋅
190	q^−3 + q^−1 + 2q + q^2 + 2q^3 + q^4 + 3q^5 + 3q^6 + 5q^7 + 3q^8 + ⋅⋅⋅	q^−2 + q^−1 + q + q^2 + q^3 + q^4 + q^5 + 2q^6 + q^7 + 3q^8 + 2q^9 + ⋅⋅⋅
195	q^−3 − q^−1 + q + q^2 + 2q^3 + q^5 + 3q^6 + 3q^7 + 3q^8 + 5q^9 + ⋅⋅⋅	q^−2 + q^−1 + q^2 + q^3 + 2q^4 + q^5 + q^6 + 2q^7 + 2q^8 + 2q^9 + ⋅⋅⋅
210	q^−3 + q^−1 + q + 3q^3 + 2q^4 + 2q^5 + 2q^6 + 3q^7 + 2q^8 + 6q^9 + ⋅⋅⋅	q^−2 + q^−1 + q + 2q^2 + q^4 + q^5 + q^6 + q^7 + q^8 + 2q^9 + 2q^10 + ⋅⋅⋅
222	q^−3  +   q + 2q^3 + q^4 + 2q^5 + 2q^6 + 2q^7 + 2q^8 + 5q^9 + ⋅⋅⋅	q^−2  +   q + q^2 + q^4 + q^6 + q^7 + q^8 + q^9 + 2q^10 + q^11 + ⋅⋅⋅
231	q^−3  +   q + 2q^3 + q^4 + q^5 + 2q^6 + 2q^7 + 3q^8 + 4q^9 + ⋅⋅⋅	q^−2  +   q + q^4 + q^5 + q^6 + q^7 + q^8 + q^9 + 2q^10 + q^11 + ⋅⋅⋅
238	q^−3  +   q + q^2 + q^3 + q^4 + 2q^5 + q^6 + 2q^7 + q^8 + 4q^9 + ⋅⋅⋅	q^−2 + q^−1 + q^3 + q^4 + q^5 + 2q^6 + q^7 + 2q^8 + q^9 + 2q^10 + ⋅⋅⋅

For each genus one group Γ₀(N)⁺, the two generators j_{1; N} and j_{2; N} of the function field have integer q-expansions.

If we set x = j_1;37(z) and y = j_2;37(z), then it can be shown that j_{1; 37} and j_{2; 37} satisfy the cubic relation: In Table 6 we present the list of cubic relations satisfied by j_{1; N} and j_{2; N} for all genus one groups Γ₀(N)⁺.

Table 6. Cubic relation satisfied by x = j_1;N(z) and y = j_2;N(z) for the genus one groups Γ₀(N)⁺.

N	Cubic relation
37	y^2 − x^3 + 6xy − 6x^2 + 41y + 49x + 300 = 0
43	y^2 − x^3 + 6xy − 4x^2 + 33y + 43x + 192 = 0
53	y^2 − x^3 + 3xy − 6x^2 + 16y + x + 42 = 0
57	y^2 − x^3 + 6xy − 2x^2 + 25y + 37x + 114 = 0
58	y^2 − x^3 + 3xy − 6x^2 + 13y − 2x + 24 = 0
61	y^2 − x^3 + 3xy − 4x^2 + 12y + 7x + 30 = 0
65	y^2 − x^3 + 3xy − 4x^2 + 12y + x + 18 = 0
74	y^2 − x^3 − 6x^2 + 3y − 11x − 4 = 0
77	y^2 − x^3 − 6x^2 + 3y − 14x − 10 = 0
79	y^2 − x^3 + 3xy − 2x^2 + 8y + 8x + 15 = 0
82	y^2 − x^3 + 3xy − 4x^2 + 9y + 2x + 14 = 0
83	y^2 − x^3 + 3xy − 2x^2 + 8y + 5x + 12 = 0
86	y^2 − x^3 − 4x^2 + 3y − 5x = 0
89	y^2 − x^3 + 3xy − 2x^2 + 8y + 7x + 14 = 0
91	y^2 − x^3 + 6xy + 17y + 23x + 42 = 0
101	y^2 − x^3 − 4x^2 + 3y − 4x + 2 = 0
102	y^2 − x^3 + 3xy − 2x^2 + 5y + 4x + 6 = 0
111	y^2 − x^3 + 6xy + 17y + 25x + 48 = 0
114	y^2 − x^3 − 2x^2 + 3y + x + 2 = 0
118	y^2 − x^3 + 3xy − 2x^2 + 5y + x + 2 = 0
123	y^2 − x^3 − 2x^2 + 3y − 2x + 2 = 0
130	y^2 − x^3 − 3xy − 4x^2 − 6y − 3x + 2 = 0
131	y^2 − x^3 − 2x^2 + 3y − 2x + 1 = 0
138	y^2 − x^3 + 3xy − 2x^2 + 5y + 3x + 4 = 0
141	y^2 − x^3 − 2x^2 + 3y + 3 = 0
142	y^2 − x^3 + 3xy − 2x^2 + 5y + 3x + 6 = 0
143	y^2 − x^3 − 2x^2 + 3y + 5 = 0
145	y^2 − x^3 + 3xy + 4y + 7x + 4 = 0
155	y^2 − x^3 − 2x^2 + 3y + 2 = 0
159	y^2 − x^3 + 3xy + 4y + 4x + 3 = 0
174	y^2 − x^3 + 3xy + y + x = 0
182	y^2 − x^3 + 3y − x + 2 = 0
190	y^2 − x^3 + 3xy − 2x^2 + 5y = 0
195	y^2 − x^3 + 3xy + 2x^2 + x = 0
210	y^2 − x^3 + 3xy − 2x^2 + 5y + 5x + 10 = 0
222	y^2 − x^3 + 3y + x + 2 = 0
231	y^2 − x^3 + 3y − 2x + 2 = 0
238	y^2 − x^3 + 3xy + y − 2x − 2 = 0

As in the genus zero setting, there are two important consequences of the successful implementation of our algorithm. First, the functions j_{1; N} and j_{2; N} can be expressed as linear combinations of elements in for some M. In particular, the Weierstrass ℘-function j_{1; N} can be written in terms of certain holomorphic Eisenstein series and powers of the Kronecker limit function. The same assertion holds for j_{2; N}. Second, since j_{1; N} and j_{2; N} generate the function field, we conclude that all holomorphic modular forms can be written in terms of certain holomorphic Eisenstein series and the Kronecker limit functions. A full documentation of these results will be given in [Jorgenson et al. in preparation-b].

10. Higher genus examples

To conclude the presentation of results, we describe some of the information we obtained for genus two and genus three groups Γ₀(N)⁺. In all instances, we derived q-expansions for the generators of the function fields out to q^κ, where κ is determined according to Remark 6. With the exception of j_1;510 and j_2;510, the q-expansions have integer coefficients. The q-expansions of j_1;510 and j_2;510 have half-integer coefficients. With a further base change, namely v = j_2;510 + j_1;510 and w = j_2;510 − j_1;510, the q-expansions for the generators v and w have integer coefficients for N = 510, also. This together with Theorem 5 yields the following result and finally proves Theorem 1.

For each group Γ₀(N)⁺ of genus up to three, the two generators j_{1; N} and j_{2; N} for N ≠ 510 and j_{2; N} + j_{1; N} and j_{2; N} − j_{1; N} for N = 510 have integer q-expansions. Moreover, the leading coefficients are one and the orders of the poles at i∞ are at most gN + 2.

As it turned out, for all groups Γ₀(N)⁺ of genus two and for all but four groups Γ₀(N)⁺ of genus three, the gap sequence consists of {q⁻¹, …, q^−gN}, and i∞ was not a Weierstrass point. For Γ₀(109)⁺, Γ₀(151)⁺, and Γ₀(179)⁺, the gap sequence consists of {q⁻¹, q⁻², q⁻⁴} and for Γ₀(282)⁺ the gap sequence consists of {q⁻¹, q⁻², q⁻⁵}. For the latter four groups, i∞ is a Weierstras point.

We list in Tables 7^{Table 8Table 9}–10 the q-expansions of the generators as well as the algebraic relation satisfied by the generators.

Table 7. q-expansion of y = j_{2; N}(z), and q-expansion of x = j_{1; N}(z), and κ for the genus two groups Γ₀(N)⁺.

N	y	x	κ
67	q^−4 + 3q^−2 + 3q^−1 + 14q + 30q^2 + 45q^3 + 84q^4 + 129q^5 + 214q^6 + ⋅⋅⋅	q^−3 + q^−2 + 3q^−1 + 9q + 13q^2 + 23q^3 + 33q^4 + 52q^5 + 75q^6 + ⋅⋅⋅	240
73	q^−4 + 3q^−2 + 3q^−1 + 12q + 25q^2 + 36q^3 + 66q^4 + 97q^5 + 156q^6 + ⋅⋅⋅	q^−3 + q^−2 + 3q^−1 + 8q + 11q^2 + 19q^3 + 26q^4 + 40q^5 + 57q^6 + ⋅⋅⋅	240
85	q^−4 + 2q^−2 + q^−1 + 8q + 15q^2 + 20q^3 + 34q^4 + 51q^5 + 85q^6 + ⋅⋅⋅	q^−3 + q^−2 + 3q^−1 + 6q + 8q^2 + 14q^3 + 19q^4 + 26q^5 + 35q^6 + ⋅⋅⋅	508
93	q^−4 + 3q^−2 + 3q^−1 + 9q + 16q^2 + 18q^3 + 36q^4 + 46q^5 + 66q^6 + ⋅⋅⋅	q^−3 + q^−1 + 3q + 3q^2 + 9q^3 + 9q^4 + 13q^5 + 21q^6 + 27q^7 + ⋅⋅⋅	240
103	q^−4 + 3q^−2 + 3q^−1 + 7q + 13q^2 + 16q^3 + 27q^4 + 34q^5 + 51q^6 + ⋅⋅⋅	q^−3 + q^−1 + 3q + 3q^2 + 6q^3 + 7q^4 + 11q^5 + 14q^6 + 20q^7 + ⋅⋅⋅	240
106	q^−4 + 2q^−2 + q^−1 + 4q + 11q^2 + 10q^3 + 24q^4 + 23q^5 + 45q^6 + ⋅⋅⋅	q^−3 + q^−2 + 2q^−1 + 4q + 5q^2 + 9q^3 + 9q^4 + 15q^5 + 17q^6 + ⋅⋅⋅	240
107	q^−4 + q^−2 + q^−1 + 4q + 8q^2 + 10q^3 + 17q^4 + 23q^5 + 35q^6 + 47q^7 + ⋅⋅⋅	q^−3 + q^−2 + 2q^−1 + 4q + 5q^2 + 8q^3 + 10q^4 + 14q^5 + 18q^6 + ⋅⋅⋅	240
115	q^−4 + 3q^−2 + 3q^−1 + 5q + 10q^2 + 14q^3 + 22q^4 + 25q^5 + 36q^6 + ⋅⋅⋅	q^−3 + q^−1 + 3q + 3q^2 + 4q^3 + 5q^4 + 8q^5 + 11q^6 + 15q^7 + ⋅⋅⋅	508
122	q^−4 + 2q^−2 + q^−1 + 3q + 9q^2 + 7q^3 + 17q^4 + 16q^5 + 30q^6 + 30q^7 + ⋅⋅⋅	q^−3 + 2q^−1 + 3q + 2q^2 + 6q^3 + 4q^4 + 9q^5 + 8q^6 + 15q^7 + ⋅⋅⋅	366
129	q^−4 + 2q^−2 + 2q^−1 + 5q + 7q^2 + 7q^3 + 15q^4 + 16q^5 + 22q^6 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 2q + 4q^2 + 6q^3 + 5q^4 + 8q^5 + 11q^6 + ⋅⋅⋅	240
133	q^−4 + 3q^−2 + 3q^−1 + 5q + 8q^2 + 9q^3 + 15q^4 + 19q^5 + 26q^6 + ⋅⋅⋅	q^−3 − q^−2 + q + 3q^3 + 2q^4 + 2q^5 + 3q^6 + 6q^7 + 7q^8 + 11q^9 + ⋅⋅⋅	764
134	q^−4 + q^−2 + q^−1 + 2q + 6q^2 + 5q^3 + 12q^4 + 11q^5 + 20q^6 + 20q^7 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 3q + 3q^2 + 5q^3 + 5q^4 + 8q^5 + 9q^6 + 13q^7 + ⋅⋅⋅	240
146	q^−4 + q^−2 + q^−1 + 2q + 5q^2 + 4q^3 + 10q^4 + 9q^5 + 16q^6 + 15q^7 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 2q + 3q^2 + 5q^3 + 4q^4 + 6q^5 + 7q^6 + 11q^7 + ⋅⋅⋅	240
154	q^−4 + q^−2 + q + 4q^2 + 3q^3 + 10q^4 + 7q^5 + 13q^6 + 13q^7 + 23q^8 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 2q + 3q^2 + 5q^3 + 3q^4 + 5q^5 + 6q^6 + 9q^7 + ⋅⋅⋅	881
158	q^−4 + 2q^−2 + q^−1 + 2q + 6q^2 + 4q^3 + 10q^4 + 8q^5 + 15q^6 + 14q^7 + ⋅⋅⋅	q^−3 + q^−1 + 2q + q^2 + 3q^3 + 2q^4 + 5q^5 + 4q^6 + 7q^7 + 6q^8 + ⋅⋅⋅	240
161	q^−4 + q^−2 + q^−1 + 3q + 4q^2 + 4q^3 + 7q^4 + 7q^5 + 10q^6 + 13q^7 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 3q^3 + 3q^4 + 4q^5 + 5q^6 + 6q^7 + 6q^8 + ⋅⋅⋅	698
165	q^−4 + 2q^−2 + q^−1 + 2q + 5q^2 + 4q^3 + 10q^4 + 8q^5 + 10q^6 + 15q^7 + ⋅⋅⋅	q^−3 + q^−1 + 2q + q^2 + 3q^3 + 2q^4 + 3q^5 + 5q^6 + 6q^7 + 5q^8 + ⋅⋅⋅	908
166	q^−4 + 2q^−2 + q^−1 + 2q + 5q^2 + 4q^3 + 9q^4 + 7q^5 + 14q^6 + 12q^7 + ⋅⋅⋅	q^−3 + q^−1 + 2q + q^2 + 3q^3 + 2q^4 + 4q^5 + 3q^6 + 6q^7 + 5q^8 + ⋅⋅⋅	240
167	q^−4 + q^−2 + q^−1 + 2q + 4q^2 + 4q^3 + 6q^4 + 7q^5 + 10q^6 + 12q^7 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 2q + 2q^2 + 3q^3 + 4q^4 + 5q^5 + 6q^6 + 7q^7 + ⋅⋅⋅	240
170	q^−4 + q^−1 + 2q + 3q^2 + 2q^3 + 6q^4 + 5q^5 + 9q^6 + 8q^7 + 14q^8 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 2q + 2q^2 + 4q^3 + 3q^4 + 4q^5 + 5q^6 + 8q^7 + ⋅⋅⋅	508
177	q^−4 + q^−2 + q^−1 + 2q + 4q^2 + 3q^3 + 5q^4 + 7q^5 + 8q^6 + 10q^7 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 3q^3 + 2q^4 + 3q^5 + 4q^6 + 4q^7 + 5q^8 + ⋅⋅⋅	240
186	q^−4 + q^−2 + q^−1 + q + 4q^2 + 2q^3 + 6q^4 + 6q^5 + 8q^6 + 7q^7 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 3q^3 + q^4 + 3q^5 + 3q^6 + 5q^7 + 4q^8 + ⋅⋅⋅	240
191	q^−4 + q^−2 + q^−1 + 2q + 3q^2 + 3q^3 + 5q^4 + 5q^5 + 7q^6 + 8q^7 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 2q^3 + 2q^4 + 3q^5 + 3q^6 + 4q^7 + 4q^8 + ⋅⋅⋅	240
205	q^−4 + q^−2 + q^−1 + 2q + 2q^2 + 3q^3 + 5q^4 + 4q^5 + 6q^6 + 6q^7 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 2q^3 + 2q^4 + 2q^5 + 2q^6 + 4q^7 + 3q^8 + ⋅⋅⋅	508
206	q^−4 + q^−2 + q^−1 + q + 3q^2 + 2q^3 + 5q^4 + 4q^5 + 7q^6 + 6q^7 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 2q^3 + q^4 + 3q^5 + 2q^6 + 4q^7 + 3q^8 + ⋅⋅⋅	240
209	q^−4 + q + 2q^2 + q^3 + 3q^4 + 3q^5 + 5q^6 + 6q^7 + 8q^8 + 8q^9 + 10q^10 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + q + 2q^2 + 3q^3 + 2q^4 + 3q^5 + 3q^6 + 4q^7 + ⋅⋅⋅	380
213	q^−4 + q^−2 + q^−1 + q + 3q^2 + 2q^3 + 4q^4 + 5q^5 + 5q^6 + 6q^7 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 2q^3 + q^4 + 2q^5 + 3q^6 + 3q^7 + 3q^8 + ⋅⋅⋅	240
215	q^−4 + 2q^−2 + 2q^−1 + q + 3q^2 + 4q^3 + 5q^4 + 5q^5 + 7q^6 + 8q^7 + ⋅⋅⋅	q^−3 − q^−2 + q + q^2 + q^5 + q^6 + 2q^7 + q^8 + 3q^9 + 2q^10 + ⋅⋅⋅	508
221	q^−4 + q^−2 + q^−1 + q + 3q^2 + 2q^3 + 4q^4 + 4q^5 + 5q^6 + 5q^7 + 8q^8 + ⋅⋅⋅	q^−3 − q^−2 + q^−1 + q + 2q^3 + q^5 + q^6 + 2q^7 + 2q^8 + 3q^9 + ⋅⋅⋅	359
230	q^−4 + q^−2 + q^−1 + q + 2q^2 + 2q^3 + 4q^4 + 3q^5 + 6q^6 + 4q^7 + 9q^8 + ⋅⋅⋅	q^−3 + q^−1 + q + q^2 + 2q^3 + q^4 + 2q^5 + q^6 + 3q^7 + 2q^8 + ⋅⋅⋅	508
255	q^−4 + 2q^−2 + q^−1 + 2q + 3q^2 + 2q^3 + 4q^4 + 3q^5 + 4q^6 + 4q^7 + ⋅⋅⋅	q^−3 + q^−2 + 2q^2 + 2q^3 + q^4 + 2q^5 + 2q^6 + 2q^7 + 3q^8 + 4q^9 + ⋅⋅⋅	508
266	q^−4 + q^−2 + q^−1 + q + 2q^2 + q^3 + 3q^4 + 3q^5 + 4q^6 + 3q^7 + 6q^8 + ⋅⋅⋅	q^−3 − q^−2 + 2q^−1 + q + q^3 + 2q^5 + q^6 + 2q^7 + q^8 + 3q^9 + ⋅⋅⋅	764
285	q^−4 + q^−2 + q + 2q^2 + q^3 + 2q^4 + 2q^5 + 3q^6 + 3q^7 + 5q^8 + 3q^9 + ⋅⋅⋅	q^−3 + q^−1 + q + 2q^3 + q^4 + q^5 + q^6 + q^7 + 2q^8 + 3q^9 + 2q^10 + ⋅⋅⋅	508
286	q^−4 + q^−2 + q + 2q^2 + q^3 + 3q^4 + q^5 + 3q^6 + 2q^7 + 5q^8 + 4q^9 + ⋅⋅⋅	q^−3 + q^−1 + q + q^3 + q^4 + 2q^5 + q^6 + 2q^7 + q^8 + 3q^9 + 2q^10 + ⋅⋅⋅	359
287	q^−4 + q^−2 + q^−1 + q + 2q^2 + 2q^3 + 2q^4 + 2q^5 + 3q^6 + 3q^7 + 4q^8 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + q + q^2 + q^3 + 2q^4 + 2q^5 + 2q^6 + 2q^7 + ⋅⋅⋅	577
299	q^−4 − q^−2 + q^−1 + q + q^3 + 2q^4 + 2q^5 + q^6 + q^7 + 2q^8 + 2q^9 + ⋅⋅⋅	q^−3 + q^−2 + q + q^2 + q^3 + q^4 + q^5 + 2q^6 + 2q^7 + 2q^8 + 3q^9 + ⋅⋅⋅	359
330	q^−4 + q^−1 + q^2 + 2q^4 + 2q^5 + 2q^6 + q^7 + 4q^8 + 3q^9 + 3q^10 + ⋅⋅⋅	q^−3 + q^−1 + q^2 + q^3 + q^5 + q^6 + 2q^7 + q^8 + 3q^9 + q^10 + ⋅⋅⋅	908
357	q^−4 + q + q^2 + q^3 + q^4 + q^5 + q^6 + q^7 + 3q^8 + q^9 + 2q^10 + 3q^11 + ⋅⋅⋅	q^−3 + q^−1 + q^3 + q^4 + q^5 + q^6 + q^7 + q^8 + 2q^9 + q^10 + 2q^11 + ⋅⋅⋅	718
390	q^−4 + 2q^−2 + q^−1 + q^2 + q^3 + 3q^4 + q^5 + 2q^6 + 2q^7 + 4q^8 + 2q^9 + ⋅⋅⋅	q^−3 − q^−2 + q + q^3 + q^7 − q^8 + q^9 + q^12 + q^13 + 2q^15 + q^16 + ⋅⋅⋅	838

Table 8. Polynomial relation satisfied by x = j_1;N(z) and y = j_2;N(z) for the genus two groups Γ₀(N)⁺.

N	Polynomial relation
67	y^3 − x^4 + 4y^2x + 5yx^2 − 15x^3 + 47y^2 + 121yx − 34x^2 + 724y + 602x + 3348 = 0
73	y^3 − x^4 + 4y^2x + 5yx^2 − 15x^3 + 43y^2 + 111yx − 39x^2 + 606y + 467x + 2508 = 0
85	y^3 − x^4 + 4y^2x + 8yx^2 − 7x^3 + 28y^2 + 88yx + 31x^2 + 277y + 367x + 864 = 0
93	y^3 − x^4 − 5yx^2 − 9x^3 + 16y^2 − 27yx − 57x^2 + 49y − 189x − 66 = 0
103	y^3 − x^4 − 5yx^2 − 9x^3 + 16y^2 − 21yx − 60x^2 + 65y − 164x + 18 = 0
106	y^3 − x^4 + 4y^2x + 4yx^2 − 7x^3 + 22y^2 + 55yx + 4x^2 + 158y + 160x + 360 = 0
107	y^3 − x^4 + 4y^2x + 7yx^2 − 5x^3 + 19y^2 + 58yx + 20x^2 + 128y + 164x + 272 = 0
115	y^3 − x^4 − 5yx^2 − 9x^3 + 16y^2 − 15yx − 59x^2 + 79y − 105x + 120 = 0
122	y^3 − x^4 + 2yx^2 − 3x^3 + 12y^2 + 3yx + 6x^2 + 48y + 12x + 64 = 0
129	y^3 − x^4 + 4y^2x − 10x^3 + 13y^2 + 21yx − 34x^2 + 48y − 18x + 36 = 0
133	y^3 − x^4 − 4y^2x − 7yx^2 − 3x^3 − 3y^2 − 77yx − 47x^2 − 84y − 243x − 198 = 0
134	y^3 − x^4 + 4y^2x + 3yx^2 − 5x^3 + 15y^2 + 35yx + 2x^2 + 72y + 66x + 108 = 0
146	y^3 − x^4 + 4y^2x + 3yx^2 − 5x^3 + 11y^2 + 27yx + 3x^2 + 40y + 43x + 48 = 0
154	y^3 − x^4 + 4y^2x + 3yx^2 − 2x^3 + 10y^2 + 29yx + 18x^2 + 33y + 49x + 36 = 0
158	y^3 − x^4 − 2yx^2 − 3x^3 + 10y^2 − 3yx − 11x^2 + 31y − 13x + 28 = 0
161	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 − 3yx − x^2 = 0
165	y^3 − x^4 − 2yx^2 − 3x^3 + 10y^2 − 3yx − 8x^2 + 27y − 3x + 18 = 0
166	y^3 − x^4 − 2yx^2 − 3x^3 + 10y^2 − 3yx − 8x^2 + 34y − 8x + 40 = 0
167	y^3 − x^4 + 4y^2x + 3yx^2 − 5x^3 + 11y^2 + 23yx − 6x^2 + 35y + 15x + 25 = 0
170	y^3 − x^4 + 4y^2x + 6yx^2 − 3x^3 + 10y^2 + 26yx + 7x^2 + 25y + 21x = 0
177	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 − x^2 + 5y + 2 = 0
186	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 + 3yx − x^2 + 5y + 3x + 2 = 0
191	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 − 2x^2 + 4y − x + 1 = 0
205	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 + x^2 + y + 4x − 2 = 0
206	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 + 3yx − 2x^2 + 7y + 2x + 4 = 0
209	y^3 − x^4 + 4y^2x + 6yx^2 + 5y^2 + 17yx + 13x^2 + 11y + 22x + 10 = 0
213	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 + 3yx − 2x^2 + 6y + 3x + 3 = 0
215	y^3 − x^4 − 4y^2x − 4yx^2 − 2x^3 − y^2 − 33yx − 32x^2 + y − 52x − 21 = 0
221	y^3 − x^4 − 4y^2x + 3yx^2 − x^3 + 5y^2 − 14yx + 6x^2 + 8y − 8x + 4 = 0
230	y^3 − x^4 + yx^2 − 3x^3 + 4y^2 + 3yx + x^2 + 7y + 7x + 4 = 0
255	y^3 − x^4 + 4y^2x − 4yx^2 − 7x^3 − 2y^2 + 7yx − 20x^2 − 17y + 7x + 18 = 0
266	y^3 − x^4 − 4y^2x + 7yx^2 − x^3 + 5y^2 − 17yx + 11x^2 + 10y − 15x + 6 = 0
285	y^3 − x^4 + yx^2 + 4y^2 − 3yx + 4x^2 + 6y − 6x + 3 = 0
286	y^3 − x^4 + yx^2 + 4y^2 − 3yx + 5y − 3x + 2 = 0
287	y^3 − x^4 + 4y^2x + 3yx^2 − 5x^3 + 7y^2 + 14yx − 9x^2 + 11y − 7x − 6 = 0
299	y^3 − x^4 + 4y^2x + 5yx^2 − x^3 + 5y^2 + 11yx + 3x^2 + 2y − 6x − 12 = 0
330	y^3 − x^4 + 4yx^2 − 3x^3 − 2y^2 + 9yx − 2x^2 + 3y + 3x + 6 = 0
357	y^3 − x^4 + 4yx^2 − 2y^2 − 3yx + x^2 − 3x + 3 = 0
390	y^3 − x^4 − 4y^2x − 4yx^2 + x^3 − 27yx − 16x^2 + 18y + 56 = 0

Table 9. q-Expansion of y = j_{2; N}(z), and q-expansion of x = j_{1; N}(z), and κ for the genus three groups Γ₀(N)⁺. For N = 510 the value of κ is for the generators v = y + x and w = y − x.

N	y	x	κ
97	q^−5 + 2q^−3 + q^−2 + 5q^−1 + 14q + 20q^2 + 38q^3 + 54q^4 + 92q^5 + ⋅⋅⋅	q^−4 + q^−3 + 3q^−2 + 4q^−1 + 11q + 18q^2 + 25q^3 + 39q^4 + ⋅⋅⋅	487
109	q^−5 + 2q^−4 + 3q^−2 + 4q^−1 + 14q + 25q^2 + 35q^3 + 61q^4 + 89q^5 + ⋅⋅⋅	q^−3 + q^−2 + 2q^−1 + 4q + 5q^2 + 8q^3 + 9q^4 + 13q^5 + 17q^6 + ⋅⋅⋅	508
113	q^−5 + 2q^−3 + 2q^−2 + 5q^−1 + 13q + 16q^2 + 28q^3 + 38q^4 + 60q^5 + ⋅⋅⋅	q^−4 + q^−3 + 2q^−2 + 3q^−1 + 7q + 12q^2 + 16q^3 + 24q^4 + 32q^5 + ⋅⋅⋅	487
127	q^−5 + q^−3 + q^−2 + 4q^−1 + 7q + 10q^2 + 17q^3 + 22q^4 + 35q^5 + ⋅⋅⋅	q^−4 + q^−3 + 2q^−2 + 2q^−1 + 6q + 9q^2 + 12q^3 + 18q^4 + 23q^5 + ⋅⋅⋅	487
139	q^−5 + q^−3 + q^−2 + 4q^−1 + 7q + 8q^2 + 14q^3 + 17q^4 + 28q^5 + 35q^6 + ⋅⋅⋅	q^−4 + q^−3 + 2q^−2 + 2q^−1 + 5q + 8q^2 + 10q^3 + 15q^4 + 18q^5 + ⋅⋅⋅	487
149	q^−5 + q^−3 + 2q^−2 + 3q^−1 + 7q + 8q^2 + 12q^3 + 16q^4 + 24q^5 + ⋅⋅⋅	q^−4 + q^−3 + q^−2 + 2q^−1 + 4q + 6q^2 + 8q^3 + 11q^4 + 14q^5 + ⋅⋅⋅	487
151	q^−5 + q^−4 + q^−2 + 3q^−1 + 6q + 8q^2 + 12q^3 + 18q^4 + 25q^5 + 33q^6 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 2q + 3q^2 + 4q^3 + 4q^4 + 6q^5 + 7q^6 + 9q^7 + ⋅⋅⋅	508
178	q^−5 + 2q^−3 + 3q^−1 + 6q + 4q^2 + 11q^3 + 8q^4 + 19q^5 + 16q^6 + ⋅⋅⋅	q^−4 + 2q^−2 + q^−1 + 2q + 5q^2 + 3q^3 + 8q^4 + 6q^5 + 11q^6 + ⋅⋅⋅	487
179	q^−5 + q^−4 + q^−2 + 2q^−1 + 4q + 6q^2 + 8q^3 + 12q^4 + 16q^5 + 20q^6 + ⋅⋅⋅	q^−3 + q^−2 + q^−1 + 2q + 2q^2 + 3q^3 + 3q^4 + 4q^5 + 5q^6 + 6q^7 + ⋅⋅⋅	508
183	q^−5 + q^−3 + 2q^−1 + 3q + 4q^2 + 7q^3 + 7q^4 + 14q^5 + 14q^6 + 19q^7 + ⋅⋅⋅	q^−4 + q^−3 + 2q^−2 + 2q^−1 + 4q + 5q^2 + 6q^3 + 9q^4 + 9q^5 + ⋅⋅⋅	635
185	q^−5 + q^−3 + q^−2 + 2q^−1 + 4q + 4q^2 + 7q^3 + 9q^4 + 14q^5 + 15q^6 + ⋅⋅⋅	q^−4 + q^−2 + 2q + 3q^2 + 3q^3 + 4q^4 + 5q^5 + 8q^6 + 9q^7 + 13q^8 + ⋅⋅⋅	487
187	q^−5 + q^−2 + 3q^−1 + 3q + 5q^2 + 6q^3 + 6q^4 + 10q^5 + 12q^6 + 18q^7 + ⋅⋅⋅	q^−4 + q^−3 + q^−2 + q^−1 + 3q + 3q^2 + 5q^3 + 7q^4 + 8q^5 + ⋅⋅⋅	515
194	q^−5 + 2q^−3 + q^−2 + 3q^−1 + 6q + 4q^2 + 10q^3 + 8q^4 + 16q^5 + 14q^6 + ⋅⋅⋅	q^−4 + q^−3 + q^−2 + 2q^−1 + 3q + 4q^2 + 5q^3 + 7q^4 + 7q^5 + ⋅⋅⋅	487
203	q^−5 + q^−3 + q^−2 + 3q^−1 + 4q + 4q^2 + 7q^3 + 7q^4 + 10q^5 + 13q^6 + ⋅⋅⋅	q^−4 + q^−3 + q^−2 + q^−1 + 2q + 4q^2 + 4q^3 + 6q^4 + 7q^5 + 8q^6 + ⋅⋅⋅	1015
217	q^−5 + 2q^−3 + q^−2 + 3q^−1 + 4q + 5q^2 + 7q^3 + 8q^4 + 11q^5 + 12q^6 + ⋅⋅⋅	q^−4 − q^−3 + q^−2 − q^−1 + q^2 + 2q^4 + 3q^5 + 3q^6 + 3q^7 + 5q^8 + ⋅⋅⋅	905
239	q^−5 + q^−3 + q^−2 + 2q^−1 + 3q + 3q^2 + 5q^3 + 5q^4 + 7q^5 + 8q^6 + ⋅⋅⋅	q^−4 + q^−3 + q^−2 + q^−1 + 2q + 3q^2 + 3q^3 + 4q^4 + 5q^5 + 6q^6 + ⋅⋅⋅	487
246	q^−5 + 2q^−1 + 2q + 2q^2 + 3q^3 + 2q^4 + 6q^5 + 4q^6 + 9q^7 + 8q^8 + ⋅⋅⋅	q^−4 + q^−2 + q + 2q^2 + q^3 + 4q^4 + 2q^5 + 4q^6 + 4q^7 + 7q^8 + ⋅⋅⋅	635
249	q^−5 + q^−3 + q^−2 + q^−1 + 3q + 2q^2 + 4q^3 + 5q^4 + 6q^5 + 7q^6 + ⋅⋅⋅	q^−4 + q^−1 + q + 2q^2 + q^3 + 2q^4 + 3q^5 + 3q^6 + 3q^7 + 6q^8 + ⋅⋅⋅	635
258	q^−5 + 2q^−3 + q^−2 + 2q^−1 + 4q + 2q^2 + 6q^3 + 5q^4 + 8q^5 + 7q^6 + ⋅⋅⋅	q^−4 + q^−3 + q^−2 + q^−1 + q + 3q^2 + 3q^3 + 4q^4 + 4q^5 + 5q^6 + ⋅⋅⋅	635
282	q^−4 + q^−2 + q + 2q^2 + q^3 + 3q^4 + q^5 + 3q^6 + 3q^7 + 5q^8 + 3q^9 + ⋅⋅⋅	q^−3 + q^−1 + q + 2q^3 + q^4 + q^5 + q^6 + 2q^7 + q^8 + 3q^9 + 2q^10 + ⋅⋅⋅	282
290	q^−5 + q^−3 + q^−1 + 2q + q^2 + 3q^3 + 2q^4 + 6q^5 + 4q^6 + 7q^7 + 5q^8 + ⋅⋅⋅	q^−4 + 2q^−2 + q^−1 + q + 3q^2 + q^3 + 4q^4 + 2q^5 + 3q^6 + 3q^7 + ⋅⋅⋅	487
295	q^−5 + q^−1 + q + q^2 + 2q^3 + 2q^4 + 4q^5 + 3q^6 + 4q^7 + 5q^8 + 6q^9 + ⋅⋅⋅	q^−4 + q^−3 + q^−2 + q^−1 + 2q + 2q^2 + 2q^3 + 3q^4 + 3q^5 + 4q^6 + ⋅⋅⋅	487
303	q^−5 + q^−2 + q^−1 + 2q + q^2 + 2q^3 + 3q^4 + 3q^5 + 3q^6 + 6q^7 + 5q^8 + ⋅⋅⋅	q^−4 + q^−3 + q^−1 + q + 2q^2 + 2q^3 + 2q^4 + 3q^5 + 3q^6 + 3q^7 + ⋅⋅⋅	635
310	q^−5 + q^−1 + q + q^2 + 2q^3 + q^4 + 4q^5 + 2q^6 + 5q^7 + 4q^8 + 6q^9 + ⋅⋅⋅	q^−4 + q^−2 + q + 2q^2 + q^3 + 2q^4 + q^5 + 3q^6 + 2q^7 + 4q^8 + ⋅⋅⋅	487
318	q^−5 + q^−3 − q^−2 + q^−1 + 2q + q^2 + 3q^3 + q^4 + 4q^5 + 2q^6 + 5q^7 + ⋅⋅⋅	q^−4 + 2q^−2 + q^−1 + q + 2q^2 + q^3 + 3q^4 + 2q^5 + 3q^6 + 3q^7 + ⋅⋅⋅	635
329	q^−5 + q^−2 + q^−1 + 2q + 2q^2 + q^3 + 2q^4 + 3q^5 + 3q^6 + 4q^7 + 4q^8 + ⋅⋅⋅	q^−4 + q^−3 + q^−1 + q + q^2 + 2q^3 + 2q^4 + 2q^5 + 3q^6 + 3q^7 + ⋅⋅⋅	905
429	q^−5 + q^−2 + q^−1 + 2q + q^2 + q^3 + q^4 + 2q^5 + q^6 + 3q^7 + 2q^8 + ⋅⋅⋅	q^−4 + q^−3 + q^2 + q^3 + q^4 + q^5 + 2q^6 + 2q^7 + 3q^8 + 2q^9 + ⋅⋅⋅	1187
430	q^−5 + q^−3 + q^−1 + q + q^2 + 2q^3 + q^4 + 3q^5 + q^6 + 3q^7 + 2q^8 + ⋅⋅⋅	q^−4 − q^−3 + q^−2 + q^4 + 2q^6 + 2q^8 + 2q^10 + q^11 + 2q^12 + ⋅⋅⋅	487
455	q^−5 − q^−3 + q^−2 + q + q^2 + 2q^5 + q^6 + q^7 + 2q^8 + q^9 + 2q^10 + ⋅⋅⋅	q^−4 − q^−3 + q^−2 + q + q^2 + q^4 + 2q^8 + q^9 + q^10 + q^11 + ⋅⋅⋅	1129
462	q^−5 + q^−3 + q^−1 + q + q^2 + 2q^3 + q^4 + 3q^5 + q^6 + 2q^7 + 2q^8 + ⋅⋅⋅	q^−4 + q^−2 + q + q^2 + q^4 + q^5 + q^6 + q^7 + 2q^8 + q^9 + 3q^10 + ⋅⋅⋅	1680
510			635

Table 10. Polynomial relation satisfied by x = j_1;N(z) and y = j_2;N(z) for the genus three groups Γ₀(N)⁺. For N = 510, we give the polynomial relation satisfied by v = y + x and w = y − x, instead.

N	Polynomial relation
97	y^4 − x^5 + 5y^3x + 12y^2x^2 + 17yx^3 − 27x^4 + 64y^3 + 280y^2x + 545yx^2 − 128x^3 + 1685y^2 + 5965yx + 2558x^2 + 22370y + 31654x + 102984 = 0
109	y^3 − x^5 − yx^3 + 13y^2x − 17x^4 + 49yx^2 + 59y^2 − 32x^3 + 494yx + 743x^2 + 1146y + 4461x + 7254 = 0
113	y^4 − x^5 + 5y^3x + 7y^2x^2 + 3yx^3 − 26x^4 + 50y^3 + 162y^2x + 129yx^2 − 252x^3 + 870y^2 + 1629yx − 878x^2 + 6279y + 1397x + 12108 = 0
127	y^4 − x^5 + 5y^3x + 11y^2x^2 + 9yx^3 − 21x^4 + 39y^3 + 160y^2x + 209yx^2 − 139x^3 + 598y^2 + 1580yx − 63x^2 + 3968y + 2600x + 7344 = 0
139	y^4 − x^5 + 5y^3x + 11y^2x^2 + 9yx^3 − 21x^4 + 34y^3 + 140y^2x + 179yx^2 − 151x^3 + 455y^2 + 1176yx − 321x^2 + 2574y + 782x + 3084 = 0
149	y^4 − x^5 + 5y^3x + 6y^2x^2 − 18x^4 + 30y^3 + 87y^2x + 33yx^2 − 130x^3 + 289y^2 + 364yx − 426x^2 + 988y − 412x + 468 = 0
151	y^3 − x^5 + 2yx^3 + 4y^2x − 4x^4 + 12yx^2 + 19y^2 + 7x^3 + 55yx + 62x^2 + 118y + 176x + 240 = 0
178	y^4 − x^5 + 2y^2x^2 + 5yx^3 − 12x^4 + 6y^3 + 8y^2x + 44yx^2 − 44x^3 + 19y^2 + 117yx − 32x^2 + 106y + 97x + 132 = 0
179	y^3 − x^5 + 2yx^3 + 4y^2x − 4x^4 + 15yx^2 + 13y^2 + 8x^3 + 49yx + 69x^2 + 62y + 142x + 108 = 0
183	y^4 − x^5 + 5y^3x + 11y^2x^2 + 13yx^3 − 10x^4 + 24y^3 + 101y^2x + 171yx^2 + 6x^3 + 228y^2 + 739yx + 376x^2 + 1045y + 1435x + 1650 = 0
185	y^4 − x^5 + y^2x^2 − 4yx^3 − 8x^4 + 12y^3 + y^2x − 11yx^2 − 38x^3 + 52y^2 − 16yx − 84x^2 + 87y − 89x + 28 = 0
187	y^4 − x^5 + 5y^3x + 10y^2x^2 + 6yx^3 − 15x^4 + 20y^3 + 71y^2x + 60yx^2 − 95x^3 + 132y^2 + 203yx − 319x^2 + 234y − 570x − 432 = 0
194	y^4 − x^5 + 5y^3x + 2y^2x^2 − 3yx^3 − 11x^4 + 28y^3 + 44y^2x − 13yx^2 − 50x^3 + 165y^2 + 77yx − 112x^2 + 342y − 42x + 216 = 0
203	y^4 − x^5 + 5y^3x + 6y^2x^2 − yx^3 − 15x^4 + 16y^3 + 47y^2x + 9yx^2 − 90x^3 + 87y^2 + 82yx − 262x^2 + 144y − 359x − 173 = 0
217	y^4 − x^5 − 5y^3x + 2y^2x^2 − 5x^4 − 5y^3 − 38y^2x − yx^2 − 4x^3 − 97y^2 − 125yx − 22x^2 − 324y − 187x − 309 = 0
239	y^4 − x^5 + 5y^3x + 6y^2x^2 − yx^3 − 11x^4 + 15y^3 + 45y^2x + 13yx^2 − 52x^3 + 73y^2 + 86yx − 115x^2 + 114y − 107x − 26 = 0
246	y^4 − x^5 + 5y^2x^2 − 8x^4 + 5y^3 + 19y^2x + 7yx^2 − 22x^3 + 15y^2 + 14yx − 24x^2 − 5y − 9x = 0
249	y^4 − x^5 − 4y^2x^2 + yx^3 − 2x^4 + 8y^3 − 6y^2x − 12yx^2 + x^3 + 22y^2 − 20yx − 7x^2 + 24y − 14x + 9 = 0
258	y^4 − x^5 + 5y^3x + 2y^2x^2 − 8yx^3 − 7x^4 + 11y^3 + 20y^2x − 21yx^2 − 29x^3 + 36y^2 + 7yx − 56x^2 + 46y − 26x + 20 = 0
282	y^3 − x^4 + yx^2 + 4y^2 − 3yx + 4x^2 + 3y − 3x = 0
290	y^4 − x^5 + 6y^2x^2 + 5yx^3 − 6x^4 − 2y^3 + 24y^2x + 36yx^2 + 5x^3 + 15y^2 + 77yx + 72x^2 + 62y + 94x + 32 = 0
295	y^4 − x^5 + 5y^3x + 10y^2x^2 + 10yx^3 − 4x^4 + 12y^3 + 46y^2x + 65yx^2 + 9x^3 + 51y^2 + 136yx + 64x^2 + 90y + 99x + 47 = 0
303	y^4 − x^5 + 5y^3x + 5y^2x^2 + yx^3 − 7x^4 + 11y^3 + 23y^2x + 12yx^2 − 17x^3 + 27y^2 + 34yx − 11x^2 + 28y + 16x + 20 = 0
310	y^4 − x^5 + 5y^2x^2 − 4x^4 + 5y^3 + 17y^2x + 11yx^2 − 2x^3 + 23y^2 + 40yx + 10x^2 + 45y + 23x + 26 = 0
318	y^4 − x^5 + 6y^2x^2 + 9yx^3 − 6x^4 − 6y^3 + 10y^2x + 56yx^2 − 2x^3 + 9y^2 + 123yx + 26x^2 + 100y + 23x − 24 = 0
329	y^4 − x^5 + 5y^3x + 5y^2x^2 + yx^3 − 7x^4 + 11y^3 + 18y^2x − 2yx^2 − 20x^3 + 18y^2 − 17yx − 31x^2 − 18y − 28x − 12 = 0
429	y^4 − x^5 + 5y^3x + 5y^2x^2 − 4yx^3 − 7x^4 + y^3 − 4y^2x − 23yx^2 − 21x^3 − 3y^2 − 21yx − 22x^2 − 6y − 8x = 0
430	y^4 − x^5 − 5y^3x + 6y^2x^2 + 2yx^3 − 4x^4 − 5y^2x + 11yx^2 − 5x^3 − 2y^2 + 8yx − 4x^2 − 8y + 6x = 0
455	y^4 − x^5 − 5y^3x + 14y^2x^2 − 16yx^3 + 7x^4 + 3y^3 − 7y^2x + 13yx^2 + 2x^3 − 6y^2 + 30yx − 47x^2 − 27y + 66x − 27 = 0
462	y^4 − x^5 + y^2x^2 − 4x^4 + 5y^3 + 3y^2x − yx^2 − 8x^3 + 5y^2 + 4yx − 20x^2 − 11y − 19x − 24 = 0
510	w^5 − 5w^4v + 4w^4 + 10w^3v^2 − 6w^3v − 3w^3 − 10w^2v^3 − 4w^2v^2 + 46w^2v − 23w^2 + 5wv^4 + 34wv^3 − 75wv^2 + 97wv − 5w − v^5 + 4v^4 + 45v^3 − 78v^2 + 50v + 30 = 0

11. Concluding remarks

In a few words, we can express a response to the problem posed by T. Gannon which we quoted in Section 1.

The generators of the function fields associated to X_N which one should consider are given by the two invariant holomorphic functions whose poles at i∞ have the smallest possible orders and with initial q-expansions which are normalized by the criteria of row-reduced echelon form.

The above statement when applied to all the groups Γ₀(N)⁺ we considered produced generators with integer q-expansions, except for the level N = 510 where a further base change was necessary. The cases we considered include all groups Γ₀(N)⁺ with square-free N of genus up to three.

In other articles, we will present further results related to the material presented here. Specifically, in [Jorgenson et al. in preprint-a], we study further details regarding genus zero groups Γ₀(N)⁺. In particular, for each level, we list the generators of the ring of holomorphic forms as well as relations for the Hauptmodul and Kronecker limit functions in terms of holomorphic Eisenstein series. Similar information for genus one groups Γ₀(N)⁺ is studied in [Jorgenson et al. in preparation-b], as well as additional aspects of the “arithmetic of the moonshine elliptic curves.”

Finally, in [Jorgenson et al. in preparation-c], we compute values of the Hauptmoduli j_N associated to all genus zero groups Γ₀(N)⁺ at elliptic fixed points and prove that all values are algebraic integers.

Funding

J. J. acknowledges grant support from NSF and PSC-CUNY grants, and H. T. acknowledges support from EPSRC grant EP/H005188/1.

Acknowledgments

The first named author (J.J.) discussed a preliminary version of this paper with Peter Sarnak which included the findings from Section 4 onward. During the discussion, Professor Sarnak suggested that we seek a proof of the integrality of all coefficients in the q-expansion once we have shown that a certain number of the first coefficients are integers. His suggestion led us to formulate and prove the material in Section 3. We thank Professor Sarnak for generously sharing with us his mathematical insight. The classical material in Section 1.4 came from a discussion between the third named author (H.T.) and Abhishek Saha. The approach using arithmetic algebraic geometry stemmed from a discussion between the first named author (J.J.) and an anonymous reader of a previous draft of this article. We thank these individuals for allowing us to include their remarks. Finally, the numerical computations for some levels were quite resource demanding. We thank the Public Enterprise Electric Utility of Bosnia and Herzegovina for generously granting us full access to one of their new 256 GB RAM computers, which enabled us to compute for all groups Γ₀(N)⁺ of genus up to three. We are very grateful for their support of our academic research.