1 Introduction
Let D be a domain in ℂ and f a meromorphic function on D. The Schwarz derivative, or the Schwarzian of f, is defined by
(1.1){f,z}=(f′′f′)′-12(f′′f′)2=f′′′f′-32(f′′f′)2.
It was named after Schwarz by Cayley. However, Schwarz himself pointed out that it was discovered by Lagrange in 1781 and it also
appeared in a paper by Kummer in 1836 [15].
The Schwarz derivative plays an important role in the study of the complex projective line, univalent functions, conformal mappings, Teichmuller spaces
and in the theory of automorphic functions and hypergeometric functions [1, 3, 6, 12, 14, 17]. Most importantly, there is a strong link with the theory of ordinary differential equations as follows.
Let R(z) be a meromorphic function on D and consider the second-order differential equation
y′′+R(z)2y=0
with two linearly independent solutions y1 and y2. Then f=y1y2 is a solution to the Schwarzian differential equation
{f,z}=R(z).
This connection with second-order ordinary differential equations leads to the most important properties
of the Schwarz derivative, see Section 3.
There is also an important link with automorphic functions for a discrete subgroup G of PSL2(ℝ), that is, a Fuchsian group of the first kind acting on the upper half-plane ℍ={τ∈ℂ∣ℑ(τ)>0} by linear fractional transformations
γτ=aτ+bcτ+d,γ=(abcd)∈Γ.
Let f be an automorphic function for G, that is, f is a meromorphic function on ℍ invariant under the action of G (in addition to a meromorphic behavior at the cusps). Using the projective invariance of the Schwarz derivative and the fact that it defines a quadratic differential, we see that {f,τ} is a weight 4 automorphic form for G. Moreover, if G is genus 0 and f is a Hauptmodul, then {f,τ} is an automorphic form for the normalizer of G in PSL2(ℝ) [12].
As an example, let λ be the Klein elliptic modular function for Γ(2). Then
{λ,τ}=π22E4(τ),
where E4 is the weight 4 Eisenstein series.
However, for a given weight 4 automorphic form F for the Fuchsian group G, a meromorphic function f satisfying {f,τ}=F is not necessary an automorphic function. Nevertheless, it defines an interesting class of functions. Indeed, there exists a representation ρ of G into GL2(ℂ) (or PGL2(ℂ)) such that
for all γ∈G and τ∈ℍ,
f(γ⋅τ)=ρ(γ)⋅f(τ),
where the action in both sides is by linear fraction and f is called ρ-equivariant. When ρ=1 is trivial, then f is an automorphic function for G. In addition, the form F is holomorphic on ℍ if and only if the solution f is locally univalent on ℍ, meaning that its derivative is nowhere vanishing.
We focus on the case G=SL2(ℤ) and F being a holomorphic weight 4 modular form. In other words, we look at the equation
{h,τ}=sE4(τ),
where s is a complex parameter. If a solution h is given, then any other solution is simply a linear fraction of h, and their corresponding representations are conjugate by a constant matrix in GL2(ℂ).
Thanks to the Frobenius method for the Fuchsian differential equation, one can write down the shape of the solution h. In particular, h will have a logarithmic singularity at infinity if and only if s=2π2n2 for an integer n. Our approach to study the general solutions consists of analyzing their corresponding representation. Indeed, h will be a modular function if and only if its invariance group Γ=kerρ has a finite index in SL2(ℤ), or equivalently, Imρ is finite. We distinguish between the irreducible representations and the reducible ones. In the former case, we find that there are only finitely many possibilities for the level of Γ defined to be the least common multiple of the cusp widths. As a solution h is necessarily locally univalent, Γ is a torsion-free, normal and finite index subgroup of SL2(ℤ) and h realizes a covering map X(Γ)→ℙ1(ℂ), where X(Γ) is the modular curve attached to Γ. Finally, applying the Riemann–Hurwitz to this covering, we deduce that the genus of Γ is zero which leads to a finite number of possibilities.
On the other hand, if ρ is reducible, it turns out that Γ cannot have finite index in SL2(ℤ) and so does not provide any modular solution. This leads to the main result in this paper stating that there are infinitely many values for the parameter s that lead to a modular function h as a solution to {h,τ}=sE4(τ). This happens precisely when s has the form s=2π2(nm)2, where m and n are positive integers
with 2≤m≤5 and gcd(m,n)=1. Here m becomes the level of kerρ, n is the ramification index at ∞ of the covering h and together, with the degree d of this covering, they satisfy
(6n-m)ν∞=12d,
where ν∞ is the parabolic class number of Γ. It is worth noting that four of our solutions corresponding to d=1 already appeared in [22] under different circumstances.
Furthermore, as a solution h to {h,τ}=sE4 has a nowhere vanishing derivative h′ on ℍ, it is possible to provide a basis of global solutions of the Fuchsian differential equation y′′+s2E4y=0 by means of y1=hh′ and y2=1h′.
Second-order modular differential equations have been the subject of study by several authors [7, 8, 9, 13, 11, 26] among others in the context of hypergeometric equations, Virasoro and vertex algebras, and rational conformal theories. In these papers, the equation takes the form
y′′+αE2y′+βE4y=0,
with specific values of α and β.
Our approach is different in the sense that our Fuchsian differential equations do not contain a term in y′ with a quasi-modular coefficient simply because they arise from a Schwarzian differential equation which is our main interest. The differential equations can rather be looked at as a Schrödinger equation with an automorphic potential in the same spirit that a Lamé equation has an elliptic potential.
The paper is organized as follows. In Section 2 we introduce the notion of equivariant functions attached to a 2-dimensional complex representation of a Fuchsian group of the first kind. We make explicit how they are closely connected to vector-valued automorphic forms with multiplier system given by the above representation. In Section 3, we list the main properties of the Schwarz derivative as well as its automorphic properties. We explain how solutions to Schwarzian equations are equivariant functions. In Section 4, we focus on the modular group, and we show that it does not matter in our context whether we work with representations of SL2(ℤ) or of PSL2(ℤ). We then use a result of G. Mason to classify the representations of the modular group having a finite image. In Section 5, we apply the Frobenius method to our Fuchsian differential equations after a change of variable to find explicit expansions of the solutions of both the Fuchsian differential equations and the Schwarzian differential equation. In Section 6, we show that if there is a solution unramified at infinity, then the kernel of the attached representation is a normal, torsion-free and genus zero congruence subgroup of the modular group. This allows us to explicitly write down solutions in terms of classical modular forms and functions. In Section 7, we determine when a Schwarzian equation admits solutions that are modular functions in the general case while assuming that the representation is irreducible. Finally, in Section 8, we show that the case of reducible representation cannot lead to a modular function as a solution to the Schwarzian differential equation.
2 Automorphic forms and equivariant functions
Let Γ be a Fuchsian group of the first kind and let ρ be a finite-dimensional representation of Γ, that is, a homomorphism of Γ into GLn(ℂ), n∈ℤ>0. An n-dimensional vector-valued automorphic form of weight k∈ℤ and multiplier system ρ is a vector F(τ)=(f1,…,fn)t, where f1,…,fn are meromorphic functions on ℍ satisfying
F(γτ)=(cτ+d)kρ(γ)F(τ),τ∈ℍ,γ=(abcd)∈Γ.
When n=2, we define for the pair (Γ,ρ) the notion of ρ-equivariant functions. It is a meromorphic function h on ℍ satisfying
h(γτ)=ρ(γ)h(τ),τ∈ℍ,γ∈Γ,
where on both sides the action of the matrices is by linear fractional transformation. When -I2∈Γ, for this definition to make sense, we need to have ρ(-I2)∈ℂ*I2. Thus ρ induces a representation ρ¯ of the image of Γ in PSL2(ℤ) into PGL2(ℂ), and the same definition for equivariance will be used interchangeably between homogeneous and inhomogeneous Fuchsian groups.
When ρ induces a character of Γ, i.e., when ρ(Γ)⊆ℂ*I2, then h is a scalar automorphic function for Γ (with a multiplier system). When ρ is the trivial (or the standard representation) of Γ, that is, for all γ∈Γ, ρ(γ)=γ, then h is simply called an equivariant function for Γ. They appeared in [2] and have been extensively studied from the automorphic, equivariant and elliptic points of view with interesting applications in [18, 4, 19, 20, 23]. Examples of such functions are provided as follows: Let f be a weight k automorphic form for Γ. Then
hf(τ)=τ+kf(τ)f′(τ)
is an equivariant function for Γ and equivariant functions arising in this way are called rational [5].
For a given pair (Γ,ρ), there is a close connection between 2-dimensional vector-valued automorphic forms of multiplier system ρ and ρ-equivariant functions for Γ. Indeed, if F=(f1,f2)t is a vector-valued automorphic form of multiplier ρ and arbitrary weight, then one can easily verify that h=f1f2 is ρ-equivariant. A less trivial fact is that every ρ-equivariant function arises in this way [20, Theorem 4.4]. Another nontrivial fact is that for an arbitrary pair (Γ,ρ) (even when Γ is a Fuchsian group of the second kind), vector-valued automorphic forms exist in any dimension [21, Theorem 6.6]. As a consequence, ρ-equivariant functions always exist for arbitrary Γ and an arbitrary 2-dimensional representation ρ of Γ.
3 Automorphic differential equations
The following properties of the Schwarz derivative will be very useful.
The functions involved are all meromorphic functions on the domain D.
Projective invariance:
{af+bcf+d,z}={f,z} for (abcd)∈GL2(ℂ).
Cocycle property: If w is a function of z, then
(3.1){f,z}={f,w}(dwdz)2+{w,z},
which means that {f,z} is a quadratic differential
{f,z}dz2={f,w}dw2+{w,z}dz2.
We have
{f,z}=0⇔f(z)=az+bcz+d for some a,b,c,d∈ℂ.
If w=az+bcz+d with
(abcd)∈GL2(ℂ), then
(3.2){f,z}={f,w}(ad-bc)2(cz+d)4.
For two meromorphic functions f and g on D,
(3.3){f,z}={g,z}⇔f(z)=ag(z)+bcg(z)+d for some (abcd)∈GL2(ℂ).
If w(z) is a function of z with w′(z0)≠0 for some z0∈D, then in a neighborhood of z0, we have
{z,w}={w,z}(dzdw)2.
These properties are a consequence of the link with the second-order linear differential equations.
Throughout this section, Γ is a Fuchsian group of the first kind and f is a weight 4 automorphic form for Γ. All the forms and functions are meromorphic unless otherwise stated. We look at the Schwarzian differential equation
(3.4){h,τ}=f(τ).
and the related Fuchsian differential equation
y′′+f2y=0.
Proposition 3.1.
Let h(τ) be a meromorphic function on H. Then {h,τ}=f(τ) for a weight 4 automorphic form f for Γ if and only if h is ρ-equivariant for some 2-dimensional representation ρ of Γ.
Proof.
Suppose that f is a weight 4 automorphic form f for Γ and {h,τ}=f.
Using the properties of the Schwarz derivative, we have, for γ=(abcd)∈Γ,
(cτ+d)4f(τ)=f(aτ+bcτ+d)
={h(aτ+bcτ+d),aτ+bcτ+d}
=(cτ+d)4{h(aτ+bcτ+d),τ}
using (3.2).
Therefore,
{h,τ}={h(aτ+bcτ+d),τ}.
Hence, according to (3.3), there exists (ABCD)∈GL2(ℂ) such that
h(aτ+bcτ+d)=Ah(τ)+BCh(τ)+D.
This defines a 2-dimensional representation ρ of Γ in PGL2(ℂ) such that h is ρ-equivariant. Conversely, using the same properties (3.2) and (3.3), it is easy to see that if h is ρ-equivariant, then f={h,τ} is a weight 4 automorphic form for Γ.
∎
In the meantime, if h1 and h2 are two solutions to {h,τ}=f, there exists σ∈GL2(ℂ) such that h2(τ)=σh1(τ). Let ρ1 (resp. ρ2) be the 2-dimensional
complex representation of Γ such that h1 is ρ1-equivariant (resp. h2 is ρ2-equivariant). It is easy to show the following:
Proposition 3.2.
The representations ρ1 and ρ2 are conjugate. More precisely,
ρ2=σρ1σ-1.
Theorem 3.3.
Suppose that f is a weight 4 automorphic form for Γ that is holomorphic on H.
If y1 and y2 are two linearly independent holomorphic solutions to y′′+12fy=0, then F=(y1y2) is a weight
-1
vector-valued automorphic form for some multiplier system
ρ
. Furthermore, h=y1y2 is a
ρ
-equivariant function satisfying {h,τ}=f.
If h is a solution to {h,τ}=f, then y1=hh′ and y2=1h′ are two linearly independent holomorphic solutions to y′′+12fy=0.
Proof.
First, since ℍ is simply connected and f is holomorphic, linearly independent global solutions to y′′+12fy=0 always exist.
To prove (1), let y be a holomorphic solution, then for γ=(abcd)∈Γ, a straightforward calculation shows that the function y*(τ)=(cτ+d)y(γτ) is also a holomorphic solution. Therefore, if y1 and y2 are two linearly independent solutions, then y1* and y2* are also two linearly independent solutions. It follows that there exists a 2-dimensional representation ρ of Γ such that (y1*y2*)=ρ(γ)(y1y2). In other words,
F=(y1y2) is a weight -1 vector-valued automorphic form of multiplier ρ (in particular, ρ(-I2)=-I2). Moreover, h=y1y2 is a ρ-equivariant function satisfying {h,τ}=f.
As for (2), let h be a solution to {h,τ}=f; then according to the expression (1.1), if h′(τ0)=0, then τ0 is a double pole of {h,τ}. But since f is holomorphic everywhere, we see that h′(τ) is nowhere vanishing on ℍ (we choose the principal branch of the square root). In other words h is locally univalent on ℍ and the square root h′(τ) is defined everywhere as a meromorphic function. Furthermore, from (1.1) again, we see that if h has a multiple pole at τ0, then {h,τ} has a double pole at τ0 which is not the case since again f is holomorphic everywhere. A simple analysis shows that if τ0 is a pole of order 1 for h, then it is a regular point for its Schwarz derivative. We deduce that all the poles of h, if there are any, are simple, and thus they are double poles for h′(τ) and simple zeros for 1h′(τ). We deduce that y1=hh′ and y2=1h′ are holomorphic on ℍ. Differentiating twice y1 gives
y1′′=-12hh′-12{h,τ}
and so y1 is a solution to y′′+12fy=0 and the same is true for y2.
∎
4 Finite image representations
If Γ is a subgroup of SL2(ℤ), we denote by Γ¯
the corresponding homogeneous subgroup of PSL2(ℤ) and let π denote the natural surjection π:SL2(ℤ)→PSL2(ℤ) as well as π:GL2(ℂ)→PGL2(ℂ).
If ρ is a representation of SL2(ℤ) such that ρ(-I2)∈ℂ*I2, then ρ induces a representation ρ¯ of PSL2(ℤ) in PGL2(ℂ) such that
(4.1)π∘ρ=ρ¯∘π.
The commutator group Γ′ of SL2(ℤ) is an index 12 normal congruence subgroup of SL2(ℤ) of level 6 and Γ¯′, the commutator group of PSL2(ℤ) is a normal level 6 congruence subgroup of PSL2(ℤ) of index 6 [16]. In addition, Γ¯′ has genus 1. Furthermore, the quotient SL2(ℤ)/Γ′ (resp. PSL2(ℤ)/Γ¯′) is a cyclic subgroup of order 12 generated by the class of T=(1101) (resp. a cyclic subgroup of order 6 generated by the class of T(τ)=τ+1). It follows that the group of characters of SL2(ℤ) (which are trivial on Γ′) is cyclic generated by a character assigning to T a primitive 12th root of unity. In the case of PSL2(ℤ), the group of characters is cyclic of order 6 generated by any character assigning a primitive 6th root of unity to T. The commutator group Γ′ can also be explicitly realized as follows. Recall the Dedekind η-function defined by
η(τ)=q124∏n≥1(1-qn),q=exp(2πiτ),τ∈ℍ.
It is a modular form of weight 12 with a multiplier system χ which is a character of the metaplectic group. However, χ2 is a character of the modular group having the commutator group Γ′ as it kernel. In addition, the group of characters of SL2(ℤ) is generated by χ2.
We now focus on representations of SL2(ℤ) or PSL2(ℤ) having finite images or equivalently finite index kernels.
Proposition 4.1.
Let ρ be a representation of SL2(Z) and let ρ¯ be the induced representation of PSL2(Z). Then Imρ is finite if and only if Imρ¯ is finite.
Proof.
Since ρ¯∘π=π∘ρ, it is clear that if Imρ is finite, then so is Imρ¯. We now suppose that Imρ¯ is finite. It follows that Imρ¯∘π is also finite and
Imρ¯∘π=π(Imρ)=(Imρ⋅ℂ*I2)/ℂ*I2=Imρ/(Imρ∩ℂ*I2).
Thus, to prove that Imρ is finite, we only need to prove that Imρ∩ℂ*I2 is finite. Let Γ=ρ-1(ℂ*I2)=Kerπ∘ρ; then ρ acts as a character on Γ with ρ(γ)=χ(γ)I2. In the meantime, detρ is a character of SL2(ℤ)
of order dividing 12, hence χ24=1. It follows that ρ(Γ)=Imρ∩ℂ*I2 is finite if we can establish that Γ is finitely generated. To see this, as Γ=Kerπ∘ρ, we have SL2(ℤ)/Γ≅Imπ∘ρ=Imρ¯∘π which is finite, and so Γ has a finite index in SL2(ℤ) and thus finitely generated by Schreier’s lemma.
∎
We now start with a representation ρ¯ of PSL2(ℤ) such that Imρ¯ is finite. We also assume that ρ¯ has a lift ρ to SL2(ℤ) such that (4.1) holds and hence ρ has also a finite image according to the above proposition. As all our representations arise from equivariant functions, this will be always the case. Indeed, if h is ρ¯-equivariant, then according to [21, Theorem 6.6], then h=f1f2, where F=[f1,f2]t is a vector-valued modular form of multiplier ρ and weight -1. In particular, ρ(-I2)=-I2.
Assume first that ρ is irreducible; then according to [10], up to a twist by a character, ρ is unitary and the order of ρ(T) is not twice an odd number. Such a finite image representation is referred to as a basic representation of SL2(ℤ) in [10]. Since multiplying ρ by a character does not affect ρ¯, we can always assume that ρ is a basic representation.
Theorem 4.2 ([10, Theorem 3.5]).
Let ρ be a basic representation of SL2(Z). Set Γ=Kerρ, G=SL2(Z)/Γ, and
let N be the order of ρ(T). There are just four possibilities for Γ, and one of the following holds:
G is the binary dihedral group of order
12,
N=4.
G is the binary tetrahedral group of order
24,
N=3.
G is the binary octahedral group of order
48,
N=8.
G is the binary icosahedral group of order
120,
N=5.
In each case Γ(N)⊆Γ.
Corollary 4.3.
Let ρ be a basic representation of SL2(Z) and Γ=Kerρ.
If N is the order of ρ(T), then Γ(N)⊆±Γ for N∈{3,4,5}.
Proof.
This is clear for the dihedral, tetrahedral and icosahedral cases. The octahedral case is explicit in the proof of the above theorem.
∎
As a consequence, since the order of ρ¯(T) divides the order of ρ(T), we have:
Proposition 4.4.
If ρ is an irreducible representation of SL2(Z) with finite image and N is the order of ρ¯(T), then N∈{2,3,4,5}.
5 The Frobenius method
In this section we focus on the case where f in (3.4) is a holomorphic automorphic form of weight 4 for the modular group SL2(ℤ). As the space of such forms is 1-dimensional, we are looking at the Schwarzian differential equation
(5.1){h,τ}=sE4(τ),
where s is a complex parameter and
E4(τ)=1+240∑n≥1σ3(n)qn,
where σ3(n) is the sum of the cubes of the positive divisors of n .
If s=0, then h is a linear fraction, and we will assume for the rest of this paper that s≠0. The corresponding second degree ODE is given by
y′′+s2E4(τ)y=0.
Write s=2π2r2, r∈ℂ with ℜ(r)≥0, and set q=e2πiτ. The ODE becomes
d2ydq2+1qdydq-r24E4(q)q2y=0
in the punctured disc {0<|q|<1}.
As E4(q)=1+O(q), the ODE is a Fuchsian differential equation with a regular singular point at q=0. We apply the Frobenius method to determine the shape of the solutions.
The indicial equation for the ODE is given by
x2-r24=0,
and we set x1=r2 and x2=-r2 so that ℜ(x1)≥ℜ(x2). We always have a solution given by
y1(q)=qr2∑n=0∞cnqn with c0≠0.
The second solution depends on x1-x2.
If x1-x2=r is not an integer, a linearly independent solution with y1(x) is given by
y2(q)=q-r2∑n=0∞cn*qn with c0*≠0.
In this case a solution to (5.1) is given by
h(τ)=y2(q)y1(q)=q-r∑n=0∞anqn with a0≠0.
If x2-x1 is an integer (which must be positive under our assumption), then a second solution is given by
y2(q)=klog(q)y1(q)+q-r2∑n=0∞Cnqn with C0≠0,k∈ℂ,
yielding a solution to (5.1) of the form
(5.2)h(τ)=klog(q)+q-r∑n=0∞bnqn with b0≠0.
The branch of the log is chosen so that log(q)=2πiτ when τ∈ℍ.
Since the set of solutions to (5.1) is invariant under linear fractional transformations, in particular under inversion and under multiplication by a scalar, we deduce the following:
Proposition 5.1.
We have:
If r∉ℤ, there is a solution to (5.1) of the form
h(τ)=qr∑n=0∞anqn with a0≠0.
If r∈ℤ, then there is a solution to (5.1) of the form
h(τ)=τ+q-r∑n=0∞bnqn with b0≠0.
Proof.
Both cases are straightforward as long as we check that the value 0 of the constant k appearing in (5.2) is not admissible. Suppose that r∈ℤ and that we have a solution to (5.1) of the form
h(τ)=q-r∑n=0∞bnqn.
Then h(τ+1)=h(τ) and so T∈kerρ. It follows that kerρ=SL2(ℤ) as the normal closure of 〈T〉 is SL2(ℤ) itself. Therefore, h is a modular function for SL2(ℤ). If h is holomorphic at i, then differentiating h(-1τ)=h(τ) and then taking τ=i yields h′(i)=0. If h has a pole at i, then it is at least a double pole (this can be seen by changing h to 1h). In either case {h,τ} has a double pole at i which is impossible. See also Proposition 6.4
∎
In these two cases, we keep in mind that any other solution is a linear fraction of the given one. The following is deduced easily from the above considerations.
Proposition 5.2.
Let h be a solution to (5.1) with s=2π2r2, and let ρ be the 2-dimensional representation of SL2(Z) attached to h. Then Tm∈kerρ for some non-zero integer m if and only if r∈Q∖Z.
We end this section by providing the recurrence relations among the coefficients of the solutions y1(q) above. Write
-r24E4=∑n=0∞αnqn,α0=-r24.
It is easy to see that the coefficients ci of qr+i in y1(q) satisfy
[(r+s)2+α0]cs+∑i=0s-1αs-ici=0,c0 being indeterminate.
Similar relations hold for the coefficients of y2(q) in the absence of the logarithmic term.
6 Modular solutions: The degree 1 case
In this section, we focus on the solutions h to (5.1) whose attached representation ρ is such that Γ=kerρ is a finite index subgroup of SL2(ℤ). In other words, h is a modular function for Γ. Since Γ is normal in SL2(ℤ), all its cusps have the same width m which is defined as the smallest positive integer m such that the matrix (1m01)∈Γ in the case of the cusp at infinity. We will refer to the integer m as the level of Γ, even if Γ is not a congruence subgroup (this is Wohlfahrt’s definition of the level as being the least common multiple of all cusp widths [25] which coincides with Klein’s definition of the level in the case of congruence subgroups). Moreover,
q=exp(2πiτm) is the local uniformizer at ∞ for the group Γ. According to Proposition 5.2, we must have s=2π2r2, where r∈ℚ∖ℤ. If we write r in lowest terms, then necessarily
r=nm for a positive integer n. Thus we have:
Proposition 6.1.
If h is a solution to (5.1) and Γ=kerρ is a finite index subgroup of SL2(Z), then
s=2π2(nm)2,
where n∈Z>0 and m∈Z>0 is the level of Γ.
Another way to look at this statement, which will be useful later, is that h, as a modular function,
has a q-expansion meromorphic at ∞ which one can write from Proposition 5.1 as
h(z)=qn+∑i=n+1∞aiqi,q=e2πiτm.
In fact, Proposition 5.1 says more: that ai=0 if i is not of the form m+jn, j≥0. Furthermore,
an easy calculation using (3.1) with w=q yields
{h,z}=2π2(nm)2[1+(-12a3+12a22)q2+O(q3)].
Since E4(z)=1+240qm+O(q2m), we deduce that s=2π2(nm)2.
For the rest of this section, we will assume that n=1 which is equivalent to say that h has a simple zero at ∞.
Theorem 6.2.
If h is a solution to (5.1) having a simple zero at ∞ and Γ=kerρ is a finite index subgroup of SL2(Z), then Γ has genus zero.
Proof.
Recall that a non-constant holomorphic mapping between two compact Riemann surfaces f:S′→S is necessarily surjective and define a covering of finite degree d. If g and g′ are the genera of S and S′, respectively, then we have the Riemann–Hurwitz formula [24]
2g′-2=d(2g-2)+∑z∈S′(ez-1),
where ez is the ramification index of the covering at z. Now,
suppose that Γ=kerρ is a finite index subgroup of SL2(ℤ). As {h,z} is holomorphic on ℍ, it follows that h takes only simple values and simple poles on ℍ. Moreover, h takes either a simple value or has a simple pole at any cusp c of Γ. Indeed, let γ∈SL2(ℤ) such that γ⋅∞=c and set g=h∘γ. The meromorphic behavior of h(τ) at c is given by the meromorphic behavior of g at ∞. Notice that since Γ is normal in SL2(ℤ), g is also invariant under Γ. Furthermore,
we have g(τ)=ρ(γ)h(τ). Therefore, since h takes a simple zero at ∞, g(τ) either takes a simple value or has a simple pole at ∞ as ρ(γ) is an automorphism of the Riemann sphere. If ℍ*=ℍ∪ℚ∪{∞} and X(Γ)=Γ∖ℍ*, then we have a covering of Riemann surfaces h:X(Γ)→ℙ1(ℂ) that is nowhere ramified. If d is the degree of this covering, the Riemann–Hurwitz formula yields 2g′-2=-2d where g′ is the genus of X(Γ). It follows that d=1 and g′=0.
∎
Remark 6.3.
The same argument as in the above proof shows that if Γ=kerρ has finite index in SL2(ℤ), then h has the same ramification index at all cusps, even when this index is not 1.
Proposition 6.4.
If h is a solution to (5.1), then Γ=kerρ is torsion-free, that is, it has no elliptic elements.
Proof.
Suppose that Γ has an elliptic element
γ=(abcd) and τ0∈ℍ is an elliptic fixed point by
γ. For τ∈ℍ, differentiating
h(γ⋅τ)=h(τ) yields h′(γ⋅τ)=(cτ+d)2h′(τ). This gives, for τ=τ0, h′(τ0)=(cτ0+d)2h′(τ0). As h′(τ0)≠0, we must have (cτ0+d)2=1 which implies that τ0 is real, which is impossible.
∎
It turns out that there are only finitely many possibilities for such subgroups.
Proposition 6.5.
If h is a solution to (5.1) having a simple zero at ∞ and Γ=kerρ is a finite index subgroup of SL2(Z) of level m, then 2≤m≤5.
Proof.
If Γ is a finite index subgroup of SL2(ℤ), we have the natural covering of compact Riemann surfaces
X(Γ)→X(1), where X(1)=X(SL2(ℤ)), for which the Riemann–Hurwitz formula reads [16]
(6.1)g=1+μ12-e24-e33-ν∞2,
where g is the genus of X(Γ), μ=[PSL2(ℤ):Γ¯], ek, k∈{2,3}, is the number of inequivalent elliptic fixed points of order k, and ν∞ is the parabolic class number, that is, the number of Γ-inequivalent cusps. In our case, Γ is genus 0 and torsion-free so that the Riemann–Hurwitz becomes
μ=6(ν∞-2).
In the meantime, the degree of the above covering is μ and it is also the sum of the ramification indices of the ν∞ cusps above ∞. These ramification indices are all equal to m as Γ is normal in SL2(ℤ). In other words, we have μ=mν∞. Hence we have mν∞=6(ν∞-2) and so m<6 and (6-m)ν∞=12. We conclude that necessarily 2≤m≤5.
∎
Define, for n∈ℤ, the group
Δ(n)=〈γTnγ-1,γ∈SL2(ℤ)〉.
If m is the (Wohlfahrt) level of a finite index normal subgroup of SL2(ℤ), then Δ(m) is a subgroup of Γ and we have the covering X(Δ(m))→X(Γ). This is true in particular for Γ=Γ(m), the principal congruence subgroup of level m. One could argue that this covering is, in fact, the universal covering of X(Γ(m)) [25]. However,, when X(Γ(m)) has a positive genus, its universal covering is infinite-sheeted and this occurs if and only if m≥6. To be more precise, we have:
Proposition 6.6 ([16]).
We have Δ(m)=Γ(m) if 1≤m≤5 , and for 6≤m, we have [Γ(m):Δ(m)]=∞.
Finally, we have:
Theorem 6.7.
If h is a solution to (5.1) such that kerρ is a finite index subgroup of SL2(Z), then kerρ is one of the four groups Γ(m), 2≤m≤5.
Proof.
Let h be a solution to (5.1) such that Γm=kerρ is a finite index subgroup of SL2(ℤ) of level m. According to Proposition 6.5, we have 2≤m≤5. In the meantime, as Γm is normal, it contains Δ(m)=Γ(m). In other words, for 2≤m≤5, we have Γ(m)⊆Γm. Both groups are genus zero and torsion-free, and so can be generated by parabolic elements only. As the cusps for both groups have the same width which is m, the two groups share the same parabolic elements, and thus must be equal.
∎
Corollary 6.8.
If h is a solution to (5.1) such that kerρ is a finite index subgroup of SL2(Z), then h is a Hauptmodul for kerρ.
Proof.
This can readily be seen from the proof of Theorem 6.2 as the covering h:X(Γ)→ℙ1(ℂ) is an isomorphism, however, for the sake of completeness, we will provide a more direct proof.
Let m be the level of kerρ and let fm be a Hauptmodul for Γ(m), 2≤m≤5. Then {fm,τ} is a weight 4 holomorphic modular form for the normalizer of Γ(m) which is SL2(ℤ). Hence fm is a solution to (5.1) with s=2π2m2 as m is also the width at ∞ in Γ(m). Now, since h and fm have the same Schwarz derivative, h is a linear fraction of fm, and so it is also a Hauptmodul for Γ(m).
∎
The following classical modular forms and functions will be useful for the rest of this paper:
For τ∈ℍ, set q=e2πiτ and t=eπiτ so that t2=q.
The discriminant Δ is defined by
Δ(τ)=η(τ)24=q∏n=1∞(1-qn)24.
We also have the Eisenstein series
E2(τ)=12πiΔ′Δ=1-24∑n=1∞σ1(n)qn,
with σk(n) being the sum of the k-th powers of the positive divisors of n.
The Jacobi (null-)theta functions are given by
θ2(τ)=2∑n=0∞t(n+12)2=2η(2τ)2η(τ),
θ3(τ)=1+2∑n=1∞tn2=η(τ)5η(τ2)2η(2τ)2,
θ4(τ)=1+2∑n=1∞(-1)ntn2=η(τ2)2η(τ).
The elliptic modular functions λ and j are given by
λ=θ24θ34,j=E43Δ.
The modularity of the above functions will be invoked subsequently when needed.
From [22], we have the following expressions for the Hauptmoduln fm which thus provide, up to a linear fractional transformation, the only solutions to (5.1) with a finite index kernels and which are unramified at the cusps:
f2(τ)=λ(τ),f3(τ)=(η(3τ)η(τ3))3,
f4(τ)=η(τ2)η(4τ)2η(τ4)2η(2τ),f5(τ)=q15∏n≥1(1-qn)(n5),
where (⋅⋅) is the Legendre symbol.
These Hauptmoduln are ρ-equivariant for SL2(ℤ) and the representations are determined by how each one transforms under the coset generators of the subgroups. As an example, for Γ(2), the representation is trivial on Γ(2) and on the coset generators, it is determined by
(6.2)λ(τ+1)=λ(τ)λ(τ)-1,λ(-1τ)=1-λ(τ),λ(ττ+1) =1λ(τ),
λ(-1τ+1)=11-λ(τ),λ(1+τ-τ)=1-1λ(τ).
Remark 6.9.
The results of this section show that all the cases of Proposition 4.4 for an irreducible representation ρ occur.
We end this section with an application at the level m=2.
Proposition 6.10.
The differential equation
(6.3)y′′+π24E4y=0
has two linearly independent solutions given by
y1=θ22θ32θ42,y2=θ32θ22θ42.
Proof.
The solution to the Schwarzian differential equation corresponding to (6.3) is
λ=θ24θ34.
Comparing weight 2 and level 2 modular forms yields [16, formula (7.2.12)]
iπθ24=λ′1-λ,iπθ34=λ′λ(1-λ),iπθ44=λ′λ.
Therefore, we get
λ′=iπθ44λ=iπθ44θ24θ34,
and we choose
λ′=(iπ)12θ42θ22θ32,
so that
λλ′=(iπ)-12θ22θ32θ42 and 1λ′=(iπ)-12θ32θ22θ42
yielding the desired expressions for y1 and y2.
∎
It is worth noting that, using the Jacobi identity θ34=θ24+θ34, we get
y1-y2=θ42θ22θ32,
which is also a solution to (6.3).
7 Modular solutions: The ramified case
In this section we continue our investigation of solutions h to (5.1) with the corresponding representation ρ having a finite index kernel Γ in SL2(ℤ) and we allow h to be ramified at the cusps. In this case we have {h,τ}=sE4 with s=2π2(nm)2, gcd(m,n)=1 and n≥1. Here, m is the level of Γ and if we choose h to vanish at ∞, then n is its order of vanishing. Denote by C(Γ) the set of inequivalent cusps modulo Γ so that ν∞=|C(Γ)| is the cusp number of Γ. We consider the covering
h:X(Γ)→ℙ1(ℂ),
and let d be its degree. We have seen that h is only ramified at the cusps with the same ramification index which is n. The Riemann–Hurwitz formula reads
2g-2=-2d+∑c∈C(Γ)(n-1)=-2d+ν∞(n-1),
which we rewrite as
(7.1)2g-2+ν∞=-2d+nν∞.
In the meantime, we have the natural covering induced by the inclusion Γ⊆SL2(ℤ):
X(Γ)→ℙ1(ℂ)
for which the Riemann–Hurwitz formula reads as
(7.2)μ6=2g-2+ν∞,
as Γ is torsion-free and where μ=[PSL2(ℤ),Γ¯] is also the degree of this covering. The second covering is also ramified exactly at the cusps and the ramification indices are all equal to the cusp width m since Γ is normal in SL2(ℤ). In particular, we have μ=mν∞.
Essentially, the integer m plays the same role for the second covering that n plays for the first one.
Equating (7.1) and (7.2) proves the following:
Proposition 7.1.
Let h be a solution to (5.1) such that Γ has a finite index in SL2(Z) with s=2π2(nm)2, gcd(m,n)=1. Let d be the degree of h as a covering map. Then
(7.3)(6n-m)ν∞=12d.
We assume for the remainder of this section that ρ is irreducible. According to Section 4, the level m satisfies
2≤m≤5 so that Γ=Γ(m).
We can state the following theorem.
Theorem 7.2.
Let m be an integer such that 2≤m≤5 and n such that gcd(m,n)=1. The solutions to
{h,τ}=2π2(nm)2E4
are modular functions for Γ(m). Moreover, each solution is a rational function of a Hauptmoduln for Γ(m) of degree d satisfying (6n-m)ν∞=12d.
For each Γ(m), the cusp number is given by (see [16])
ν∞=μm=m22∏p∣m,p prime(1-1p2).
The following table illustrate the relation (7.3) between the degree d and the ramification index n for the values of m of interest.
| m | Relation |
| 2 | 2d+1=3n |
| 3 | d+1=2n |
| 4 | d+2=3n |
| 5 | d+5=6n |
In the following, t will indicates a Hauptmodul for Γ(m), and we provide solutions for few values of the triples (m,n,d).
Proposition 7.3.
Let t=λ be a Hauptmodul for Γ(2). Then
h=t3(t-2)t4-2t3+4t-2
is a solution to {h,τ}=2π2(32)2E4.
Proof.
Since λ(∞)=0, we can deduce the values at the other two cusps 0 and 1 from relations (6.2): λ(0)=1 and λ(1)=∞. From the same relations we obtain
h(τ+1)=-h(τ),h(-1τ)=h(τ)+13h(τ)-1,h(ττ+1) =-h(τ)+13h(τ)+1,
h(-1τ+1)=h(τ)-13h(τ)+1,h(1+τ-τ)=h(τ)+1-3h(τ)+1.
Therefore h is ρ-equivariant with ρ trivial on Γ(2) and given by the above formulas on the coset generators. Furthermore,
h′(τ)=12t2(t-1)2t′(t4-2t3+4t-2)2,
so that h′ does not vanish on ℍ as the values 0 and 1 are attained at the cusps ∞ and 0, respectively, and t′=λ′ does not vanish on ℍ. As t=16q12-128q+o(q), we see that
h=4096q32+o(q32).
Therefore {h,τ} is a weight 4 modular form for SL2(ℤ) which is holomorphic on ℍ and at the cusps and clearly {h,τ}=2π2(32)2E4.
∎
It is worth mentioning that h′λ′ is readily a square, and so it is possible to write down explicit solutions to the ODE y′′+2π2(32)2E4y=0.
The following table provides other solutions for small values of n. They were obtained with the help of Maple Software. The proof of each case is essentially similar to the level 2 case but more tedious. Here again t stands for a Hauptmodul for the relevant Γ(m).
| (m,n,d) | h |
| (2,3,4) | t3(t-2)t4-2t3+4t-2 |
| (2,5,7) | t5(2t2-7t+7)(t-2)(2t6-3t5+t4+2t3+4t2-6t+2) |
| (3,2,3) | t2(3t+1)78t3+26t2+9t+1 |
| (3,4,7) | t4(9t3+21t2+21t+7)(t+1)(27t6+36t5+27t4-6t3-15t2-6t-1) |
| (4,3,7) | t3(t4+7t3+21t2+28t+14)(t4+t3+3t2+4t+2)(t+2)3 |
| (4,5,13) | t5(t8+13t7+78t6+286t5+689t4+1092t3+1092t2+624t+156)(t+2)5(t8+3t7+8t6+6t5-11t4-28t3-28t2-16t-4) |
| (5,2,7) | t2(t5-7)7t5+1 |
| (5,3,13) | t3(t10-39t5-26)26t10-39t5-1 |
8 The reducible case
We now go back to our equation (5.1) and let h be a solution and ρ its 2-dimensional representation. If kerρ has finite index in SL2(ℤ) and ρ is irreducible, then according to Section 4, we are in one of the cases of Section 6 and Section 7. In this section we will focus on the case of ρ being reducible. There exists a constant matrix σ∈GL2(ℂ) such σρσ-1 is (upper) triangular, and according to Proposition 3.2, it corresponds to the equivariant function g=σ⋅f which satisfies the same Schwarzian equation as f. In addition, the kernels of both representations are the same. Therefore, without loss of generality, we will assume that ρ is triangular.
Now we turn our attention to the question whether there exists a solution with the representation ρ being triangular and kerρ having a finite index in SL2(ℤ).
Lemma 8.1.
If h is a solution with ρ reducible and kerρ is of finite index, then kerρ=Γ′, the commutator group of SL2(Z).
Proof.
Suppose that h is a such solution with ρ upper-triangular. Write, for γ∈SL2(ℤ), ρ(γ)=(a(γ)b(γ)0d(γ)). Then a and b are characters of SL2(ℤ) and so they are trivial on the commutator Γ′. In other words, for each γ∈Γ′, ρ(γ) is unipotent and since it has finite order (as Imρ is finite) it must be the identity. Therefore we have Γ′⊆kerρ. Recall that Γ′ is a genus 1, level 6, torsion-free and normal subgroup of SL2(ℤ) of index 12. In the meantime, according to [16, p. 36], there are only seven proper normal subgroups between Γ¯(6) and PSL2(ℤ) as shown in Figure 1.
In particular, there are only two proper normal subgroups containing the commutator, namely Γ¯2 of level 2 and index 2 generated of the squares of elements of PSL2(ℤ) and the subgroup Γ¯3 of level 3 and index 3 generated by the cubes of elements of PSL2(ℤ). However, none of these two groups is torsion-free as can easily be seen from the Riemann–Hurwitz formula (6.1). Therefore, kerρ=Γ′.
∎
This leads us to the following conclusion.
Theorem 8.2.
There is no solution to (5.1) such that ρ is reducible and kerρ having a finite index in SL2(Z).
Proof.
Suppose there is such ρ-equivariant function h. According to the above lemma, kerρ=Γ′ whose image in PSL2(ℤ) has index μ=6, level m=6 and genus g=1. Recall that μ=mν∞ so that Γ′ has only one cusp. Let d be the degree of the covering h:X(Γ′)→ℙ1 which is unramified on ℍ and (possibly) ramified at the lone cusp ∞.
Combining (7.1) and (7.2), which are still valid in our case, yields
n=2d+1,
where n is the ramification index of h at ∞. However, this is impossible since d≥n.
∎
In conclusion, to supplement Theorem 7.2 without the irreducibility condition, we can state:
Theorem 8.3.
The solutions to (5.1) are modular functions if and only if s=2π2(nm)2 with m and n two positive integers satisfying 2≤m≤5 and gcd(m,n)=1 in which case the invariance group of the solutions is given by Γ(m).
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and
Hicham Saber
