Modular Ordinary Differential Equations on ${\rm SL}(2,\mathbb{Z})$ of Third Order and Applications

In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form $y'''+Q_2(z)y'+Q_3(z)y=0$, $z\in\mathbb{H}=\{z\in\mathbb{C} \,|\,\operatorname{Im}z>0 \}$, where $Q_2(z)$ and $Q_3(z)-\frac12 Q_2'(z)$ are meromorphic modular forms on ${\rm SL}(2,\mathbb{Z})$ of weight $4$ and $6$, respectively. We show that any quasimodular form of depth $2$ on ${\rm SL}(2,\mathbb{Z})$ leads to such a MODE. Conversely, we introduce the so-called Bol representation $\hat{\rho}\colon {\rm SL}(2,\mathbb{Z})\to{\rm SL}(3,\mathbb{C})$ for this MODE and give the necessary and sufficient condition for the irreducibility (resp. reducibility) of the representation. We show that the irreducibility yields the quasimodularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain ${\rm SU}(3)$ Toda systems. Note that the ${\rm SU}(N+1)$ Toda systems are the classical Pl\"ucker infinitesimal formulas for holomorphic maps from a Riemann surface to $\mathbb{CP}^N$.


Introduction
Let Ly = 0 be a Fuchsian ordinary differential equation of third order defined on the upper half plane H = {z ∈ C | Im z > 0}: where := d dz . Near a regular point z 0 of Ly = 0, a local solution y(z) can be obtained by giving the initial values y (k) (z 0 ), k = 0, 1, 2, and then y(z) could be globally defined through analytic continuation. However, globally y(z) might be multi-valued. If Ly = 0 is defined on C, then the monodromy representation from π 1 (C \ {singular points}) to SL(3, C) is introduced to characterize the multi-valueness of solutions. If the potentials Q 2 (z) and Q 3 (z) are elliptic functions with periods 1 and τ (Im τ > 0) and any solution of Ly = 0 is single-valued and meromorphic, then the monodromy representation reduces to a homomorphism from π 1 (E τ ) to SL (3, C), where E τ := C/(Z + Zτ ) is the elliptic curve. The well-known examples are the integral Lamé equations and its generalizations; see, e.g., [4,5,6] for some recent developments of this subject. Note that π 1 (E τ ) is abelian. In this paper, we consider the case that Ly = 0 is defined on H and instead of the monodromy representation defined on π 1 (H \ {singular points}), we study a new representation defined on a discrete non-abelian group Γ that is related to the modular property of Γ acting on H. It is called the Bol representation in this paper as in [25] where the Bol representation was first introduced for second order differential equations.
Given a MODE (1.1), it is natural to ask whether there are solutions satisfying some modular property. The main goal of this paper is to study when MODE (1.1) has solutions that lead to modular forms or quasimodular forms. The approach is to calculate the Bol representation, which will be explained below.
First we recall some basic notions from the ODE aspect. Equation (1.1) is called Fuchsian if the order of any pole of Q j (z) is at most j, j = 2, 3. At the cusp ∞, we let q N = e 2πiz/N , where N is the width of ∞ in Γ. Then d dz = 2πi N q N d dq N and so (1.1) becomes From here we see that (1.1) is Fuchsian at ∞ if and only if Q 2 (z) and Q 3 (z) are holomorphic at ∞, and similar conclusions hold for other cusps of Γ. By (1.2), the indicial equation at the cusp ∞ is given by the roots of which are called the local exponents of (1.1) at ∞, denoted by κ ∞ and κ ∞ , satisfying j κ (j) ∞ = 0. In this paper, we always assume that the exponent differences κ (j) are integers for j = 2, 3, Then j κ (j) 3 Z for all j and so we may assume κ (1) ∞ . Similar assumptions are made for other cusps. On the other hand, let z 0 ∈ H be a singular point of (1.1) and write at z 0 , then the indicial equation at z 0 is given by the roots of which are the local exponents of (1.1) at z 0 , denoted by κ (1) z 0 , κ (2) z 0 and κ (3) z 0 , satisfying j κ (j) z 0 = 3. In this paper, we always assume that the exponent differences κ (j) z 0 are integers for j = 2, 3. Then j κ (j) z 0 = 3 implies κ (j) z 0 ∈ 1 3 Z for all j and so we may assume κ (1) z 0 ≤ κ (2) z 0 ≤ κ (3) z 0 . Since the exponent differences are integers, (1.1) might have solutions with logarithmic singularities at z 0 . See Appendix A for all possibilities of the solution structure of (1.1) at z 0 . The singularity z 0 is called apparent if (1.1) has no solutions with logarithmic singularities at z 0 . In this case, the three local exponents must be distinct, i.e., κ (1) z 0 < κ (2) z 0 < κ (3) z 0 ; see, e.g., Appendix A. In this paper, we always assume that L is apparent at any singularity z 0 ∈ H. More precisely, we assume that the MODE (1.1) satisfies (H1) The MODE (1.1) is Fuchsian on H ∪ {cusps}; (H2) At any singular point z 0 ∈ H, κ (1) z 0 < κ (2) z 0 < κ (3) z 0 satisfy κ (1) z 0 ∈ 1 3 Z ≤0 and κ (j) z 0 − κ (1) z 0 ∈ Z for j = 2, 3. Furthermore, z 0 is apparent.
(1. 6) It is easy to see that Q 2 (z) and Q 3 (z) are single-valued, and g 1 , g 2 are also solutions of (1.1).
Our first result reads as follows.
As an example, in Section 6, we will work out the MODE in the case f (z) is an extremal quasimodular form on SL(2, Z), introduced first in [18]; see Theorem 6.2. Now we introduce the notion of the Bol representation of Γ associated to the MODE (1.1), which was first introduced in [25] for second order MODEs. It is well known that any (local) solution y(z) of (1.1) can be extended to a multi-valued function in H through analytic continuation. Fix a point z 0 ∈ H that is not a singular point of (1.1) and let U be a simply-connected neighborhood of z 0 that contains no singularities of (1.1). For γ = a b c d ∈ Γ, choose a path σ from z 0 to γz 0 and consider the analytic continuation of y(z), z ∈ U , along the path. Then y(γz) is well-defined in U . Define then by a direct computation or by using Bol's identity [3], we see that (y| −2 γ)(z) is also a solution of (1.1). Thus, given a fundamental system of solutions Y (z) = (y 1 (z), y 2 (z), y 3 (z)) t , there isγ ∈ SL(3, C) such that where the fact detγ = 1 follows from that the Wronskians of Y and (Y −2 γ) are the same. Obviously, this matrixγ depends on the choice of the path σ. However, under the above assumptions, all local monodromy matrices are εI 3 with ε 3 = 1, so different choices of σ will only possibly changeγ to e ± 2πi 3γ . From here, we see that there is a well-defined homomorphism ρ : Γ → PSL(3, C) such that where y j (γz) are always understood to take analytic continuation along the same path for j = 1, 2, 3. This homomorphism ρ will be called the Bol representation as in [25]. For the convenience of computations, it is better to lift ρ to a homomorphismρ : Γ → GL(3, C) as follows. Suppose that we can find a multi-valued meromorphic function F (z) such that: (i) The analytic continuation ofŷ(z) := F (z)y(z), where y(z) is any solution of (1.1), gives rise to a single-valued holomorphic function on H, and (ii) F (z) 3 is a modular form on Γ of weight 3k with some character, where k ∈ N. Such F (z) can be constructed explicitly when Γ is a triangle group. Then by lettingŶ (z) : This homomorphismρ : Γ → GL(3, C), as a lift of ρ, will also be called the Bol representation since there is no confusion arising. Naturally we consider the following problem: Question. Can we characterize, in terms of local exponents, the MODEs (1.1) whose Bol representations are irreducible?
One purpose of this paper is to answer this question for the case Γ = SL(2, Z). For Γ = SL(2, Z), the above F (z) can be taken to be where are the Eisenstein series of weight 4 and 6, respectively, are the elliptic points of SL(2, Z), {z 1 , . . . , z m } {i, ρ, ∞} denotes the set of singular points of the MODE (1.1) mod SL(2, Z), t j := E 4 (z j ) 3 /E 6 (z j ) 2 and F j (z) := E 4 (z) 3 − t j E 6 (z) 2 . Then F (z) 3 is a modular form of weight 3( + 2), where the integer = k − 2 is given by In other words, besides the assumptions (H1)-(H3), we need to assume further that κ (1) ρ ∈ Z such that ∈ Z. Consequently, we will see from Lemma 3.2 that the Bol representationρ is indeed a group homomorphism from SL(2, Z) to SL(3, C).
is not unique since we can multiply F (z) by a holomorphic modular form to obtain a new one. Different choices of F (z)'s may give different weights k (and so ) but keepingρ(γ) invariant. For example, when the MODE (1.1) comes from a quasimodular form f (z) of depth 2 on Γ as shown in Theorem 1.1, then one choice is to take 3 (1.3). Note that for Γ = SL(2, Z), 3 W f (z) might be different from the F (z) given by (1.7). To obtain that 3 W f (z) equals to the F (z) given by (1.7), we need to assume that f 0 , f 1 , f 2 have no common zeros.
Note from j κ (j) i = 3 that we have either 3κ (1) i , 3κ (2) i , 3κ i , 3κ Note that all irreducible representations of SL(2, Z) of rank up to 5 have been classified by Tuba and Wenzl [31]. 1 One may use their results, the work of Westbury [32], and Lemma 3.10 below to give another proof of Theorem 1.3 different from that given in Section 3. See Remark 3.12.
As an application of Theorem 1.3, we can show that the converse statement of Theorem 1.1 holds. More precisely, we have Theorem 1.4. Let Γ = SL(2, Z) and suppose that the MODE (1.1) satisfies (H1)-(H3) and is a quasimodular form of weight + 2 and depth 2.
In the reducible case, Theorem 1.4 (2) can be applied to construct solutions of the SU(3) Toda system. See Section 4 for the precise statement. The Toda system is an important integrable system in mathematical physics. In algebraic geometry, the SU(N + 1) Toda system is exactly the classical infinitesimal Plücker formula associated with holomorphic maps from Riemann surfaces to CP N ; see, e.g., [7,23,24] and references therein for the recent development of the Toda system. The rest of this paper is organized as follows. In Section 2, we give the proof of Theorem 1.1, namely we will prove that every quasimodular form of depth 2 leads to a MODE (1.1) satisfying the conditions (H1)-(H3). We focus on the case Γ = SL(2, Z) from Section 3. Theorems 1.3-1.4 and Corollary 1.5 will be proved in Section 3. In Section 4, we discuss the reducible case and prove the converse statement of Theorem 1.4 (2). We also give an application to the SU(3) Toda system. In Section 5, we discuss the criterion on the existence of the MODE (1.1) which is Fuchsian and apparent throughout H with prescribed local exponents at singularities and at cusps. In Section 6, as examples of MODEs, we will work out the explicit expressions of Q j (z)'s for an extremal quasimodular form f (z). Finally in Appendix A, we recall the theory of the solution structure of third order ODEs at a regular singular point.

Quasimodular forms of depth and its associated 3rd order MODE
The main purpose of this section is to prove Theorem 1.1. Let Γ be a discrete subgroup of SL(2, R) that is commensurable with SL(2, Z) and χ : Γ → C × be a character of Γ of finite order. A holomorphic function f (z) defined on the upper half plane H is a modular form of weight k with character χ if the following conditions hold: (1) (f k γ)(z) := (cz + d) −k f (γz) = χ(γ)f (z) for any γ ∈ a b c d ∈ Γ; (2) f is holomorphic at any cusp s of Γ.
For example, the Eisenstein series E 4 (z) and E 6 (z) in (1.8) are modular forms of weight 4 and 6 on SL(2, Z), respectively. We let M k (Γ, χ) denote the space of modular forms of weight k with character χ on Γ. For example, for Γ = SL(2, Z), ∞ is the only cusp and we assume that the character is trivial, i.e., χ ≡ 1. Then condition (1) implies f (z + 1) = f (z), which implies that f (z) can be viewed as a function of q = e 2πiz , and condition (2) just means that f is holomorphic at q = 0.
The notion of quasimodular forms was introduced by Kaneko and Zagier [20]. Originally, they are defined as the holomorphic parts of nearly holomorphic modular forms. For our purpose, it suffices to know that a holomorphic function f (z) is a quasimodular form of weight k and depth r with character χ on Γ if and only if f (z) can be expressed as for all γ = a b c d ∈ Γ for some nonzero complex number α and φ(z) is holomorphic at cusps of Γ. This φ(z) is called a quasimodular form of weight 2 and depth 1 on Γ. For example, if Γ is a subgroup of SL(2, Z), we can always let and so α = 6 πi . We let M ≤r k (Γ, χ) denote the space of quasimodular forms of weight k and depth ≤ r with character χ. One basic property is that the quasi-modularity is invariant under the differentiation, namely if f (z) ∈ M ≤r k (Γ, χ), then f (z) ∈ M ≤r+1 k+2 (Γ, χ); see [33,Proposition 20]. We refer the reader to [8,20,33] for the general theory of quasimodular forms. Now we consider the Wronskian W f (z) (see [28]) associated to As in [28], we set and define W f (z) as in (1.4) to be the Wronskian associated to f .
Proof . For Γ = SL(2, Z), this result is proved in [28] and can be also derived from Mason [26,Lemma 3.1], and the approaches in [26,28] can be easily applied to general discrete subgroups Γ.
Here we provide an elementary proof for general Γ for completeness. By (2.2) and ad − bc = 1 we have Together with the fact det A = (ad − bc) 3 = 1, it is easy to see that In particular, for the case γ = 1 N 0 1 , where N is the width of the cusp ∞, the transformation law shows that W f (z) is a polynomial only in f 0 , f 1 , f 2 , φ, and their derivatives. Thus, by the computation above and the fact that the ring of quasimodular forms is invariant under differentiation, W f (z) is a quasimodular form that is actually modular. In other words, W f (z) ∈ M 3k Γ, χ 3 . This completes the proof. Now we define g j (z) := h j (z)/ 3 W f (z). Then (1.5) holds, so g 3 (z) is a solution of where Q 2 (z) := g 1 g 2 − g 1 g 2 g 1 g 2 − g 1 g 2 , Q 3 (z) := g 1 g 2 − g 2 g 1 g 1 g 2 − g 1 g 2 . (2.5) Note that (i) g 1 (z) and g 2 (z) are also solutions of (2.4) and g 1 , g 2 , g 3 are linearly independent.
(ii) Although g j might be multi-valued, g j /g j is single-valued. Thus Q j (z) is single-valued and meromorphic on H for j = 1, 2. We will see from Theorem 2.2 below that any pole of Q j (z) comes from zeros of W f (z).
z 0 to be the local exponents of (2.4) at z 0 .
Theorem 2.2. Under the above notations, Q 2 (z) and Q 3 (z)− 1 2 Q 2 (z) are meromorphic modular forms on Γ (with trivial character) of weight 4 and 6, respectively. Furthermore, all singular points of (2.4) on H comes from the zeros of W f (z), (H1)-(H3) hold, and every cusp of Γ is completely not apparent for (2.4).
Let z 0 be any singular point. It follows from g j (z) = where ord z 0 W f denotes the zero order of W f (z) at z 0 . Since (g 1 , g 2 , g 3 ) is a fundamental system of solutions of (2.4) and g j 's have no logarithmic singularities at z 0 , we conclude from (2.7) and Remark A.8 that (1) the local exponents κ (j) z 0 ∈ 1 3 Z and are all distinct; (2) the exponent differences are all nonzero integers, namely m (j) z 0 − 1 are nonnegative integers for j = 1, 2; (3) z 0 is an apparent singularity of (2.4). Since then the indicial equation of (2.4) at z 0 is This proves (H2). Remark that if z 0 is not a zero of W f (z), i.e., ord z 0 W f = 0, then it follows from (2.7) that κ z 0 = 0, i.e., the local exponents at z 0 are {0, 1, 2}. This already implies A 2 = A 3 = 0. Together with the fact that z 0 is apparent, we easily deduce from the Frobenius method that both Q 2 (z) and Q 3 (z) are holomorphic at z 0 , a contradiction with that z 0 is a singular point. Thus z 0 is a zero of W f (z). This proves that all singular points of (2.4) on H come from the zeros of W f (z).
Let N be the width of the cusp ∞ and q N = e 2πiz/N . Since modular forms f j (z), W f (z) are holomorphic in terms of q N and z = N 2πi ln q N , we see from (2.3) that so ∞ is also a regular singular point of (2.4), i.e., (2.4) is Fuchsian at ∞ and so Q j (z) is holomorphic at ∞ for j = 2, 3. Since (ln q N ) 2 appears in the expression of g 1 (z), we see from Remark A.9 that ∞ is completely not apparent and the local exponents κ Note that the indicial equation at ∞ is . We now consider other cusps. Assume that s is another cusp of Γ different from ∞. Let σ = a b c d ∈ SL(2, Z) be a matrix such that σ∞ = s. Regarding f (z) as a quasimodular form on Γ = ker χ ∩ SL(2, Z), we can α = 6/πi and is a third root of unity. Except for cz + d, every term in the expression has a q M -expansion, where M is the width of the cusp σ and q M = e 2πiz/M . Since s = ∞, we have c = 0. This shows that there is a local solution at the cusp s having a factor z 2 . According to the solution structure discussed in the appendix, the point s must be completely not apparent and we have κ By the same reasoning as in the case of the cusp ∞, the sum κ s is equal to 0 and hence κ Recalling (2.8) and (2.9). we only need to prove Ã 2 , Now we consider Γ = SL(2, Z) and discuss the local exponents of (2.4) at the elliptic points i and ρ, namely to complete the proof of Theorem 1.1. For this purpose, we note that the remark below could be used to simplify some computations.
to be a holomorphic modular form such that it has only one simple zero at z 0 . Then f j (z)/M (z) are holomorphic modular forms and f (z)/M (z) is a quasimodular form.
By applying this remark, we have Proof . Suppose that there are three distinct right cosets Γ ∞ γ in Γ ∞ \Γ such that f (γz 0 ) = 0. Without loss of generality, we assume that one of them is the coset of I, i.e., f (z 0 ) = 0. Now we have Since the MODE associated to f is the same as that associated to f /η m , by considering f /η m if necessary, we can always assume χ(T ) = 1 and so χ(S) = χ(R) = 1, i.e., we can always assume that the character χ is trivial for Γ = SL(2, Z).
Let z 0 ∈ {i, ρ}. By Lemma 2.5, we have f (γz 0 ) = 0 for some γ ∈ SL(2, Z). Then it follows from Proposition 2.3 that Since W f (z) is a modular form of weight 3k on SL(2, Z), it follows from the valence formula for modular forms (see, e.g., [30]) that This implies κ Recall that we may assume that f 0 , f 1 , and f 2 have no common zeros. When k ≡ 0 mod 4, we have and we are done. When k ≡ 2 mod 4, the weights of f 0 and f 2 are congruent to 2 modulo 4 and their expansions in w = (z − i)/(z + i) are of the form while the expansion of f 1 (z) is of the form (See Proposition 5.1 and Remark 5.2 below.) Let h j (z), j = 1, 2, 3, be given by (2.3). Then the local exponent of ah 1 (z) + bh 2 (z) + ch 3 (z) at z = i must be one of for any (a, b, c) ∈ C 3 . Consider the function We compute that z = i(1 + w)/(1 − w) and hence we find that for some power series d 2n+1 w 2n+1 . It follows that, by (2.11), From this, we see that for some e j . This, together with (2.10), implies that either κ is odd. This proves the assertion (3) that 3κ The proof of (4) is similar. We consider the function which implies that either κ and obtain the same conclusion.
For the case k ≡ 0 mod 3, we need to make the computation more precise. Let be the expansions of f j . A computation similar to (2.12) yields and hence On the other hand, by (2.13), we have We then check that the expansion of h 1 − ρh 2 + ρ 2 h 3 + 3f is of the form which again implies that either κ ρ is not congruent to 0 modulo 3 and hence κ The purpose of this section is to prove Theorems 1.3 and 1.4, and Corollary 1.5. Let Q 2 (z) and Q 3 (z) − 1 2 Q 2 (z) be meromorphic modular forms on SL(2, Z) of weight 4 and 6 respectively, i.e., is a MODE. In this section, we use the notations

Proof of Theorem 1.4(1)
First we want to give the proof of Theorem 1.4(1), which is long and will be separated into several lemmas. For this purpose, throughout this section we always assume that . . , z m } {i, ρ} and j = 1, 2, and m (j) Remark 3.1. Note that (S1) and (S2) are equivalent to the assumptions (H1)-(H3) and κ (1) ρ ∈ Z ≤0 in Theorem 1.3, while (S3) is needed to obtain quasimodular forms as stated in Theorem 1.3(1). In view of Theorem 1.1, the above assumptions (S1)-(S3) are necessary for the validity of Theorem 1.4(1). We will prove below that they are also sufficient.
Then this modular form F j (z) has only one simple zero at z j , up to SL(2, Z)-equivalence. Define Then F (z) 3 is a modular form of weight 3( + 2). Clearly for any solution y(z) of (3.1), is single-valued and holomorphic on H. Furthermore, its order at z ∈ {z 1 , . . . , z m } {i, ρ} is one of 0, m ∞ . Fix a fundamental system of solutions Y (z) = (y 1 (z), y 2 (z), y 3 (z)) t of (3.1) and letŶ (z) := F (z)Y (z). Then for any γ ∈ SL(2, Z), there is a matrixρ(γ) ∈ GL(3, C) such that This is a lifting of the Bol representation. We will use freely the notationγ =ρ(γ) just for convenience.
Proof . The proof is similar to that of [25,Lemma 4.2], where the second-order MODE was studied. Let This proves detρ(γ) = 1 because F (z) 3 is a modular form of weight 3( + 2) on SL(2, Z).
Remark that under our assumption κ 3 is a singlevalued and meromorphic function on H for any solution y(z) of (3.1).

Lemma 3.3.
Under the assumptions (S1)-(S3), there is at least one solution y(z) of (3.1) such that y(z) 3 is not a meromorphic modular form of weight −6.
Proof . Suppose the conclusion is not true, namely y(z) 3 is a meromorphic modular form of weight −6 for any solution y(z). Then by the well-known valence formula for modular forms (see, e.g., [30]), we obtain so ord i y 3 is odd. Since ord i y 3 can be chosen as any one of 3κ i , 3κ i , these three numbers are all odd, clearly a contradiction with our assumption (S3). To prove Lemma 3.4, we need the following well-known lemma due to Beukers and Heckman [2]. Let {a 1 , . . . , a n } and {b 1 , . . . , b n } be the eigenvalues of A and B respectively. The following lemma, due to Levelt, is to recover A and B by their eigenvalues. See [2] for a proof.
Lemma 3.6 (cf. [2]). Suppose that rank(A − B) = 1 and a 1 , . . . , a n , b 1 , . . . , b n are all nonzero complex numbers with a i = b j for any i, j. Then up to a common conjugation in GL(n, C), A and B can be uniquely determined by where A j 's and B j 's are given by and so A 2 = I 3 , a contradiction with the assumption A 2 = I 3 .
Thus A and B have common eigenvalues, and it follows from Lemma 3.5 that H acts on C 3 reducibly.
Proof of Lemma 3.4. Suppose that the statement is not true. Then rank T − I 3 ≤ 1 (notê T = I 3 if and only if ∞ is apparent). ThenR =ŜT implies rank Ŝ −R ≤ 1.
ThusŜ is a conjugate of diag(1, −1, −1). IfR = λI 3 for some λ 3 = 1, then by λ = −1 we obtain SoR is conjugate to diag 1, ε, ε 2 . By Corollary 3.7, there is a subspace V C 3 which is invariant under the actionsŜ andR. If dim V = 2, then there is an invertible matrix P such that where A 1 and B 1 are 2 × 2 matrices. This implies rank a contradiction. So dim V = 1, which implies the existence of an invertible matrix P such that where A 1 and B 1 are 2 × 2 matrices. Clearly the same argument as (3.3) also yields a contradiction. This completes the proof.

4)
where β = 0 is a constant andm * 1 (z),m 2 (z) are of the form Proof . Under our assumption, Lemma 3.4 says that ∞ is completely not apparent, so it follows from Remark A.9 that (3.1) has a basis of solutions of the form (y − , y ⊥ , y + ), where y + , y ⊥ , and y − are given by (A. 19) and (A.20).
Recalling (3.4), we setm 1 (z) andm 0 (z) to bê The following result can be seen as the converse statement of Theorem 2.2.
3.2 Proofs of Theorems 1.3 and 1.4, and Corollary 1.5 In this section, we complete the proof of Theorems 1.3 and 1.4, and Corollary 1.5. First we need the following general observation.
The proof of (3.20) is similar and is omitted here.
First suppose 3κ (1) i , 3κ (2) i , 3κ (3) i ≡ {0, 0, 1} mod 2. Then Theorem 3.9 holds, in particular, R = I 3 and the eigenvalues can not be all the same. This together with (3.19) imply that κ Conversely, suppose (3.21) holds. If 3κ i , 3κ ρ mod 3, a contradiction with (3.21). Thus 3κ Remark 3.12. We note the under the assumption that the eigenvalues ofρ(T ) are all 1, Proposition 2.5 and Corollary to Theorem 2.9 of [31] and results of [32] imply thatρ is irreducible if and only if the eigenvalues ofρ(S) andρ(R) are 1, −1, −1 and 1, e 2πi/3 , e −2πi/3 , respectively. Our Theorem 1.3 shows that the irreducibility property ofρ is solely determined by the local exponents at i. This link between the results of [31,32] and Theorem 1.3 is provided by Lemma 3.10. In other words, one may also use results of [31,32] and Lemma 3.10 to give an alternative proof of Theorem 1.3. Our approach has the advantage that it directly shows thatŷ + (z) is a quasimodular form of depth 2. (Note that Westbury's paper [32] does not seem to be easily available. We refer the reader to the introduction section of [22] for a quick review of Westbury's results.)

Reducibility and SU(3) Toda systems on SL(2, Z)
In view of Theorem 3.11 or equivalently Theorem 1.4 (2), it is natural to ask whether the converse statement holds or not. The purpose of this section is to establish such a converse statement and apply it to the SU(3) Toda system. Let Γ be a discrete subgroup of SL(2, R) commensurable with SL(2, Z). In general, there are at least three sources of modular forms and quasimodular forms that will give rise to third-order MODEs on Γ: k (Γ, χ), then f (z)/ 3 W f (z) satisfies a third-order MODE on Γ. This case has been studied in Section 2.
k (Γ, χ 1 ) and g(z) ∈ M k−1 (Γ, χ 2 ) with χ 1 (−I 2 )χ 2 (−I 2 ) = −1, then a similar argument as Theorem 2.2 shows that are solutions of some third-order MODE on Γ. Here To simplify the situation, we impose the condition that the values of χ j (T ) in the second and the third cases are all the same (so that ρ(T ) has only one eigenvalue with multiplicity 3). In the case Γ = SL(2, Z), this condition implies that χ j are all the same, say, χ j = χ for all j, so Case (ii) will not occur. Moreover, in Case (iii), we can divide f , g, h by an eta-power η(z) m satisfying e 2πim/24 = χ(T ). The differential equation corresponding to f (z)/η(z) m is the same as that corresponding to f (z). Thus, in the case Γ = SL(2, Z), we may assume that χ is trivial.
Lemma 4.1. Let f, g, h ∈ M k (SL(2, Z)) be three linearly independent modular forms of weight k on SL(2, Z). Define W f,g,h (z) by (4.1). Then W f,g,h is a modular form of weight 3(k + 2) on SL(2, Z).
Proof . Let F (z) = (f (z), g(z), h(z)) t , which satisfies F (γz) = (cz + d) k F (z) for all γ = a b c d ∈ SL(2, Z). Then the assertion follows from the basic properties of the determinant function.
Proof . The proof is similar as that of Theorem 2.2. The only difference is that ∞ is also apparent because f / 3 W f,g,h , g/ 3 W f,g,h and h/ 3 W f,g,h are linearly independent solutions. Proposition 4.3. Let f, g, h ∈ M k (SL(2, Z)) be linearly independent modular forms and Ly = 0 be the differential equation satisfied by f / 3 W f,g,h , g/ 3 W f,g,h , and h/ 3 W f,g,h . Then the local exponents of Ly = 0 at the elliptic points i and ρ = −1 + √ 3i /2 satisfy ρ ∈ Z for all j, and κ Proof . Note that every solution y(z) of Ly = 0 can be written as for some a, b, c ∈ C. As a, b, and c vary, the order of y(z) at i (respectively, ρ) will go through all possible local exponents of Ly = 0 at i (respectively, ρ). Since hold for any (a, b, c) = (0, 0, 0). From here and for all j, we obtain the assertion (1) and κ ρ ∈ Z for all j and so the assertion (2) holds.
The above result is precisely the converse statement of Theorem 1.4 (2).

Example 4.4.
Recall that the smallest weight k such that dim M k (SL(2, Z)) = 3 is 24. Let f (z) = E 4 (z) 6 , g(z) = E 4 (z) 3 ∆(z), and h(z) = ∆(z) 2 , which form a basis for M 24 (SL(2, Z)). To determine the differential equation and compute that W f,g,h (z) = cE 4 (z) 6 E 6 (z) 3 ∆(z) 3 for some nonzero number c. Noticing that W f,g,h (z) has a zero of order 3 at i and a zero of order 6 at ρ, we know that κ ord ∞ W f,g,h = 0, and ord ∞ h − 1 3 ord ∞ W f,g,h = 1, which implies that the indicial equation at ∞ is (x + 1)x(x − 1). Therefore, according to Lemmas 5.7, 5.8, and 5.9, the meromorphic modular forms Q(z) and R(z) in Ly are (note that this can also be computed directly using (2.5)) and for some complex number s (1) i . Using the apparentness condition at i, we can show that s (1) i = 0. In other words, the differential equation is D 3 q y(z)+Q(z)D q y(z)+ 1 2 D q Q(z)y(z) = 0. We remark that the reason why the differential equation is of this special form is due to the fact that it is the symmetric square of some second order MODE.
It is worth to point out that Theorem 4.2 can be applied to construct solutions of the SU(3) Toda system is the Laplace operator and δ p denotes the Dirac measure at p. We always use the complex variable w = x 1 + ix 2 . Then the Laplace operator ∆ = 4∂ ww .

Polynomial systems derived from the conditions (H1)-(H3)
In view of Theorem 1.3 proved in Section 3, a natural question is whether given a prescribed set of singular points and the local exponents at singularities and at the cusps, there exist MODEs (1.1) satisfying the conditions (H1)-(H3). We will see in this section that this problem of existence is equivalent to that of solving a certain system of polynomial equations. Note that in view of Theorems 1.1 and 4.2 such a MODE (1.1) exists for certain sets of data.

Solution expansions for MODEs
Let Γ be a discrete subgroup of SL(2, R) commensurable with SL(2, Z) and (1.1) be a MODE on Γ. To verify the apparentness of a singular point z 0 of (1.1), we use the classical Frobenius method. However, since (1.1) is modular, it will be more convenient that all functions are expanded in terms of w as introduced in [25] rather than z − z 0 . Fix z 0 ∈ H, we let w = (z − z 0 )/(z − z 0 ) and where φ(z) is the quasimodular form of weight 2 and depth 1 on Γ, i.e., for some nonzero complex number α 0 (note α 0 = 2πiα for α given in (2.1)), and φ * (z) : The advantage of the expansion in terms of w is the following result. Furthermore, if z 0 is an elliptic point with the stabilizer subgroup Γ z 0 of order N , then a n = 0 whenever k + 2n ≡ 0 mod N .

Remark 5.2.
Note that when f (z) is a holomorphic modular form, the coefficients a n in the series can be expressed in terms of the Serre derivatives of f (z). See [25,Proposition A.4] for the precise statement. Also note that when z 0 is an elliptic point, we have A = 0 and hence w = w. This is because φ * transforms like a modular form of weight 2, but any modular form of weight 2 will vanish at every elliptic point.
By Proposition 5.1, we can write ∞ n=−2 a n (−4π(Im z 0 ) w) n , Then (1.1) is equivalent to For later usage, we recall Bol's identity in the following form.
Lemma 5.4. Let γ = a b c d ∈ GL(2, C) and r be a positive integer. Set Then for a (r + 1)-th differentiable function g(z), we have Proof . Bol's identity states that Noticing that we find that the factor (det γ) 1/2 /(cz + d) appearing in the slash operator can be written as which yields the version of Bol's identity stated in the lemma.
To apply Frobenius' method by using the expansion in Proposition 5.1, we need the following result.
Lemma 5.5. Let Q(z) and R(z) be meromorphic modular forms of weight 4 and 6, respectively, on Γ. Assume that Q(x) = n≥n 0 a n x n and R(x) = n≥n 0 b n x n are the power series such that Then is a solution of (5.2) if and only if the series y(x) = ∞ n=0 c n x n+α satisfies Thus, by (5.3), y(z) and y(x) are related by Hence, applying Lemma 5.4 with r = 2, we obtain Also, a direct computation yields, by (5.3), Hence, Putting everything together, we find that Thus, y(z) is a solution of (5.2) if and only if y(x) is a solution of (5.6).
We will see from Lemma 5.10 below that Corollary 5.3 and Lemma 5.5 can be applied to prove that (1.1) or equivalently (5.2) is apparent at elliptic points of order e ≥ 3. This is a great advantage of using expansions in terms of w rather than z − z 0 . Remark 5.6. In practice, the power series Q(x) and R(x) can be computed using Proposition A.4 of [25]. For example, for the Eisenstein series E 4 (z) and E 6 (z) on SL(2, Z), we find that BCu 3 + · · · , where u = −4π(Im z 0 ) w, B = E 4 (z 0 ), and C = E 6 (z 0 ).

Existence of Q(z) and R(z)
In this section we shall discuss the criterion of the existence of meromorphic modular forms Q(z) and R(z) of weight 4 and 6, respectively, on SL(2, Z), such that the differential equation (5.2) is Fuchsian and apparent throughout H with prescribed local exponents at singularities and at cusps. Throughout the section, we let z j , j = 1, . . . , m, be SL(2, Z)-inequivalent point on H, none of which is an elliptic point. We let i = √ −1 and ρ = −1 + √ 3i /2 be the unique elliptic point of order 2 and 3 of SL(2, Z), respectively. For z = z j , i, or ρ, we assume that κ z , are rational numbers in 1 3 Z such that κ i , 3κ Also, when z = ρ, we note that the assumptions on κ (j) ρ above imply that κ (j) ρ ∈ Z for all j. We further assume that Finally, for the cusp ∞ of SL(2, Z), we let κ ∞ be three rational numbers in 1 3 Z such that κ ∞ ∈ Z. We shall consider the problem whether there exist meromorphic modular forms Q(z) and R(z) of weight 4 and 6, respectively, on SL(2, Z), such that (3.1) is Fuchsian and apparent throughout H and the local exponents κ's are given as above.
Lemma 5.7. Let notations i and ρ be as above. Then meromorphic modular forms Q(z) and R(z) of weight 4 and 6, respectively, on SL(2, Z) that have poles of order at most 2 and 3, respectively, at points SL(2, Z)-equivalent to z j , i, or ρ and are holomorphic at other points and cusps are of the form where ∆ 0 (z) = 1728∆(z) = (E 4 (z) 3 − E 6 (z) 2 ) and F j (z) = E 4 (z) Proof . By Corollary 5.3, there are no meromorphic modular forms of weight 4 having a pole at i or ρ with a nonzero residue. Also, the order of a meromorphic modular form of weight 6 on SL(2, Z) at i is necessarily odd, while that at ρ is congruent to 0 modulo 3. Thus, we can take r (2) i , r (2) ρ , r is a holomorphic modular form of weight 4 on SL(2, Z), so it must be a multiple of E 4 (z). The proof for R(z) is similar.
Lemma 5.8. Suppose that Q(z) and R(z) are meromorphic modular forms given by (5.10).
Then the indicial equation of (5.2) at the cusp ∞ is Proof . It is clear that Q(z) = r ∞ +O(q) and R(z) = s ∞ +O(q). Assume that there is a solution of (5.2) of the form y(z) = q α (1 + O(q)), α ∈ R. We compute that We now consider the indicial equation of (5.2) at a point in H. Let z 0 be one of z j , i, or ρ and w be defined by (5.1) with φ * (z) = E * 2 (z) := E 2 (z)+6/(πi(z −z)). Recall that in Lemma 5.5 we have proved that if Q(x) and R(x) are the Laurent series in x such that (5.4) holds. Then is a solution of (5.2) if and only if the series y(x; α) = ∞ n=0 c n (α)x n+α satisfies d 3 where each a n and b n is linear in the parameters r's and s's. Then following the computation in Appendix A, we see that (5.11) is equivalent to for all n ≥ 0, where f (t) := t(t − 1)(t − 2) + a −2 t + b −3 . In particular, R 0 (α) : f (α) = 0 is the indicial equation at z 0 , i.e., Using (5.7), we can work out the coefficients a −2 and b −3 .
Lemma 5.9. Suppose that Q(z) and R(z) are meromorphic modular forms given by (5.10).

respectively.
Proof . For the elliptic point i, we know that E 6 (i) = 0. Hence, using (5.7), we find that Therefore, we have a −2 = 4r (2) i and b −3 = −4r (2) i − 8s (2) i . Similarly, for the elliptic point ρ, using E 4 (ρ) = 0 and (5.7) again, we find that a −2 = −9r (2) ρ and b −3 = 9r (2) ρ + 27s (3) ρ . For the point z j , letting B = E 4 (z j ) and C = E 6 (z j ), we compute that we can only solve for c n (α) up to n = κ Proof . By Corollary 5.3, the coefficients a n in the expansion of Q(x) vanish whenever n ≡ −2 mod e. Likewise, the coefficients b n in R(x) vanish whenever n ≡ −3 mod e. Using these facts, we find that the condition (5.13) reduces to f (α + n)c n (α) + k≡n mod e, k≤n−1 [(α + k)a n−k−2 + b n−k−3 ]c k (α) = 0. (5.15) Then we can easily prove by induction up to n = n − 1 that c n (α) = 0 whenever n ≡ 0 mod e. Now if n ≡ 0 mod e, then (5.14) automatically holds because every summand is 0 due to the facts that a n −k−2 and b n −k−3 are nonzero only when k ≡ n mod e and c k (α) is nonzero only when k ≡ 0 mod e, but k cannot be congruent to n and 0 at the same time. This proves the lemma.
In the following, we let P i (r, s) denote the only nonzero polynomial P i,k 1 ,k 2 (r, s) in the lemma. The discussion above shows that there are 3m + 1 polynomial equations P i (r, s) = 0, P z j ,k 1 ,k 2 (r, s) = 0, j = 1, . . . , m, (k 1 , k 2 ) ∈ {(1, 2), (1, 3), (2, 3)} in 3m+1 variables such that (5.2) is Fuchsian and apparent throughout H and all SL(2, Z)-inequivalent singularities belong to {i, ρ, z 1 , . . . , z m } with the given local exponents if and only if the parameters r and s are common roots of the polynomials. We now consider the degree of these polynomials.
Proposition 5.11. We have 1, 2) for all z ∈ H, this number is 1, as expected.) However, because the polynomials have intersection of positive dimension at infinity in general, we are not able to use the Bézout theorem to obtain this conclusion. (Even the existence of (Q, R) with an arbitrary set of given data is not established yet.) We leave this problem for future study.

Extremal quasimodular forms
We note that by using some results in Section 5, Theorem 2.2 can be improved in some special case. More precisely, the main result of this section is Theorem 6.2 below, which states that Q 2 (z), Q 3 (z) in Theorem 2.2 can be explicitly written down in the case f (z) is an extremal quasimodular form on Γ = SL(2, Z).

Definition 6.1 ([18]).
A quasimodular form f ∈ M ≤r k (SL(2, Z)) is said to be extremal if its vanishing order at ∞ is equal to dim M ≤r k (SL(2, Z)) − 1. We say f is normalized if its leading Fourier coefficient is 1.
Pellarin [28] proved that if r ≤ 4, then a normalized extremal quasimodular form in M ≤r k (SL(2, Z)) exists and is unique.
Theorem 6.2. Let f (z) be an extremal quasimodular form of weight k and depth 2 on SL(2, Z) and D 3 q y(z) + Q(z)D q y(z) + 1 2 D q Q(z) + R(z) y(z) = 0 (6.1) be the differential equation satisfied by f (z)/ 3 W f (z) as derived in Section 2.
(i) If k ≡ 0 mod 4, then (ii) If k ≡ 2 mod 4, then and R(z) = − (k − 2) 3 864 E 6 (z) + 5 54 Indeed, by letting f (z) = ∆(z) k 12 y(z), a direct computation shows that y(z) solves To prove Theorem 6.2, we need the following general lemma, in which the quasimodular form f (z) is not assumed to be extremal.
∞ be the local exponents of (2.4) at ∞.
Proof . Let r = 1 3 ord ∞ W f . Since up to scalars, g 3 (z) = f (z)/ 3 W f (z) is the unique solution of (2.4) without logarithmic singularity near ∞, according to Frobenius' method for complex ordinary differential equations (see, e.g., Appendix A), we must have κ are the other two linearly independent solutions of (2.4), we have Analyzing case by case, we obtain the claimed conclusions.
Proof of Theorem 6.2. First of all, recall that dim M ≤2 k (SL(2, Z)) = 1 + (SL(2, Z)), be an extremal quasimodular form in M ≤2 k (SL(2, Z)). Note that f j (z) cannot have a common zero on H. To see this, say, assume that f j (z) has a common zero at z 0 . Let F (z) be a modular form of weight k with a simple zero at z 0 and nonvanishing elsewhere. Then f (z)/F (z) ∈ M ≤2 k−k (SL(2, Z)) has order k/4 at ∞, which is impossible by (6.2) and the facts that k ≥ 4 and that extremal quasimodular forms of depth 2 exist for any weight and are unique up to scalars. Therefore, f j (z) have no common zeros on H. Now according to Pellarin's argument [28], one has ord ∞ W f = ord ∞ f = k/4 . Hence, we have for some nonzero complex number c. Also, by Lemma 6.4, we must have ord ∞ (f 1 + 2f 2 E 2 ) = 0 and the local exponents at ∞ must be −r/3, −r/3, and 2r/3, where r = k/4 . In other words, the indicial equation of (2.4) at ∞ is Consider first the case k ≡ 0 mod 4. In this case, since ∆(z) has no zeros on H, (2.4) has no singularities on H. Hence, Q(z) is a multiple of E 4 (z), while R(z) is a multiple of E 6 (z). In view of (6.3) and Lemma 5.8, we see that We now consider the case k ≡ 2 mod 4. In this case, W f (z) = c∆(z) (k−2)/4 E 6 (z) has a simple zero at i. Thus, the local exponents of (2.4) at i are −1/3, 2/3, and 8/3 since the differences must be positive integers and the sum must be equal to 3, and the indicial equation at i is We will use this information, together with the apparentness property, to determine Q(z) and R(z).
Remark A.9. Clearly all the above arguments work when we study whether the regular singularity ∞ is apparent or not for y (z) + Q 2 (z)y (z) + Q 3 (z)y(z) = 0, z ∈ H, (A. 17) when the local exponents κ (1) Since Q j (z)'s have Fourier expansions in terms of q N = e 2πiz N (where N is the width of the cusp ∞ on Γ and N = 1 for Γ = SL(2, Z)), this is equivalent to whether the regular singularity q N = 0 is apparent or not for Note that c 0,j = 0 may happen for any j; see Theorem A.3 for example.