Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications

In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form ′′′ + ₂()′ + ₃() = 0, ∈ ℍ = { ∈ ℂ | Im > 0}, where ₂() and ₃() − 1/2′₂() are meromorphic modular forms on SL(2, ℤ) of weight 4 and 6, respectively. We show that any quasimodular for...

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Veröffentlicht in:	Symmetry, Integrability and Geometry: Methods and Applications
Datum:	2022
Hauptverfasser:	Chen, Zhijie, Lin, Chang-Shou, Yang, Yifan
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author	Chen, Zhijie Lin, Chang-Shou Yang, Yifan
author_facet	Chen, Zhijie Lin, Chang-Shou Yang, Yifan
citation_txt	Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 18 (2022), 013, 50 pages
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container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form ′′′ + ₂()′ + ₃() = 0, ∈ ℍ = { ∈ ℂ \| Im > 0}, where ₂() and ₃() − 1/2′₂() are meromorphic modular forms on SL(2, ℤ) of weight 4 and 6, respectively. We show that any quasimodular form of depth 2 on SL(2, ℤ) leads to such a MODE. Conversely, we introduce the so-called Bol representation ^: SL(2, ℤ) → SL(3, 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 SU(3) Toda systems. Note that the SU( + 1) Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to ℂℙᴺ.
first_indexed	2026-03-18T10:46:51Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 013, 50 pages Modular Ordinary Differential Equations on SL(2,Z) of Third Order and Applications Zhijie CHEN a, Chang-Shou LIN b and Yifan YANG c a) Department of Mathematical Sciences, Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China E-mail: zjchen2016@tsinghua.edu.cn b) Center for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan E-mail: cslin@math.ntu.edu.tw c) Department of Mathematics, National Taiwan University and National Center for Theoretical Sciences, Taipei 10617, Taiwan E-mail: yangyifan@ntu.edu.tw Received June 24, 2021, in final form February 13, 2022; Published online February 22, 2022 https://doi.org/10.3842/SIGMA.2022.013 Abstract. In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form y′′′+Q2(z)y ′+Q3(z)y = 0, z ∈ H = {z ∈ C \| Im z > 0}, where Q2(z) and Q3(z)− 1 2Q ′ 2(z) are meromorphic modular forms on SL(2,Z) of weight 4 and 6, respectively. We show that any quasimodular form of depth 2 on SL(2,Z) leads to such a MODE. Conversely, we introduce the so-called Bol representation ρ̂ : SL(2,Z) → SL(3,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 quasimod- ularity of some solution of this MODE, while the reducibility yields the modularity of all solutions and leads to solutions of certain SU(3) Toda systems. Note that the SU(N + 1) Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to CPN . Key words: modular differential equations; quasimodular forms; Toda system 2020 Mathematics Subject Classification: 11F11; 34M03 1 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}: Ly := y′′′ +Q2(z)y ′ +Q3(z)y = 0, z ∈ H, (1.1) where ′ := d dz . Near a regular point z0 of Ly = 0, a local solution y(z) can be obtained by giving the initial values y(k)(z0), 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 π1(C \ {singular points}) to SL(3,C) is introduced to characterize the multi-valueness of solutions. If the potentials Q2(z) and Q3(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 π1(Eτ ) to SL(3,C), where Eτ := 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 π1(Eτ ) is abelian. In this paper, we consider the case that Ly = 0 is mailto:zjchen2016@tsinghua.edu.cn mailto:cslin@math.ntu.edu.tw mailto:yangyifan@ntu.edu.tw https://doi.org/10.3842/SIGMA.2022.013 2 Z. Chen, C.-S. Lin and Y. Yang defined on H and instead of the monodromy representation defined on π1(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. Let Γ be a discrete subgroup of SL(2,R) that is commensurable with SL(2,Z). Equa- tion (1.1) is called a modular ordinary differential equation (MODE for short) on Γ if Q2(z) and Q3(z)− 1 2Q ′ 2(z) are meromorphic modular forms on Γ of weight 4 and 6, respectively. Mod- ular ordinary differential equations (of general order) appear prominently in the study of rational conformal field theories (see, e.g., [1, 10, 12, 14, 19, 21, 27, 34]). They provide a practical tool for classifying rational conformal field theories. As an object in the theory of modular forms, modular differential equations have also been studied by mathematicians. See, for example, [9, 11, 13, 17, 18, 29]. 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 Qj(z) is at most j, j = 2, 3. At the cusp ∞, we let qN = e2πiz/N , where N is the width of ∞ in Γ. Then d dz = 2πi N qN d dqN and so (1.1) becomes( qN d dqN )3 y + ( N 2πi )2 Q2(z)qN d dqN y + ( N 2πi )3 Q3(z)y = 0. (1.2) From here we see that (1.1) is Fuchsian at ∞ if and only if Q2(z) and Q3(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 κ3 + ( N 2πi )2 Q2(∞)κ+ ( N 2πi )3 Q3(∞) = 0, the roots of which are called the local exponents of (1.1) at ∞, denoted by κ (1) ∞ , κ (2) ∞ and κ (3) ∞ , satisfying ∑ j κ (j) ∞ = 0. In this paper, we always assume that the exponent differences κ (j) ∞ −κ (1) ∞ are integers for j = 2, 3, Then ∑ j κ (j) ∞ = 0 implies κ (j) ∞ ∈ 1 3Z for all j and so we may assume κ (1) ∞ ≤ κ (2) ∞ ≤ κ (3) ∞ . Similar assumptions are made for other cusps. On the other hand, let z0 ∈ H be a singular point of (1.1) and write Qj(z) = Aj(z − z0) −j +O ( (z − z0) −j+1 ) at z0, then the indicial equation at z0 is given by κ(κ− 1)(κ− 2) +A2κ+A3 = 0, the roots of which are the local exponents of (1.1) at z0, denoted by κ (1) z0 , κ (2) z0 and κ (3) z0 , satisfying∑ j κ (j) z0 = 3. In this paper, we always assume that the exponent differences κ (j) z0 − κ (1) z0 are integers for j = 2, 3. Then ∑ j κ (j) z0 = 3 implies κ (j) z0 ∈ 1 3Z for all j and so we may assume κ (1) z0 ≤ κ (2) z0 ≤ κ (3) z0 . Since the exponent differences are integers, (1.1) might have solutions with logarithmic singularities at z0. See Appendix A for all possibilities of the solution structure of (1.1) at z0. The singularity z0 is called apparent if (1.1) has no solutions with logarithmic Modular Ordinary Differential Equations on SL(2,Z) of Third Order 3 singularities at z0. In this case, the three local exponents must be distinct, i.e., κ (1) z0 < κ (2) z0 < κ (3) z0 ; see, e.g., Appendix A. In this paper, we always assume that L is apparent at any singularity z0 ∈ 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 z0 ∈ H, κ (1) z0 < κ (2) z0 < κ (3) z0 satisfy κ (1) z0 ∈ 1 3Z≤0 and κ (j) z0 −κ (1) z0 ∈ Z for j = 2, 3. Furthermore, z0 is apparent. (H3) At any cusp s of Γ, κ (1) s ≤ κ (2) s ≤ κ (3) s satisfies κ (1) s ∈ 1 3Z≤0 and κ (j) s − κ (1) s ∈ Z for j = 2, 3. The motivation of all these assumptions will be clear from Theorem 1.1 below. Note ln qN = 2πi N z. Under our assumption (H3), (1.1) might have solutions containing (ln qN )2 = −4π2 N2 z 2 terms; see Remark A.9. In this case, we call the cusp ∞ to be completely not apparent or maximally unipotent, because under the Bol representation ρ̂ that will be introduced below, the corresponding matrix ρ̂(T ) ∈ SL(3,C) of T = ( 1 N 0 1 ) ∈ Γ has eigenvalues {1, 1, 1} but rank(ρ̂(T ) − I3) = 2, i.e., ρ̂(T ) is maximally unipotent. Here I3 denotes the 3 × 3 identity matrix. One class of the MODEs can be derived from quasimodular forms of depth 2. The no- tion of quasimodular forms was first introduced by Kaneko and Zagier [20]. See Section 2 for a brief overview of basic properties of quasimodular forms. In particular, given a holomorphic function ϕ(z) satisfying (ϕ ∣∣ 2 γ)(z) := (cz + d)−2ϕ(γz) = ϕ(z) + αc cz + d for all γ = ( a b c d ) for some nonzero complex number α, any quasimodular form f(z) of weight k and depth 2 with character χ can be expressed as f(z) = f0(z) + f1(z)ϕ(z) + f2(z)ϕ(z) 2, where fj(z) is a modular form on Γ of weight k − 2j with character χ and f2 ̸= 0. Considerh1(z) h2(z) h3(z)  := z2f(z) + αz(f1(z) + 2f2(z)ϕ(z)) + α2f2(z) 2zf(z) + α(f1(z) + 2f2(z)ϕ(z)) f(z)  (1.3) and define Wf (z) := det h1 h′1 h′′1 h2 h′2 h′′2 h3 h′3 h′′3  (1.4) to be the Wronskian associated to f . Then Wf (z) is a modular form on Γ of weight 3k with character χ3; see Lemma 2.1 for a proof. This Wf (z) was first introduced by Pellarin [28] for Γ = SL(2,Z) and ϕ(z) = E2(z). Now we define gj(z) := hj(z) 3 √ Wf (z) , then det g1 g′1 g′′1 g2 g′2 g′′2 g3 g′3 g′′3  = 1, and a further differentiation leads to det g1 g′1 g′′′1 g2 g′2 g′′′2 g3 g′3 g′′′3  = 0, (1.5) 4 Z. Chen, C.-S. Lin and Y. Yang so g3(z) is a solution of (1.1) with Q2(z) := g′′′1 g2 − g1g ′′′ 2 g1g′2 − g′1g2 , Q3(z) := g′1g ′′′ 2 − g′2g ′′′ 1 g1g′2 − g′1g2 . (1.6) It is easy to see that Q2(z) and Q3(z) are single-valued, and g1, g2 are also solutions of (1.1). Our first result reads as follows. Theorem 1.1. Let Q2(z) and Q3(z) be given by (1.6). Then (1) (1.1) is a MODE, i.e., Q2(z) and Q3(z) − 1 2Q ′ 2(z) are meromorphic modular forms on Γ (with trivial character) of weight 4 and 6, respectively. (2) (H1)–(H3) hold for (1.1). Furthermore, for Γ = SL(2,Z), we have that (3) At the elliptic point i, { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2. (4) At the elliptic point ρ = −1+ √ 3i 2 , κ (j) ρ ∈ Z for all j and { κ (1) ρ , κ (2) ρ , κ (3) ρ } ≡ {0, 1, 2} mod 3. We emphasize that (1) and (2) in Theorem 1.1 hold for any Γ, not only for Γ = SL(2,Z). They will lay the ground for our future study of general MODEs on other congruence subgroups. 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 contin- uation. Fix a point z0 ∈ H that is not a singular point of (1.1) and let U be a simply-connected neighborhood of z0 that contains no singularities of (1.1). For γ = ( a b c d ) ∈ Γ, choose a path σ from z0 to γz0 and consider the analytic continuation of y(z), z ∈ U , along the path. Then y(γz) is well-defined in U . Define (y\|−2γ)(z) := (cz + d)2y(γz), z ∈ U, then by a direct computation or by using Bol’s identity [3], we see that (y\|−2γ)(z) is also a solution of (1.1). Thus, given a fundamental system of solutions Y (z) = (y1(z), y2(z), y3(z)) t, there is γ̂ ∈ SL(3,C) such that (Y ∣∣ −2 γ)(z) = γ̂Y (z), where the fact det γ̂ = 1 follows from that the Wronskians of Y and (Y ∣∣ −2 γ) are the same. Obviously, this matrix γ̂ depends on the choice of the path σ. However, under the above assumptions, all local monodromy matrices are εI3 with ε3 = 1, so different choices of σ will only possibly change γ̂ to e± 2πi 3 γ̂. From here, we see that there is a well-defined homomorphism ρ : Γ → PSL(3,C) such that (Y ∣∣ −2 γ)(z) = e 2πik 3 ρ(γ)Y (z), k ∈ {0,±1}, where yj(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 homomorphism ρ̂ : Γ → GL(3,C) as follows. Suppose that we can find a multi-valued meromorphic function F (z) such that: (i) The analytic continuation of ŷ(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 Modular Ordinary Differential Equations on SL(2,Z) of Third Order 5 with some character, where k ∈ N. Such F (z) can be constructed explicitly when Γ is a triangle group. Then by letting Ŷ (z) := F (z)Y (z), there is ρ̂(γ) ∈ GL(3,C) such that( Ŷ ∣∣ ℓ γ ) (z) = ρ̂(γ)Ŷ (z), where ℓ = k − 2. This homomorphism ρ̂ : 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 rep- resentations 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 F (z) := ∆(z)−κ (1) ∞ E4(z) −κ (1) ρ E6(z) −κ (1) i m∏ j=1 Fj(z) −κ (1) zj , (1.7) where E4(z) = 1 + 240 ∞∑ n=1 n3qn 1− qn , E6(z) = 1− 504 ∞∑ n=1 n5qn 1− qn , q = e2πiz, (1.8) are the Eisenstein series of weight 4 and 6, respectively, ∆(z) = E4(z) 3 − E6(z) 2 1728 = q − 24q2 + 252q3 − 1472q4 + · · · , i = √ −1 and ρ = ( −1 + √ 3i ) /2 are the elliptic points of SL(2,Z), {z1, . . . , zm} ⊔{i, ρ,∞} denotes the set of singular points of the MODE (1.1) mod SL(2,Z), tj := E4(zj) 3/E6(zj) 2 and Fj(z) := E4(z) 3 − tjE6(z) 2. Then F (z)3 is a modular form of weight 3(ℓ+2), where the integer ℓ = k − 2 is given by ℓ := −2− 12κ(1)∞ − 4κ(1)ρ − 6κ (1) i − 12 m∑ j=1 κ(1)zj . In other words, besides the assumptions (H1)–(H3), we need to assume further that κ (1) ρ ∈ Z such that ℓ ∈ Z. Consequently, we will see from Lemma 3.2 that the Bol representation ρ̂ is indeed a group homomorphism from SL(2,Z) to SL(3,C). Remark 1.2. The choice of F (z) 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 ℓ) but keeping ρ̂(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 √ Wf (z) as F (z), i.e., Ŷ = (h1, h2, h3) t defined in (1.3). Note that for Γ = SL(2,Z), 3 √ Wf (z) might be different from the F (z) given by (1.7). To obtain that 3 √ Wf (z) equals to the F (z) given by (1.7), we need to assume that f0, f1, f2 have no common zeros. Note from ∑ j κ (j) i = 3 that we have either { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2 or { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {1, 1, 1} mod 2. Our second result of this paper is Theorem 1.3. Let Γ = SL(2,Z) and suppose that the MODE (1.1) satisfies (H1)–(H3) and κ (1) ρ ∈ Z. Then the Bol representation ρ̂ is irreducible if and only if { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2. 6 Z. Chen, C.-S. Lin and Y. Yang 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 Re- mark 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 κ (1) ρ ∈ Z. Let y+(z) be the solution of (1.1) of the form y+(z) = qκ (3) ∞ ∑∞ j=0 cjq j, c0 = 1, and F (z) be defined by (1.7). (1) If { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2, then ŷ+(z) := F (z)y+(z) is a quasimodular form of weight ℓ+ 2 and depth 2. (2) If { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {1, 1, 1} mod 2, then the Bol representation ρ̂ is trivial, i.e., ρ̂(γ) = I3 for all γ ∈ SL(2,Z). In particular, 12\|ℓ and ŷ(z) := F (z)y(z) is a modular form of weight ℓ for any solution y(z) of (1.1). Together with Theorems 1.4 and 1.1, we can obtain Corollary 1.5. Let Γ = SL(2,Z) and suppose (H1)–(H3) and κ (1) ρ ∈ Z hold for the MODE (1.1). Then (3) and (4) in Theorem 1.1 are equivalent. 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 CPN ; 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 Qj(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. 2 Quasimodular forms of depth 2 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)−kf(γz) = χ(γ)f(z) for any γ ∈ ( a b c d ) ∈ Γ; (2) f is holomorphic at any cusp s of Γ. 1We thank the referee for providing the reference. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 7 For example, the Eisenstein series E4(z) and E6(z) in (1.8) are modular forms of weight 4 and 6 on SL(2,Z), respectively. We let Mk(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 = e2πiz, 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 f(z) = r∑ j=0 fj(z)ϕ(z) j , where fj(z) ∈ Mk−2j(Γ, χ) with fr ̸≡ 0 and ϕ(z) is a holomorphic function satisfying that (ϕ ∣∣ 2 γ)(z) := (cz + d)−2ϕ(γz) = ϕ(z) + αc cz + d (2.1) 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 ϕ(z) = E2(z) := 1 2πi ∆′(z) ∆(z) = 1− 24 ∞∑ n=1 nqn 1− qn , and so α = 6 πi . We let M̃≤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) ∈ M̃≤r k (Γ, χ), then f ′(z) ∈ M̃≤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 Wf (z) (see [28]) associated to f(z) = f0(z) + f1(z)ϕ(z) + f2(z)ϕ(z) 2 ∈ M̃≤2 k (Γ, χ), where fj(z) ∈ Mk−2j(Γ, χ) with f2 ̸= 0. Then( f ∣∣ k γ ) (z) := (cz + d)−kf(γz) (2.2) = χ(γ) 2∑ j=0 fj(z) ( ϕ(z) + αc cz + d )j , γ = ( a b c d ) ∈ Γ. As in [28], we set Pf (t) := 2∑ j=0 fj(z) (ϕ(z) + αt)j , Qf (t) := t2Pf (1/t) = f(z)t2 + α(f1(z) + 2f2(z)ϕ(z))t+ α2f2(z), Ff (z) :=  Qf (t) ∂ ∂t Qf (t) 1 2 ∂2 ∂t2 Qf (t)  \|t=z 8 Z. Chen, C.-S. Lin and Y. Yang = z2f(z) + αz(f1(z) + 2f2(z)ϕ(z)) + α2f2(z) 2zf(z) + α(f1(z) + 2f2(z)ϕ(z)) f(z)  =: h1(z) h2(z) h3(z)  . (2.3) and define Wf (z) as in (1.4) to be the Wronskian associated to f . Lemma 2.1. Wf (z) is a modular form on Γ of weight 3k with character χ3. 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 f(γz) = χ(γ)(cz + d)k [ f + αc cz + d (f1 + 2f2ϕ) + α2c2 (cz + d)2 f2 ] = χ(γ)(cz + d)k−2 ( c2h1 + cdh2 + d2h3 ) , h2(γz) = 2 az + b cz + d f(γz) + α(f1(γz) + 2f2(γz)ϕ(γz)) = χ(γ)(cz + d)k−2 [ 2 az + b cz + d ((cz + d)2f + αc(cz + d)(f1 + 2f2ϕ) + α2c2f2) + α(f1 + 2f2ϕ) + 2α2c cz + d f2 ] = χ(γ)(cz + d)k−2[2ach1 + (ad+ bc)h2 + 2bdh3], h1(γz) = (az + b)2 (cz + d)2 f(γz) + α az + b cz + d (f1(γz) + 2f2(γz)ϕ(γz)) + α2f2(γz) = χ(γ)(cz + d)k−2 [ (az + b)2 (cz + d)2 ( (cz + d)2f + αc(cz + d)(f1 + 2f2ϕ) + α2c2f2 ) + az + b cz + d [ α(f1 + 2f2ϕ) + 2α2c cz + d f2 ] + α2 (cz + d)2 f2 ] = χ(γ)(cz + d)k−2 [ a2h1 + abh2 + b2h3 ] . Thus Ff (γz) = χ(γ)(cz + d)k−2AFf (z), where A :=  a2 ab b2 2ac ad+ bc 2bd c2 cd d2  . Together with the fact detA = (ad− bc)3 = 1, it is easy to see that Wf (γz) = det(Ff (γz), F ′ f (γz), F ′′ f (γz)) = χ(γ)3(cz + d)3kWf (z). In particular, for the case γ = ( 1 N 0 1 ) , where N is the width of the cusp ∞, the transformation law shows that Wf (z) is a polynomial only in f0, f1, f2, ϕ, and their derivatives. Thus, by the computation above and the fact that the ring of quasimodular forms is invariant under differentiation, Wf (z) is a quasimodular form that is actually modular. In other words, Wf (z) ∈ M3k ( Γ, χ3 ) . This completes the proof. ■ Now we define gj(z) := hj(z)/ 3 √ Wf (z). Then (1.5) holds, so g3(z) is a solution of Ly := y′′′ +Q2(z)y ′ +Q3(z)y = 0, (2.4) Modular Ordinary Differential Equations on SL(2,Z) of Third Order 9 where Q2(z) := g′′′1 g2 − g1g ′′′ 2 g1g′2 − g′1g2 , Q3(z) := g′1g ′′′ 2 − g′2g ′′′ 1 g1g′2 − g′1g2 . (2.5) Note that (i) g1(z) and g2(z) are also solutions of (2.4) and g1, g2, g3 are linearly independent. (ii) Although gj might be multi-valued, g′j/gj is single-valued. Thus Qj(z) is single-valued and meromorphic on H for j = 1, 2. We will see from Theorem 2.2 below that any pole of Qj(z) comes from zeros of Wf (z). For z0 ∈ H, we denote κ (1) z0 ≤ κ (2) z0 ≤ κ (3) z0 to be the local exponents of (2.4) at z0. Theorem 2.2. Under the above notations, Q2(z) and Q3(z)− 1 2Q ′ 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 Wf (z), (H1)–(H3) hold, and every cusp of Γ is completely not apparent for (2.4). Proof. To prove the modularity of Qj , we consider ỹ(z) := ( y ∣∣ −2 γ ) (z) = (cz + d)2y(γz), γ ∈ Γ. Then ỹ′(z) = y′(γz) + 2c(cz + d)y(γz), ỹ′′′(z) = (cz + d)−4y′′′(γz), so Lỹ = (cz + d)−4 { y′′′(γz) + (cz + d)4Q2(z)y ′(γz) + [ (cz + d)6Q3(z) + 2c(cz + d)5Q2(z) ] y(γz) } . (2.6) Recalling g3(z) = f(z) 3 √ Wf (z) , we have ( g3 ∣∣ −2 γ ) (z) = ( f 3 √ Wf ∣∣∣ −2 γ ) (z) = (cz + d)2f(γz) 3 √ Wf (γz) = ε 3 √ Wf (z) ( c2h1 + cdh2 + d2h3 ) = ε ( c2g1(z) + cdg2(z) + d2g3(z) ) , where ε3 = 1. Thus ( g3 ∣∣ −2 γ ) (z) is also a solution of (2.4). From this and (2.6), we have Q2(γz) = (cz + d)4Q2(z), Q3(γz) = (cz + d)6Q3(z) + 2c(cz + d)5Q2(z), so ( Q2 ∣∣ 4 γ ) = Q2 and ( (Q3 − 1 2Q ′ 2) ∣∣ 6 γ ) = Q3 − 1 2Q ′ 2. This proves the modularity of Q2 and Q3. To prove (H1)–(H3), we let z0 be any pole of Qj(z) for some j = 1, 2. Clearly gj(z) = (z − z0) αj (cj + O(z − z0) j) near z0 for some αj ∈ 1 3Z and cj ̸= 0. By replacing g2 by g2 − c2 c1 g1 if necessary, we may assume α1 ̸= α2. Then we easily deduce from (2.5) that Q2(z) = α1(α1 − 1)(α1 − 2)− α2(α2 − 1)(α2 − 2) (α2 − α1)(z − z0)2 +O ( (z − z0) −1 ) , Q3(z) = α1α2[(α2 − 1)(α2 − 2)− (α1 − 1)(α1 − 2)] (α2 − α1)(z − z0)3 +O ( (z − z0) −2 ) , so z0 is a regular singular point of (2.4). Thus, (2.4) is Fuchsian on H. 10 Z. Chen, C.-S. Lin and Y. Yang Let z0 be any singular point. It follows from gj(z) = hj(z) 3 √ Wf (z) that gj(z) = (z − z0) −ordz0Wf 3 ∑ l≥0 dj(z − z0) j , (2.7) where ordz0Wf denotes the zero order of Wf (z) at z0. Since (g1, g2, g3) is a fundamental system of solutions of (2.4) and gj ’s have no logarithmic singularities at z0, we conclude from (2.7) and Remark A.8 that (1) the local exponents κ (j) z0 ∈ 1 3Z and are all distinct; (2) the exponent differences are all nonzero integers, namely m (j) z0 := κ (j+1) z0 −κ (j) z0 − 1 are nonnegative integers for j = 1, 2; (3) z0 is an apparent singularity of (2.4). Since Qj(z) = Aj(z − z0) −j +O ( (z − z0) −j+1 ) , j = 2, 3, (2.8) then the indicial equation of (2.4) at z0 is κ(κ− 1)(κ− 2) +A2κ+A3 = 0, (2.9) which implies ∑3 j=1 κ (j) z0 = 3 and so κ (1) z0 = −2m (1) z0 +m (2) z0 3 ∈ 1 3Z≤0. This proves (H2). Remark that if z0 is not a zero of Wf (z), i.e., ordz0Wf = 0, then it follows from (2.7) that κ (1) z0 ∈ Z≥0 and so κ (1) z0 = m (1) z0 = m (2) z0 = 0, i.e., the local exponents at z0 are {0, 1, 2}. This already implies A2 = A3 = 0. Together with the fact that z0 is apparent, we easily deduce from the Frobenius method that both Q2(z) and Q3(z) are holomorphic at z0, a contradiction with that z0 is a singular point. Thus z0 is a zero of Wf (z). This proves that all singular points of (2.4) on H come from the zeros of Wf (z). Let N be the width of the cusp ∞ and qN = e2πiz/N . Since modular forms fj(z), Wf (z) are holomorphic in terms of qN and z = N 2πi ln qN , we see from (2.3) that gj(z) = hj(z) 3 √ Wf (z) = 3−j∑ k=0 (ln qN )kq − ord∞ Wf 3 N ∞∑ t=0 cj,k,tq t N , j = 1, 2, 3, so ∞ is also a regular singular point of (2.4), i.e., (2.4) is Fuchsian at ∞ and so Qj(z) is holomorphic at ∞ for j = 2, 3. Since (ln qN )2 appears in the expression of g1(z), we see from Remark A.9 that ∞ is completely not apparent and the local exponents κ (1) ∞ ≤ κ (2) ∞ ≤ κ (3) ∞ satisfy κ (j) ∞ ∈ 1 3Z and m(1) ∞ := κ(2)∞ − κ(1)∞ ∈ Z≥0, m(2) ∞ := κ(3)∞ − κ(2)∞ ∈ Z≥0. Note that the indicial equation at ∞ is κ3 + ( N 2πi )2 Q2(∞)κ+ ( N 2πi )3 Q3(∞) = 0, which implies ∑ κ (j) ∞ = 0 and so κ (1) ∞ = −2m (1) ∞ +m (2) ∞ 3 . 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′ = kerχ ∩ SL(2,Z), we can express f(z) as f(z) = f̃0(z) + f̃1(z)E2(z) + f̃2(z)E2(z) 2 for some f̃j(z) ∈ Mk−2j(Γ ′). We check that (g3\|−2σ) (z) = (cz + d)2f(σz) 3 √ Wf (σz) = ϵ (cz + d)2(f \|kσ)(z) 3 √ (Wf \|3kσ)(z) Modular Ordinary Differential Equations on SL(2,Z) of Third Order 11 = ϵ (cz + d)2p1(z) + αc(cz + d)p2(z) + α2c2p3(z) 3 √ (Wf \|3kσ)(z) , where p1(z) = ( f̃0\|kσ ) (z) + ( f̃1\|k−2σ ) (z)E2(z) + ( f̃2\|k−4σ ) (z), p2(z) = ( f̃1\|k−2σ ) (z) + 2 ( f̃2\|k−4σ ) (z)E2(z), p3(z) = ( f̃2\|k−4σ ) (z), α = 6/πi and ϵ is a third root of unity. Except for cz + d, every term in the expression has a qM -expansion, where M is the width of the cusp σ and qM = e2πiz/M . Since s ̸= ∞, we have c ̸= 0. This shows that there is a local solution at the cusp s having a factor z2. According to the solution structure discussed in the appendix, the point s must be completely not apparent and we have κ (2) s −κ (1) s , κ (3) s −κ (1) s ∈ Z. By the same reasoning as in the case of the cusp ∞, the sum κ (1) s + κ (2) s + κ (3) s is equal to 0 and hence κ (j) s ∈ 1 3Z for all j. This proves (H1), (H3), and that every cusp is completely not apparent. ■ For a MODE, the local exponents are invariant under z0 → γz0 for any γ ∈ Γ. Proposition 2.3. Let z0 be a singular point of (2.4). Then the local exponents of (2.4) at γz0 are the same for all γ = ( a b c d ) ∈ Γ. Proof. Let Qj(z) = Ãj(z − γz0) −j +O ( (z − γz0) −j+1 ) , j = 2, 3. Recalling (2.8) and (2.9). we only need to prove ( Ã2, Ã3 ) = (A2, A3). Since γz − γz0 = z − z0 (cz + d)(cz0 + d) = z − z0 (cz0 + d)2 (1 +O(z − z0)) as z → z0, we have for z → z0 that Q2(z) = (cz + d)−4Q2(γz) = (cz0 + d)−4(1 +O(z − z0)) [ Ã2(γz − γz0) −2 +O ( (γz − γz0) −1 )] = Ã2(z − z0) −2 +O ( (z − z0) −1 ) , so Ã2 = A2. Since Q3 − 1 2Q ′ 2 is a modular form of weight 6, a similar argument implies Ã3 + Ã2 = A3 +A2 and so Ã3 = A3. ■ 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. Remark 2.4. If f0, f1, f2 have a common zero z0 ∈ H∪{∞}, we take M(z) to be a holomorphic modular form such that it has only one simple zero at z0. Then fj(z)/M(z) are holomorphic modular forms and f(z)/M(z) is a quasimodular form. Since Wf (z) = M(z)3Wf/M (z) and so gj(z) = hj(z) 3 √ Wf (z) = hj(z)/M(z) 3 √ Wf/M (z) , namely all gj(z)’s are invariant by replacing f(z) by f(z)/M(z), so the differential equation (2.4) derived from f(z) and that from f(z)/M(z) are the same (Note that gj(z) are solutions of (2.4) but neither f(z) nor f(z)/M(z) are solutions of (2.4), so we do not mean that f(z)/M(z) satisfies the same differential equation as f(z)). Therefore, without loss of generality, we assume throughout the section that f0, f1, f2 have no common zeros on H ∪ {∞}. 12 Z. Chen, C.-S. Lin and Y. Yang By applying this remark, we have Lemma 2.5. Let Γ∞ = ±⟨T ⟩ be the stabilizer subgroup of ∞ in Γ. Then there are at most two right cosets Γ∞γ in Γ∞\Γ (the set of right cosets of Γ∞ in Γ) such that f(γz0) = 0. Proof. Suppose that there are three distinct right cosets Γ∞γ in Γ∞\Γ such that f(γz0) = 0. Without loss of generality, we assume that one of them is the coset of I, i.e., f(z0) = 0. Now we have (f \|kγ) (z) = χ(γ) ( f(z) + αc cz + d (f1(z) + 2f2(z)ϕ(z)) + ( αc cz + d )2 f2(z) ) for γ ∈ Γ. If f(γ1z0) = f(γ2z0) = 0 for γ1 and γ2 in two different cosets in Γ∞\Γ, then we have f1(z0) + 2f2(z0)ϕ(z0) = f2(z0) = 0. However, this implies that f0(z0) ̸= 0 since fj(z) are assumed to have no common zeros on H, and hence f(z0) ̸= 0, a contradiction. ■ Now we let Γ = SL(2,Z). Given a character χ, we have χ(T ) = e2πim/24 for some integer m ∈ [0, 23]. Recall the Dedekind eta function η(z) = e2πiz/24 ∞∏ n=1 ( 1− e2nπiz ) . Since the MODE associated to f is the same as that associated to f/ηm, by considering f/ηm 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). Proof of Theorem 1.1. The conclusions (1) and (2) are proved in Theorem 2.2. It suffices to consider Γ = SL(2,Z) and prove (3) and (4). Let z0 ∈ {i, ρ}. By Lemma 2.5, we have f(γz0) ̸= 0 for some γ ∈ SL(2,Z). Then it follows from Proposition 2.3 that κ(1)z0 = κ(1)γz0 = ordγz0 f(z) 3 √ Wf (z) = −1 3 ordz0 Wf (z). Since Wf (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 ordiWf 2 + ordρWf 3 ≡ k 4 mod 1. This implies κ (1) ρ = −ordρ Wf 3 ∈ Z≤0 and 3κ (1) i = − ordiWf ≡ k/2 mod 2. Recall that we may assume that f0, f1, and f2 have no common zeros. When k ≡ 0 mod 4, we have m (2) i ≡ 3κ (1) i ≡ k 2 ≡ 0 mod 2, and we are done. When k ≡ 2 mod 4, the weights of f0 and f2 are congruent to 2 modulo 4 and their expansions in w = (z − i)/(z + i) are of the form f0(z) = (1− w)k ∞∑ n=0 a2n+1w 2n+1, f2(z) = (1− w)k−4 ∞∑ n=0 c2n+1w 2n+1, Modular Ordinary Differential Equations on SL(2,Z) of Third Order 13 while the expansion of f1(z) is of the form f1(z) = (1− w)k−2 ∞∑ n=0 b2nw 2n. (See Proposition 5.1 and Remark 5.2 below.) Let hj(z), j = 1, 2, 3, be given by (2.3). Then the local exponent of ah1(z) + bh2(z) + ch3(z) at z = i must be one of{ 0, κ (2) i − κ (1) i , κ (3) i − κ (1) i } (2.10) for any (a, b, c) ∈ C3. Consider the function h1 + ih2 − h3 = z2 ( f0 + f1E2 + f2E 2 2 ) + αz(f1 + 2f2E2) + α2f2 + 2iz ( f0 + f1E2 + f2E 2 2 ) + iα(f1 + 2f2E2)− ( f0 + f1E2 + f2E 2 2 ) = (z + i)2f0 + (z + i)((z + i)E2 + α)f1 + ((z + i)E2 + α)2f2. We compute that z = i(1 + w)/(1− w) and hence z + i = 2i 1− w , dw dz = 2i (z + i)2 = (1− w)2 2i . (2.11) Also, since E2 = 1 2πid log∆(z)/dz and the expansion of ∆(z) is of the form ∆(z) = (1− w)12 ∞∑ n=0 d2nw 2n, we find that E2(z) = −(1− w)2 4π ( − 12 1− w + ∑ 2nd2nw 2n−1∑ d2nw2n ) = 3 π (1− w)− (1− w)2 4π ∞∑ n=0 d′2n+1w 2n+1 (2.12) for some power series ∑ d′2n+1w 2n+1. It follows that, by (2.11), (z + i)E2(z) + 6 πi = 1− w 2πi ∞∑ n=0 d′2n+1w 2n+1. From this, we see that h1 + ih2 − h3 = (1− w)k−2 ∞∑ n=0 e2n+1w 2n+1 for some ej . This, together with (2.10), implies that either κ (2) i − κ (1) i or κ (3) i − κ (1) i is odd. This proves the assertion (3) that { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2. The proof of (4) is similar. We consider the function h1 − ρh2 + ρ2h3 = (z − ρ)2f0 + (z − ρ)((z − ρ)E2 + α)f1 + ((z − ρ)E2 + α)2f2. Setting w = (z − ρ)/(z − ρ), we have z = (ρ− ρw)/(1− w), z − ρ = √ 3i 1− w , dw dz = (1− w)2√ 3i , 14 Z. Chen, C.-S. Lin and Y. Yang and (z − ρ)E2(z) + 6 πi = 1− w 2πi ∞∑ n=0 d′3n+2w 3n+2 (2.13) (since the expansion of ∆(z) is (1 − w)12 ∑ d3nw 3n for some d3n). When k ≡ 1 mod 3, the expansions of fj are of the form f0 = (1 − w)k ∑ a3n+1w 3n+1, f1 = (1 − w)k−2 ∑ b3n+2w 3n+2, and f2 = (1− w)k−4 ∑ c3nw 3n. Therefore, the expansion of h1 − ρh2 + ρ2h3 is of the form h1 − ρh2 + ρ2h3 = (1− w)k−2 ∞∑ n=0 e3n+1w 3n+1, which implies that either κ (2) ρ − κ (1) ρ or κ (3) ρ − κ (1) ρ is congruent to 1 modulo 3. Since the sum of κ (j) ρ is 3, we deduce that the set { κ (1) ρ , κ (2) ρ , κ (3) ρ } is congruent to {0, 1, 2} modulo 3. Likewise, when k ≡ 2 mod 3, we can show that h1 − ρh2 + ρ2h3 = (1 − w)k−2 ∑ e3n+2w 3n+2 and obtain the same conclusion. For the case k ≡ 0 mod 3, we need to make the computation more precise. Let f0(z) = (1− w)k ∞∑ n=0 a3nw 3n, f1(z) = (1− w)k−2 ∞∑ n=0 b3n+1w 3n+1, f2(z) = (1− w)k−4 ∞∑ n=0 c3n+2w 3n+2 be the expansions of fj . A computation similar to (2.12) yields E2(z) = 2 √ 3 π (1− w)− (1− w)2 2π √ 3 ∞∑ n=0 d′3n+2w 3n+2 and hence f(z) = (1− w)k ∞∑ n=0 a3nw 3n + (1− w)k−1 ( 2 √ 3 π − 1− w 2π √ 3 ∞∑ n=0 d′3n+2w 3n+2 )( ∞∑ n=0 b3n+1w 3n+1 ) + (1− w)k−2 ( 2 √ 3 π − 1− w 2π √ 3 ∞∑ n=0 d′3n+2w 3n+2 )2( ∞∑ n=0 c3n+2w 3n+2 ) . On the other hand, by (2.13), we have h1 − ρh2 + ρ2h3 = (1− w)k−2 ( −3 ∞∑ n=0 a3nw 3n + √ 3 2π ( ∞∑ n=0 d′3n+2w 3n+2 )( ∞∑ n=0 b3n+1w 3n+1 ) − 1 4π2 ( ∞∑ n=0 d′3n+2w 3n+2 )2( ∞∑ n=0 c3n+2w 3n+2 ) . Modular Ordinary Differential Equations on SL(2,Z) of Third Order 15 We then check that the expansion of h1 − ρh2 + ρ2h3 + 3f is of the form (1− w)k−2 ∞∑ n=0 ( e3n+1w 3n+1 + e3n+2w 3n+2 ) , which again implies that either κ (2) ρ − κ (1) ρ or κ (3) ρ − κ (1) ρ is not congruent to 0 modulo 3 and hence { κ (1) ρ , κ (2) ρ , κ (3) ρ } ≡ {0, 1, 2} mod 3. This completes the proof. ■ 3 The MODE on SL(2,Z) The purpose of this section is to prove Theorems 1.3 and 1.4, and Corollary 1.5. Let Q2(z) and Q3(z)− 1 2Q2(z) be meromorphic modular forms on SL(2,Z) of weight 4 and 6 respectively, i.e., Ly := y′′′(z) +Q2(z)y ′(z) +Q3(z)y(z) = 0, z ∈ H (3.1) is a MODE. In this section, we use the notations S = ( 0 −1 1 0 ) , T = ( 1 1 0 1 ) and R = ST = ( 0 −1 1 1 ) such that S2 = R3 = −I2. Suppose the MODE (3.1) has regular singularities at {z1, . . . , zm} ⊔ {i, ρ,∞} mod SL(2,Z), where ρ = ( −1 + √ 3i ) /2 is the fixed point of R. 3.1 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 (S1) κ (1) ∞ , κ (1) i , κ (1) zj ∈ 1 3Z≤0 and κ (1) ρ ∈ Z≤0 such that ℓ := −2− 12κ(1)∞ − 4κ(1)ρ − 6κ (1) i − 12 m∑ j=1 κ(1)zj ∈ Z. (Note ℓ ∈ Z implies ℓ ∈ 2Z.) Furthermore, m (j) z := κ (j+1) z − κ (j) z − 1 ∈ Z≥0 for z ∈ {z1, . . . , zm} ⊔ {i, ρ} and j = 1, 2, and m (j) ∞ := κ (j+1) ∞ − κ (j) ∞ ∈ Z≥0 for j = 1, 2. (S2) ODE (3.1) is apparent at any singular point z ∈ {z1, . . . , zm} ⊔ {i, ρ}. (S3) { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2. 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. Set tj := E4(zj) 3/E6(zj) 2 ̸∈ {0, 1,∞} and Fj(z) := E4(z) 3 − tjE6(z) 2. Then this modular form Fj(z) has only one simple zero at zj , up to SL(2,Z)-equivalence. Define F (z) := ∆(z)−κ (1) ∞ E4(z) −κ (1) ρ E6(z) −κ (1) i m∏ j=1 Fj(z) −κ (1) zj . Then F (z)3 is a modular form of weight 3(ℓ+ 2). Clearly for any solution y(z) of (3.1), ŷ(z) := F (z)y(z) 16 Z. Chen, C.-S. Lin and Y. Yang is single-valued and holomorphic on H. Furthermore, its order at z ∈ {z1, . . . , zm}⊔{i, ρ} is one of { 0,m (1) z + 1,m (1) z +m (2) z + 2 } , and at ∞ is one of { 0,m (1) ∞ ,m (1) ∞ +m (2) ∞ } . Fix a fundamental system of solutions Y (z) = (y1(z), y2(z), y3(z)) t of (3.1) and let Ŷ (z) := F (z)Y (z). Then for any γ ∈ SL(2,Z), there is a matrix ρ̂(γ) ∈ GL(3,C) such that( Ŷ ∣∣ ℓ γ ) (z) = ρ̂(γ)Ŷ (z). (3.2) This is a lifting of the Bol representation. We will use freely the notation γ̂ = ρ̂(γ) just for convenience. Lemma 3.2. There holds det ρ̂(γ) = 1 for any γ ∈ SL(2,Z). That is, ρ̂ is a group homomor- phism from SL(2,Z) to SL(3,C). Proof. The proof is similar to that of [25, Lemma 4.2], where the second-order MODE was studied. Let W (z) = det y1 y′1 y′′1 y2 y′2 y′′2 y3 y′3 y′′3  , Ŵ (z) = det ŷ1 ŷ′1 ŷ′′1 ŷ2 ŷ′2 y′′2 ŷ3 ŷ′3 ŷ′′3  . Then W (z) ≡ C is a nonzero constant. By (3.2) we have( Ŵ ∣∣ 3(ℓ+2) γ ) (z) = det ρ̂(γ)Ŵ (z). Since Ŵ (z) = F (z)3W (z) = CF (z)3, we also have Ŵ ∣∣ 3(ℓ+2) γ = C ( F 3 ∣∣ 3(ℓ+2) γ ) = ( F 3 ∣∣ 3(ℓ+2) γ ) (z) F (z)3 Ŵ (z). Thus det ρ̂(γ) = ( F 3 ∣∣ 3(ℓ+2) γ ) (z) F (z)3 . This proves det ρ̂(γ) = 1 because F (z)3 is a modular form of weight 3(ℓ+ 2) on SL(2,Z). ■ Remark that under our assumption κ (1) ∞ , κ (1) i , κ (1) zj ∈ 1 3Z≤0 and κ (1) ρ ∈ Z≤0, y(z) 3 is a single- valued 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 ordi ( y3 ) 2 + ordρ ( y3 ) 3 ≡ 1 2 mod Z, so ordi ( y3 ) is odd. Since ordi ( y3 ) can be chosen as any one of 3κ (1) i , 3κ (2) i , 3κ (3) i , these three numbers are all odd, clearly a contradiction with our assumption (S3). ■ Lemma 3.4. Under the assumptions (S1)–(S3), (3.1) is completely not apparent at ∞. To prove Lemma 3.4, we need the following well-known lemma due to Beukers and Heck- man [2]. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 17 Lemma 3.5 ([2]). Let n ≥ 2 and H ⊂ GL(n,C) be a subgroup generated by two matrices A, B such that rank(A−B) ≤ 1. Then H acts irreducibly on Cn if and only if A and B have distinct eigenvalues. Let {a1, . . . , an} and {b1, . . . , bn} 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 a1, . . . , an, b1, . . . , bn are all nonzero complex numbers with ai ̸= bj for any i, j. Then up to a common conjugation in GL(n,C), A and B can be uniquely determined by A =  0 0 · · · 0 −An 1 0 · · · 0 −An−1 0 1 · · · 0 −An−2 ... ... . . . ... ... 0 0 · · · 1 −A1  , B =  0 0 · · · 0 −Bn 1 0 · · · 0 −Bn−1 0 1 · · · 0 −Bn−2 ... ... . . . ... ... 0 0 · · · 1 −B1  , where Aj’s and Bj’s are given by n∏ j=1 (t− aj) = tn +A1t n−1 + · · ·+An, n∏ j=1 (t− bj) = tn +B1t n−1 + · · ·+Bn. Corollary 3.7. Let A, B be 3× 3 matrices such that rank(A− B) ≤ 1. Suppose that A2 = I3 and the eigenvalues of A are {1,−1,−1}. Then A, B have common eigenvalues and so the group H generated by A and B acts on C3 reducibly. Proof. This corollary is trivial if rank(A−B) = 0, i.e., A = B. So we may assume rank(A−B) = 1. Assume by contradiction that A andB have no common eigenvalues, then by (t−1)(t+1)2 = t3 + t2 − t− 1, we see from Lemma 3.6 that A is conjugate to0 0 1 1 0 1 0 1 −1  and so A2 ̸= I3, a contradiction with the assumption A2 = I3. Thus A and B have common eigenvalues, and it follows from Lemma 3.5 that H acts on C3 reducibly. ■ Proof of Lemma 3.4. Suppose that the statement is not true. Then rank ( T̂ − I3 ) ≤ 1 (note T̂ = I3 if and only if ∞ is apparent). Then R̂ = ŜT̂ implies rank ( Ŝ − R̂ ) ≤ 1. By R̂3 = ρ̂ ( R3 ) = ρ̂(−I2) = (−1)ℓI3 = I3 and det R̂ = 1, we have either R̂ = λI3 for some λ3 = 1 or R̂ is conjugate to diag ( 1, ε, ε2 ) where ε = e2πi/3. Similarly, by Ŝ2 = ρ̂ ( S2 ) = ρ̂(−I2) = I3 and det Ŝ = 1, we have either Ŝ = I3 or Ŝ is a conjugate of diag(1,−1,−1). If Ŝ = I3, then by rank ( Ŝ− R̂ ) ≤ 1 we obtain R̂ = I3. This implies that for any solution y(z) of (3.1), ŷ(z)3 = F (z)3y(z)3 is a modular form of weight 3ℓ and so y(z)3 is a meromorphic modular form of weight −6, a contradiction with Lemma 3.3. Thus Ŝ is a conjugate of diag(1,−1,−1). If R̂ = λI3 for some λ3 = 1, then by λ ̸= −1 we obtain 1 ≥ rank ( Ŝ − R̂ ) = rank(diag(1− λ,−1− λ,−1− λ)) ≥ 2, a contradiction. 18 Z. Chen, C.-S. Lin and Y. Yang So R̂ is conjugate to diag ( 1, ε, ε2 ) . By Corollary 3.7, there is a subspace V ⫋ C3 which is invariant under the actions Ŝ and R̂. If dimV = 2, then there is an invertible matrix P such that PŜP−1 = ( A1 0 ∗ a1 ) , P R̂P−1 = ( B1 0 ∗ b1 ) , where A1 and B1 are 2× 2 matrices. This implies rank ( A1 −B1 0 ∗ a1 − b1 ) = rank ( R̂− Ŝ ) ≤ 1. (3.3) Note a1 ∈ {1,−1} and b1 ∈ { 1, ε, ε2 } . If a1 ̸= b1, then (3.3) implies A1 = B1, namely Ŝ and R̂ have two common eigenvalues, a contradiction. So a1 = b1 = 1. Then the eigenvalues of A1 are {−1,−1}, so A1 = −I2. Similarly, B1 is conjugate to diag ( ε, ε2 ) . Thus 1 ≥ rank(A1 −B1) = rank ( diag ( −1− ε,−1− ε2 )) = 2, a contradiction. So dimV = 1, which implies the existence of an invertible matrix P such that PŜP−1 = ( a1 0 ∗ A1 ) , P R̂P−1 = ( b1 0 ∗ B1 ) , where A1 and B1 are 2 × 2 matrices. Clearly the same argument as (3.3) also yields a contra- diction. This completes the proof. ■ Let ŷ3(z) := ŷ+(z) = F (z)y+(z), ŷ1(z) := (ŷ3 ∣∣ ℓ S)(z) and ŷ2(z) := (ŷ3 ∣∣ ℓ R)(z), and yj(z) := ŷj(z)/F (z) for j = 1, 2, 3. Lemma 3.8. Under the assumptions (S1)–(S3), ŷ1(z), ŷ2(z) and ŷ3(z) are linearly independent and ŷ1(z) can be written as ŷ1(z) = βz2ŷ3(z) + zm̂∗ 1(z) + m̂2(z), (3.4) where β ̸= 0 is a constant and m̂∗ 1(z), m̂2(z) are of the form m̂∗ 1(z) F (z) = m∗ 1(z) = qκ (2) ∞ ∑ j≥0 cj,1q j , m̂2(z) F (z) = m2(z) = qκ (1) ∞ ∑ j≥0 cj,2q j . (3.5) 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). Step 1. We show that ŷ1(z) is linearly independent with ŷ3(z). Suppose not, i.e., there is some constant α ̸= 0 such that (ŷ3 ∣∣ ℓ S)(z) = ŷ1(z) = αŷ3(z). Then with respect to (Fy−, Fy⊥, ŷ3(z)) t, we have ρ̂(S) = ( S1 ∗ 0 α ) , where S1 is a 2× 2 matrix. Then ρ̂(S)2 = I3 implies S2 1 = I2 and α2 = 1, i.e., α = ±1. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 19 On the other hand, it follows from the expressions of (y−, y⊥, y+) in Remark A.9 that with respect to (Fy−, Fy⊥, ŷ3(z)) t, ρ̂(T ) = 1 2 ∗ 0 1 1 0 0 1  , so ρ̂(R) = ρ̂(S)ρ̂(T ) = ( S1 ∗ 0 α )1 2 ∗ 0 1 1 0 0 1  . From this and ρ̂(R)3 = I3, we obtain (S1R1) 3 = I2, where R1 := ( 1 2 0 1 ) , (3.6) and α3 = 1, so α = 1. Then detS1 = det ρ̂(S) = 1, which together with S2 1 = I2 easily implies S1 = ±I2, and then (3.6) yields ± ( 1 6 0 1 ) = I2, a contradiction. Step 2. By Step 1, there exists (β, δ) ̸= (0, 0) and ϵ such that ŷ1 = βFy− + δFy⊥ + ϵŷ3. (3.7) We claim that β ̸= 0 and (3.4) holds. Assume by contradiction that β = 0, i.e., ŷ1 = δFy⊥ + ϵŷ3 with δ ̸= 0. Then with respect to (Fy−, ŷ1, ŷ3) we have ρ̂(T ) = 1 2 δ a 0 1 δ 0 0 1  for some a ∈ C, and ρ̂(S) = −1 b b 0 0 1 0 1 0  for some b ∈ C, where we used ρ̂(S)2 = I3, det ρ̂(S) = 1, ŷ1(z) = (ŷ3 ∣∣ ℓ S)(z) and (ŷ1 ∣∣ ℓ S)(z) = (ŷ3 ∣∣ ℓ (−I2))(z) = (−1)−ℓŷ3(z) = ŷ3(z). (3.8) Thus ρ̂(R) = ρ̂(S)ρ̂(T ) = −1 b− 2 δ b+ bδ − a 0 0 1 0 1 δ  . But this implies that −1 is an eigenvalue of ρ̂(R), a contradiction with ρ̂(R)3 = I3. This proves β ̸= 0 and so it follows from the expression of (y−, y⊥, y+) in Remark A.9 that (3.4) and (3.5) hold. Step 3. We show that ŷ1(z), ŷ2(z) and ŷ3(z) are linearly independent. In fact, we see from R = ST that ŷ2(z) = (ŷ+ ∣∣ ℓ R)(z) = (ŷ1 ∣∣ ℓ T )(z) = ŷ1(z + 1), from which and (3.4) we obtain ŷ2 = ŷ1 + 2βzŷ3 + m̂∗ 1 + βŷ3. (3.9) 20 Z. Chen, C.-S. Lin and Y. Yang By (3.7) and (3.9) we haveŷ1 ŷ2 ŷ3  = A Fy− Fy⊥ ŷ3  with A = β δ ϵ β 2β + δ ∗ 0 0 1  . Since detA = 2β2 ̸= 0, we obtain that ŷ1(z), ŷ2(z) and ŷ3(z) are also linearly independent. This completes the proof. ■ Recalling (3.4), we set m̂1(z) and m̂0(z) to be m̂∗ 1(z) = m̂1(z) + πi 3 m̂2(z)E2(z), (3.10) ŷ3(z) = ( πi 6 )2 m̂2(z)E2(z) 2 + πi 6 m̂1(z)E2(z) + m̂0(z). (3.11) The following result can be seen as the converse statement of Theorem 2.2. Theorem 3.9. Under the assumptions (S1)–(S3), the following hold. (a) β = 1. (b) m̂j(z) are meromorphic modular forms of weight ℓ+2− 2j for j = 0, 1, 2, that is, ŷ3(z) is a quasimodular form of weight ℓ+ 2 with depth 2. Proof. (a) By Lemma 3.8, we can take Ŷ (z) = (ŷ1(z), ŷ2(z), ŷ3(z)) t to be a basis and let T̂ , Ŝ, R̂ denote the associated matrices ρ̂(T ), ρ̂(S) and ρ̂(R) of the Bol representation. Recalling (3.8) that (ŷ3 ∣∣ ℓ S) = ŷ1(z) and (ŷ1 ∣∣ ℓ S) = ŷ3(z), we have Ŝ = 0 0 1 λ −1 λ 1 0 0  , for some λ ∈ C, (3.12) where Ŝ2 = I3 is used. Note from SR = S2T = −T that ŷ1 ∣∣ ℓ R = ŷ3 ∣∣ ℓ SR = ŷ3 ∣∣ ℓ (−T ) = ŷ3, and from R2 = T−1S that ŷ2 ∣∣ ℓ R = ŷ3 ∣∣ ℓ R2 = ŷ3 ∣∣ ℓ ( T−1S ) = ŷ3 ∣∣ ℓ S = ŷ1, so R̂ = 0 0 1 1 0 0 0 1 0  . (3.13) Therefore, T̂ = Ŝ−1R̂ = ŜR̂ = 0 0 1 λ −1 λ 1 0 0 0 0 1 1 0 0 0 1 0  =  0 1 0 −1 λ λ 0 0 1  . (3.14) On the other hand, by (3.9) we have ŷ2 ∣∣ ℓ T = ŷ1 ∣∣ ℓ T + 2βzŷ3 + m̂∗ 1 + 3βŷ3 = −ŷ1 + 2ŷ2 + 2βŷ3. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 21 Hence (3.14) yields λ = 2 and β = 1. This proves (a). In particular, it is easy to see from (3.12)–(3.14) that the representation ρ̂ is irreducible, i.e., there is no proper nontrivial subspace of Cŷ1 + Cŷ2 + Cŷ3 that is invariant under ρ̂(SL(2,Z)). (b) To prove (b), we note that (3.4) and (3.9) become ŷ1(z) = z2ŷ3(z) + zm̂∗ 1(z) + m̂2(z), (3.15) ŷ2 = ŷ1 + 2zŷ3 + m̂∗ 1 + ŷ3, (3.16) which implies that y∗(z) := 2zy3(z) +m∗ 1(z) is also a solution of (3.1) and ŷ2 = ŷ1 + ŷ∗ + ŷ3, where ŷ∗ = Fy∗ = 2zŷ3 + m̂∗ 1. By (3.12) and λ = 2, we have 2 (−1 z ) ŷ1 + m̂∗ 1 ∣∣ ℓ S = ŷ∗ ∣∣ ℓ S = (ŷ2 − ŷ1 − ŷ3) ∣∣ ℓ S = ŷ1 − ŷ2 + ŷ3 = −ŷ∗ = −2zŷ3 − m̂∗ 1. From here and (3.15), we obtain z(m̂∗ 1 ∣∣ ℓ S)(z) = zm̂∗ 1(z) + 2m̂2(z). (3.17) On the other hand, by (3.15), ŷ3 = ŷ1 ∣∣ ℓ S = (−1 z )2 ŷ1 + (−1 z ) (m̂∗ 1 ∣∣ ℓ S) + m̂2 ∣∣ ℓ S, which implies ŷ1 = z2ŷ3 + z(m̂∗ 1 ∣∣ ℓ S)− z2(m̂2 ∣∣ ℓ S). Again by (3.15), we have z(m̂∗ 1 ∣∣ ℓ S)(z) = zm̂∗ 1(z) + m̂2(z) + z2(m̂2 ∣∣ ℓ S)(z). (3.18) Thus (3.17) and (3.18) imply m̂2 ∣∣ ℓ−2 S = z2(m̂2 ∣∣ ℓ S) = m̂2. This proves that m̂2(z) is a modular form of weight ℓ− 2. Recalling E2 ∣∣ 2 S = E2 + 6/πiz, it follows from (3.10) and (3.17) that z ( m̂1 + πi 3 m̂2E2 ) + 2m̂2 = z(m̂∗ 1 ∣∣ ℓ S) = z { m̂1 ∣∣ ℓ S + πi 3 (m̂2 ∣∣ ℓ−2 S) ( E2 + 6 πiz )} = z(m̂1 ∣∣ ℓ S) + πiz 3 m̂2E2 + 2m̂2, which yields m̂1 ∣∣ ℓ S = m̂1. This proves that m̂1(z) is a modular form of weight ℓ. Finally, to prove the modularity of m̂0(z), we use (3.15) and (3.11) to obtain z2ŷ3 + z ( m̂1 + πi 3 m̂2E2 ) + m̂2 = ŷ1 = ŷ3 ∣∣ ℓ S = ( πiz 6 )2 (m̂2 ∣∣ ℓ−2 S) ( E2 + 6 πiz )2 + πiz2 6 (m̂1 ∣∣ ℓ S) ( E2 + 6 πiz ) + m̂0 ∣∣ ℓ S = z2ŷ3 + z ( m̂1 + πi 3 m̂2E2 ) + m̂2 + m̂0 ∣∣ ℓ S − z2m̂0, which implies m̂0 ∣∣ ℓ+2 S = z−2(m̂0 ∣∣ ℓ S) = m̂0. This proves that m̂0(z) is a modular form of weight ℓ+ 2. The proof is complete. ■ Clearly the above arguments imply Theorem 1.4(1). 22 Z. Chen, C.-S. Lin and Y. Yang 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. Lemma 3.10. Let (S1)–(S2) hold. Then the eigenvalues of ρ̂(R) are precisely e− πi 3 (ℓ+2κ), κ ∈ { 0, κ(2)ρ − κ(1)ρ , κ(3)ρ − κ(1)ρ } , (3.19) and the eigenvalues of ρ̂(S) are precisely i−ℓ−2κ, κ ∈ { 0, κ (2) i − κ (1) i , κ (3) i − κ (1) i } . (3.20) Proof. Under our assumption (S2) that ρ is apparent, it follows from Remark 5.2 and Lem- ma 5.5 below that any solution y(z) of (3.1) has an expansion of the form 1 (1− w)2 ∞∑ n=0 anw n+κ, w = w(z) = z − ρ z − ρ at ρ, where a0 ̸= 0 and κ ∈ { κ (j) ρ : j = 1, 2, 3 } . Hence, ŷ(z) has an expansion of the form (1− w)ℓ ∞∑ n=0 bnw n+κ, κ ∈ { 0, κ(2)ρ − κ(1)ρ , κ(3)ρ − κ(1)ρ } with b0 ̸= 0. Recalling that ρ = ( −1 + √ 3i ) /2 is a fixed point of R = ( 0 −1 1 1 ) , we denote ζ := cρ+ d = ρ+ 1 = eπi/3. Then a direct computation gives w(Rz) = ζ−2w(z), 1− w(Rz) = ζ−1(z + 1)(1− w(z)), so ( ŷ ∣∣ ℓ R ) (z) = ζ−ℓ(1− w)ℓ ∞∑ n=0 bn ( ζ−2w )n+k . Therefore, ŷ(z) is an eigenfunction of ρ̂(R) if and only if the series expansion of ŷ(z) is of the form (1− w)ℓ ∞∑ n=0 b3nw 3n+κ, κ ∈ { 0, κ(2)ρ − κ(1)ρ , κ(3)ρ − κ(1)ρ } and the corresponding eigenvalue is ζ−ℓ−2κ = e− πi 3 (ℓ+2κ). Note from ρ̂(R)3 = (−1)ℓI3 = I3 that ρ̂(R) can be diagonizalied. We claim that the eigenvalues of ρ̂(R) are precisely those in (3.19). Indeed, for any κ ∈ { 0, κ (2) ρ − κ (1) ρ , κ (3) ρ − κ (1) ρ } , we define Nκ := # { κ̃ ∈ { 0, κ(2)ρ − κ(1)ρ , κ(3)ρ − κ(1)ρ } ∣∣ ζ−ℓ−2κ̃ = ζ−ℓ−2κ } . Clearly (3.19) holds if Nκ = 3 for some κ (and so for all κ). So we only consider the case Nκ ∈ {1, 2} for all κ. Assume by contradiction that there areNκ+1 ∈ {2, 3} linearly independent eigenfunctions ŷj = (1− w)ℓ ∞∑ n=0 bj,3nw 3n+κ, b1,0 = b2,0 = bNκ+1,0 ̸= 0, 1 ≤ j ≤ Nκ + 1, Modular Ordinary Differential Equations on SL(2,Z) of Third Order 23 corresponding to the same eigenvalue ζ−ℓ−2κ. Then ŷ1 − ŷ2 = (1− w)ℓ ∞∑ n=n0 (b1,3n − b2,3n)w 3n+κ is also an eigenfunction of ζ−ℓ−2κ, where n0 ≥ 1 is the smallest integer such that b1,3n0 − b2,3n0 ̸= 0. This implies κ, κ+ 3n0 ∈ { 0, κ (2) ρ − κ (1) ρ , κ (3) ρ − κ (1) ρ } , already a contradiction if Nκ = 1. If Nκ = 2, then by using the linear combination of ŷ1, ŷ2, ŷ3, there is another n1 ≥ 1 satisfying n1 ̸= n0 such that κ, κ + 3n0, κ + 3n1 ∈ { 0, κ (2) ρ − κ (1) ρ , κ (3) ρ − κ (1) ρ } , again a contradiction with Nκ = 2. Thus, for any κ, the dimension of eigenfunctions of ζ−ℓ−2κ is at most Nκ. This implies the assertion (3.19). The proof of (3.20) is similar and is omitted here. ■ Note that if (S3) does not hold, it follows from κ (1) i +κ (2) i +κ (3) i = 3 that { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {1, 1, 1} mod 2. We have Theorem 3.11. Let (S1)–(S2) hold and suppose { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {1, 1, 1} mod 2. Then 12\|ℓ and for any solution y(z) of (3.1), ŷ(z) is a modular form of weight ℓ. In particular, the representation ρ̂ is trivial. Proof. By (3.20) and { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {1, 1, 1} mod 2, we see that the eigenvalues of Ŝ are all the same, so we see from Ŝ2 = I3 that Ŝ = I3. Consequently, T̂ = R̂ and then T̂ 3 = R̂3 = I3. Since the eigenvalues of T̂ are {1, 1, 1}, we obtain R̂ = T̂ = I3, i.e., the representation ρ̂ is trivial and ŷ(z) is a modular form of weight ℓ for any solution y(z). Furthermore, it follows from Lemma 3.10 that e− πi 3 ℓ = i−ℓ = 1, so ℓ ≡ 0 mod 12. ■ Proof of Theorems 1.3 and 1.4. Theorem 1.3 follows from Theorems 3.9 and 3.11. ■ Proof of Corollary 1.5. Under the assumptions (H1)–(H3) and κ (1) ρ ∈ Z, we have κ (j) ρ ∈ Z for all j. Together with κ (1) ρ + κ (2) ρ + κ (3) ρ = 3, we have either κ (1) ρ ≡ κ (2) ρ ≡ κ (3) ρ mod 3 or{ κ (1) ρ , κ (2) ρ , κ (3) ρ } ≡ {0, 1, 2} mod 3. First suppose { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2. Then Theorem 3.9 holds, in particular, R̂ ̸= I3 and the eigenvalues can not be all the same. This together with (3.19) imply that κ (1) ρ ≡ κ (2) ρ ≡ κ (3) ρ mod 3 is impossible, so{ κ(1)ρ , κ(2)ρ , κ(3)ρ } ≡ {0, 1, 2} mod 3. (3.21) Conversely, suppose (3.21) holds. If { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {1, 1, 1} mod 2, then Theorem 3.11 implies R̂ = I3, which together with (3.19) imply κ (1) ρ ≡ κ (2) ρ ≡ κ (3) ρ mod 3, a contradiction with (3.21). Thus { 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2. ■ Remark 3.12. We note the under the assumption that the eigenvalues of ρ̂(T ) are all 1, Propo- sition 2.5 and Corollary to Theorem 2.9 of [31] and results of [32] imply that ρ̂ is irreducible if and only if the eigenvalues of ρ̂(S) and ρ̂(R) are 1, −1, −1 and 1, e2πi/3, e−2πi/3, respec- tively. Our Theorem 1.3 shows that the irreducibility property of ρ̂ 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 that ŷ+(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.) 24 Z. Chen, C.-S. Lin and Y. Yang 4 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: (i) If f(z) ∈ M̃≤2 k (Γ, χ), then f(z)/ 3 √ Wf (z) satisfies a third-order MODE on Γ. This case has been studied in Section 2. (ii) If f(z) = f1(z)ϕ(z) + f0(z) ∈ M̃≤1 k (Γ, χ1) and g(z) ∈ Mk−1(Γ, χ2) with χ1(−I2)χ2(−I2) = −1, then a similar argument as Theorem 2.2 shows that f(z)/ 3 √ Wf,g(z), (zf + αf1)/ 3 √ Wf,g(z) and g(z)/ 3 √ Wf,g(z) are solutions of some third-order MODE on Γ. Here Wf,g(z) = det  f f ′ f ′′ zf + αf1 (zf + αf1) ′ (zf + αf1) ′′ g g′ g′′  . (iii) If f(z) ∈ Mk(Γ, χ1), g(z) ∈ Mk(Γ, χ2), and h(z) ∈ Mk(Γ, χ3) for some characters χj of Γ, then f(z)/ 3 √ Wf,g,h(z), g(z)/ 3 √ Wf,g,h(z), h(z)/ 3 √ Wf,g,h(z) are solutions of some third- order MODE on Γ. Here Wf,g,h(z) = det f f ′ f ′′ g g′ g′′ h h′ h′′  . (4.1) To simplify the situation, we impose the condition that the values of χj(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 χj are all the same, say, χj = 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 e2πim/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 ∈ Mk(SL(2,Z)) be three linearly independent modular forms of weight k on SL(2,Z). Define Wf,g,h(z) by (4.1). Then Wf,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)kF (z) for all γ = ( a b c d ) ∈ SL(2,Z). Then the assertion follows from the basic properties of the determinant function. ■ Theorem 4.2. Let f, g, h ∈ Mk(SL(2,Z)) be linearly independent modular forms and Ly = 0 be the differential equation satisfied by f/ 3 √ Wf,g,h, g/ 3 √ Wf,g,h, and h/ 3 √ Wf,g,h. Then Q2(z) and Q3(z) − 1 2Q ′ 2(z) are meromorphic modular forms of weight 4 and 6 respectively. Furthermore, (H1)–(H3) hold and ∞ is also apparent. Proof. The proof is similar as that of Theorem 2.2. The only difference is that ∞ is also apparent because f/ 3 √ Wf,g,h, g/ 3 √ Wf,g,h and h/ 3 √ Wf,g,h are linearly independent solutions. ■ Proposition 4.3. Let f, g, h ∈ Mk(SL(2,Z)) be linearly independent modular forms and Ly = 0 be the differential equation satisfied by f/ 3 √ Wf,g,h, g/ 3 √ Wf,g,h, and h/ 3 √ Wf,g,h. Then the local exponents of Ly = 0 at the elliptic points i and ρ = ( −1 + √ 3i ) /2 satisfy Modular Ordinary Differential Equations on SL(2,Z) of Third Order 25 1) κ (2) i − κ (1) i , κ (3) i − κ (1) i ≡ 0 mod 2, and 2) κ (j) ρ ∈ Z for all j, and κ (1) ρ ≡ κ (2) ρ ≡ κ (3) ρ mod 3. Proof. Note that every solution y(z) of Ly = 0 can be written as (af(z) + bg(z) + ch(z))/ 3 √ Wf,g,h(z) 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 ordi(af + bg + cz) 2 + ordρ(af + bg + cz) 3 ≡ ordi(f) 2 + ordρ(f) 3 mod Z, so ordi(af + bg + cz)− ordi(f) ≡ 0 mod 2, ordρ(af + bg + cz)− ordρ(f) ≡ 0 mod 3 hold for any (a, b, c) ̸= (0, 0, 0). From here and κ (j) i + 1 3 ordiWf,g,h(z) ∈ {ordi(af + bg + cz) \| (a, b, c) ̸= (0, 0, 0)}, κ(j)ρ + 1 3 ordρWf,g,h(z) ∈ {ordρ(af + bg + cz) \| (a, b, c) ̸= (0, 0, 0)} for all j, we obtain the assertion (1) and κ (2) ρ − κ (1) ρ , κ (3) ρ − κ (1) ρ ≡ 0 mod 3. This together with κ (1) ρ + κ (2) ρ + κ (3) ρ = 3 imply κ (j) 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 dimMk(SL(2,Z)) = 3 is 24. Let f(z) = E4(z) 6, g(z) = E4(z) 3∆(z), and h(z) = ∆(z)2, which form a basis for M24(SL(2,Z)). To determine the differential equation Ly := D3 qy(z) +Q(z)Dqy(z) + ( 1 2 DqQ(z) +R(z) ) y(z) = 0 satisfied by f/ 3 √ Wf,g,h, g/ 3 √ Wf,g,h, and h/ 3 √ Wf,g,h, we use Ramanujan’s identities DqE2 = E2 2 − E4 12 , DqE4 = E2E4 − E6 3 , DqE6 = E2E6 − E2 4 2 , and compute that Wf,g,h(z) = cE4(z) 6E6(z) 3∆(z)3 for some nonzero number c. Noticing that Wf,g,h(z) has a zero of order 3 at i and a zero of order 6 at ρ, we know that κ (1) i = −1, κ (1) ρ = −2, which, by Proposition 4.3, implies that κ (2) i = 1, κ (3) i = 3, κ (2) ρ = 1, and κ (3) ρ = 4. In other words, the indicial equations at i and at ρ are (x+ 1)(x− 1)(x− 3) = 0 and (x+ 2)(x− 1)(x− 4) = 0, respectively. Also, we have ord∞ f − 1 3 ord∞Wf,g,h = −1, ord∞ g − 1 3 ord∞Wf,g,h = 0, and ord∞ h − 1 3 ord∞Wf,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 formsQ(z) andR(z) in Ly are Q(z) = −E4(z)− 3 4 E4(z)(E4(z) 3 − E6(z) 2) E6(z)2 + 8 9 E4(z) 3 − E6(z) 2 E4(z)2 26 Z. Chen, C.-S. Lin and Y. Yang (note that this can also be computed directly using (2.5)) and R(z) = s (1) i E4(z) 3 − E6(z) 2 E6(z) 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 D3 qy(z)+Q(z)Dqy(z)+ 1 2DqQ(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 ∆v1 + 2ev1 − ev2 = 4π N∑ k=1 n1,kδpk , ∆v2 + 2ev2 − ev1 = 4π N∑ k=1 n2,kδpk in R2, where ∆ = ∂2 ∂x2 1 + ∂2 ∂x2 2 is the Laplace operator and δp denotes the Dirac measure at p. We always use the complex variable w = x1 + ix2. Then the Laplace operator ∆ = 4∂ww̄. As in Theorem 4.2, we let f, g, h ∈ Mk(SL(2,Z)) be linearly independent modular forms and y′′′(z) +Q2(z)y ′(z) +Q3(z)y(z) = 0, z ∈ H (4.2) be the MODE satisfied by y1(z) := f(z)/ 3 √ Wf,g,h, y2(z) := g(z)/ 3 √ Wf,g,h, and y3(z) := h(z)/ 3 √ Wf,g,h. Denote the set of regular singular points of (4.2) modulo SL(2,Z) on H by S = {z1, . . . , zm, i, ρ}. For each z ∈ S, it follows from Theorem 4.2 that there are m (1) z ,m (2) z ∈ Z≥0 such that the local exponents of (4.2) at γz are the same as those at z and are given by κ(1)z = −2m (1) z +m (2) z 3 , κ(2)z = κ(1)z +m(1) z + 1, κ(3)z = κ(2)z +m(2) z + 1 for any γ ∈ SL(2,Z). Similarly, there are m (1) ∞ ,m (2) ∞ ∈ N such that the local exponents of (4.2) at the cusp ∞ are given by κ(1)∞ = −2m (1) ∞ +m (2) ∞ 3 , κ(2)∞ = κ(1)∞ +m(1) ∞ , κ(3)∞ = κ(2)∞ +m(2) ∞ . Given any λ, µ > 0, we define e−U1;λ,µ(z) := 1 4 ( λ2µ−1\|y1\|2 + µ2λ−1\|y2\|2 + λ−1µ−1\|y3\|2 ) , e−U2;λ,µ(z) := 1 4 [ λµ\|W (y1, y2)\|2 + λ−2µ\|W (y2, y3)\|2 + λµ−2\|W (y3, y1)\|2 ] , where W (yi, yj) := y′iyj − y′jyi. Note that e−Uk;λ,µ(z) is single-valued for any z ∈ H and 0 < e−Uk;λ,µ(z) < ∞ as long as z /∈ SL(2,Z)S. We have Lemma 4.5. Given any λ, µ > 0, there holds{ ∆U1;λ,µ + e2U1;λ,µ−U2;λ,µ = 0, ∆U2;λ,µ + e2U2;λ,µ−U1;λ,µ = 0 in H \ (SL(2,Z)S). (4.3) Modular Ordinary Differential Equations on SL(2,Z) of Third Order 27 Proof. The proof can be easily adopted from [7, 23, 24]; we sketch the proof here for the reader’s convenience. Given any λ, µ > 0, we define Wλ,µ := λ 3 2 y1 µ 3 2 y2 y3 λ 3 2 y′1 µ 3 2 y′2 y′3 λ 3 2 y′′1 µ 3 2 y′′2 y′′3  . Since the Wroksian W (y1, y2, y3) = 1, we have detWλ,µ = (λµ) 3 2 . Define a positive definite matrix Rλ,µ := (λµ)−1Wλ,µWλ,µ T . then detRλ,µ = 1. For 1 ≤ m ≤ 3, we let Rλ,µ;m denote the leading principal minor of Rλ,µ of dimension m. Since yj(z) is holomorphic in H \ (SL(2,Z)S), a direct computation leads to (see, e.g., [24]) Rλ,µ;m(∂zz̄Rλ,µ;m)− (∂zRλ,µ;m)(∂z̄Rλ,µ;m) = Rλ,µ;m−1Rλ,µ;m+1, m = 1, 2, (4.4) for z ∈ H \ (SL(2,Z)S), where Rλ,µ;0 := 1. On the other hand, 1 4 Rλ,µ;1 = 1 4 (λµ)−1 ( λ3\|y1\|2 + µ3\|y2\|2 + \|y3\|2 ) = e−U1;λ,µ(z). Define e−Vλ,µ(z) := 1 4Rλ,µ;2. we will prove that e−Vλ,µ(z) = e−U2;λ,µ(z). Note that Rλ,µ;3 = detRλ,µ = 1. Letting m = 1 in (4.4) leads to (note 0 < e−U1;λ,µ , e−Vλ,µ < +∞ in H \ (SL(2,Z)S)) 4e−Vλ,µ = Rλ,µ;2 = 16 [ e−U1;λ,µ ( ∂zz̄e −U1;λ,µ ) − ( ∂ze −U1;λ,µ )( ∂z̄e −U1;λ,µ )] = −16e−2U1;λ,µ∂zz̄U1;λ,µ = −4e−2U1;λ,µ∆U1;λ,µ in H \ (SL(2,Z)S), (4.5) and letting m = 2 in (4.4) leads to 4e−U1;λ,µ = Rλ,µ;1 = 16 [ e−Vλ,µ(∂zz̄e −Vλ,,µ)− ( ∂ze −Vλ,µ )( ∂z̄e −Vλ,µ )] = −16e−2Vλ,µ∂zz̄Vλ,µ = −4e−2Vλ,µ∆Vλ,µ in H \ (SL(2,Z)S). Furthermore, we insert e−U1;λ,µ = 1 4 ∑ \|ajyj \|2 (where a1 = λµ−1/2, a2 = µλ−1/2 and a3 = (λµ)−1/2) into (4.5), which leads to 1 4 e−Vλ,µ = e−U1;λ,µ ( ∂zz̄e −U1;λ,µ ) − ( ∂ze −U1;λ,µ )( ∂z̄e −U1;λ,µ ) = 1 16 [(∑ \|ajyj \|2 )(∑ \|ajy′j \|2 ) − (∑ a2jy ′ jyj )(∑ a2jyjy ′ j )] = 1 16 [ \|W (a1y1, a2y2)\|2 + \|W (a2y2, a3y3)\|2 + \|W (a3y3, a1y1)\|2 ] , so e−Vλ,µ(z) = e−U2;λ,µ(z). This proves that (U1;λ,µ(z), U2;λ,µ(z)) solves the Toda system (4.3). ■ Now any z̃ ∈ S, it follows from the local behavior of yj ’s that near γz̃, U1;λ,µ(z) = −2κ (1) z̃ ln \|z − γz̃\|+O(1), U2;λ,µ(z) = −2 ( 2− κ (3) z̃ ) ln \|z − γz̃\|+O(1). 28 Z. Chen, C.-S. Lin and Y. Yang Similarly, at the cusp ∞, we have U1;λ,µ(z) = −2κ(1)∞ ln \|q\|+O(1), U2;λ,µ(z) = 2κ(3)∞ ln \|q\|+O(1), where q = e2πiz. Since f , g, h, Wf,g,h are modular forms of weights k, k, k and 3(k + 2) respectively, we easily obtain \|yj(γz)\|2 = \|yj(z)\|2 \|cz + d\|4 , \|W (yi, yj)(γz)\|2 = \|W (yi, yj)(z)\|2 \|cz + d\|4 for any γ = ( a b c d ) ∈ SL(2,Z), so Uj;λ,µ(γz) = Uj;λ,µ(z) + 4 ln \|cz + d\|, j = 1, 2. Now we define (u1;λ,µ, u2;λ,µ) := (2U1;λ,µ − U2;λ,µ, 2U2;λ,µ − U1;λ,µ). Then we have{ ∆u1;λ,µ + 2eu1;λ,µ − eu2;λ,µ = 0, ∆u2;λ,µ + 2eu2;λ,µ − eu1;λ,µ = 0 in H \ (SL(2,Z)S), and near γz̃, u1;λ,µ(z) = 2m (1) z̃ ln \|z − γz̃\|+O(1), u2;λ,µ(z) = 2m (2) z̃ ln \|z − γz̃\|+O(1), while at the cusp ∞, u1;λ,µ(z) = 2m(1) ∞ ln \|q\|+O(1), u2;λ,µ(z) = 2m(2) ∞ ln \|q\|+O(1). Therefore, (u1;λ,µ, u2;λ,µ) is a solution of the following SU(3) Toda system ∆u1 + 2eu1 − eu2 = 4π ∑ γ ( m (1) i δγi +m(1) ρ δγρ + m∑ j=1 m(1) zj δγzj ) on H, ∆u2 + 2eu2 − eu1 = 4π ∑ γ ( m (2) i δγi +m(2) ρ δγρ + m∑ j=1 m(2) zj δγzj ) on H, uk(z) = 2m(k) ∞ ln \|q\|+O(1) as Im z → ∞, uj(γz) = uj(z) + 4 ln \|cz + d\|, ∀γ ∈ SL(2,Z). (4.6) Now consider the modular function w : H → C defined by w = w(z) := E4(z) 3 E4(z)3 − E6(z)2 . It is well known that w(z) is holomorphic, surjective and w(i) = 1, w(ρ) = 0, w(∞) = ∞. A direct computation gives w′(z) = −2πi E4(z) 2E6(z) E4(z)3 − E6(z)2 . Modular Ordinary Differential Equations on SL(2,Z) of Third Order 29 Denote pj := w(γzj). Then all points of {p1, . . . , pm, 1, 0} ⊂ C are distinct. Now we define (v1(w), v2(w)) for w ∈ C by uk(z) = vk(w(z)) + 2 ln \|w′(z)\|, z ∈ H. Since w(γz) = w(z) gives w′(γz) = (cz + d)2w′(z), it follows from uk(γz) = uk(z) + 4 ln \|cz + d\| that vk(w) is well-defined for w ∈ C. Now outside {p1, . . . , pm, 1, 0}, we have ∆uk(z) = 4∂zz̄uk(z) = 4∂ww̄vk(w)\|w′(z)\|2 = \|w′(z)\|2∆vk(w) = euk′ (z) − 2euk(z) = \|w′(z)\|2 ( evk′ (w) − 2evk(w) ) , so ∆vk(w) + 2evk(w) − evk′ (w) = 0, where {k, k′} = {1, 2}. Furthermore, since w′(zj) ̸= 0, we have that at w = pj = w(zj), vk(w) = uk(z)− 2 ln \|w′(z)\| = 2m(k) zj ln \|z − zj \|+O(1) = 2m(k) zj ln \|w − pj \|+O(1). At w = w(i) = 1, since ordi(w − 1) = 2 and ordiw ′ = 1, we have vk(w) = uk(z)− 2 ln \|w′(z)\| = 2 ( m (k) i − 1 ) ln \|z − i\|+O(1) = ( m (k) i − 1 ) ln \|w − 1\|+O(1). At w = w(ρ) = 0, since ordρw = 3 and ordρw ′ = 2, we have vk(w) = uk(z)− 2 ln \|w′(z)\| = 2 ( m(k) ρ − 2 ) ln \|z − ρ\|+O(1) = 2 ( m (k) ρ − 2 ) 3 ln \|w\|+O(1). At w = w(∞) = ∞, since w(z) = Cq−1(1 +O(q)), w′(z) = −2πiCq−1(1 +O(q)), where q = e2πiz and C ̸= 0 is a constant, so vk(w) = uk(z)− 2 ln \|w′(z)\| = 2 ( m(k) ∞ + 1 ) ln \|q\|+O(1) = −2 ( m(k) ∞ + 1 ) ln \|w\|+O(1). Therefore, (4.6) is equivalent to ∆v1 + 2ev1 − ev2 = 4π m (1) i − 1 2 δ1 + 4π m (1) ρ − 2 3 δ0 + 4π m∑ j=1 m(1) zj δpj in R2, ∆v2 + 2ev2 − ev1 = 4π m (2) i − 1 2 δ1 + 4π m (2) ρ − 2 3 δ0 + 4π m∑ j=1 m(2) zj δpj in R2, vk(w) = −2 ( m (k) ∞ + 1 ) ln \|w\|+O(1) as \|w\| → ∞. (4.7) Note from Proposition 4.3 that m (k) i + 1 = κ (k+1) i − κ (k) i ≡ 0 mod 2 and m(k) ρ + 1 = κ(k+1) ρ − κ(k)ρ ≡ 0 mod 3, so m (k) i −1 2 ∈ Z≥0 and m (2) ρ −2 3 ∈ Z≥0 for k = 1, 2. In conclusion, starting from any given linearly independent modular forms f, g, h ∈ Mk(SL(2,Z)), we can construct a two-parametric family of solutions to certain Toda system (4.7): Theorem 4.6. (v1;λ,µ(w), v2;λ,µ(w)) defined by vk;λ,µ(w(z)) = uk;λ,µ(z)− 2 ln \|w′(z)\|, z ∈ H are a two-parametric family of solutions of the SU(3) Toda system (4.7), where λ, µ > 0 can be arbitrary. 30 Z. Chen, C.-S. Lin and Y. Yang 5 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. 5.1 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 z0 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 w̃ as introduced in [25] rather than z − z0. Fix z0 ∈ H, we let w = (z − z0)/(z − z0) and w̃ = w 1 +Aw , with A = 4πϕ∗(z0) Im z0 α , (5.1) where ϕ(z) is the quasimodular form of weight 2 and depth 1 on Γ, i.e., ( ϕ ∣∣ 2 γ ) (z) = ϕ(z) + α0c 2πi(cz + d) , γ = ( a b c d ) ∈ Γ, for some nonzero complex number α0 (note α0 = 2πiα for α given in (2.1)), and ϕ∗(z) := ϕ(z) + α0 2πi(z−z̄) . Clearly ϕ∗(z) satisfies( ϕ∗∣∣ 2 γ ) (z) = ϕ∗(z), γ ∈ Γ. The advantage of the expansion in terms of w̃ is the following result. Proposition 5.1 ([25, Propositions A.4 and A.7]). Let f(z) be a meromorphic modular form of weight k on Γ. Then f(z) admits an expansion of the form f(z) = (1− (1 +A)w̃)k ∞∑ n=n0 an(−4π(Im z0)w̃) n. Furthermore, if z0 is an elliptic point with the stabilizer subgroup Γz0 of order N , then an = 0 whenever k + 2n ̸≡ 0 mod N . Remark 5.2. Note that when f(z) is a holomorphic modular form, the coefficients an 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 z0 is an elliptic point, we have A = 0 and hence w̃ = 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 Q(z) := Q2(z) (2πi)2 = (1− (1 +A)w̃)4 ∞∑ n=−2 an(−4π(Im z0)w̃) n, R(z) := Q3(z)− 1 2Q ′ 2(z) (2πi)3 = (1− (1 +A)w̃)6 ∞∑ n=−3 bn(−4π(Im z0)w̃) n. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 31 Then (1.1) is equivalent to D3 qy(z) +Q(z)Dqy(z) + ( 1 2 DqQ(z) +R(z) ) y(z) = 0, (5.2) where Dq := q d dq = 1 2πi d dz . In particular, Proposition 5.1 yields Corollary 5.3. Suppose −I2 ∈ Γ and let z0 be the elliptic point of order e of Γ. Then N = 2e and so an = 0 if n ̸≡ e− 2 mod e and bn = 0 if n ̸≡ e− 3 mod e. 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 w = γz = (az + b)/(cz + d). Then for a (r + 1)-th differentiable function g(z), we have dr+1 dzr+1 ( (det γ)r/2 (a− cw)r g(w) ) = (a− cw)r+2 (det γ)r/2+1 dr+1 dwr+1 g(w). Proof. Bol’s identity states that( y ∣∣ −r γ )(r+1) (z) = ( y(r+1) ∣∣ r+2 γ ) (z). Noticing that a− cw = a− c az + b cz + d = det γ cz + d , we find that the factor (det γ)1/2/(cz + d) appearing in the slash operator can be written as (det γ)1/2 cz + d = a− cw (det γ)1/2 , (5.3) 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 Q̃(x) = ∑ n≥n0 anx n and R̃(x) = ∑ n≥n0 bnx n are the power series such that Q(z) = (1− (1 +A)w̃)4Q̃(−4π(Im z0)w̃), R(z) = (1− (1 +A)w̃)6R̃(−4π(Im z0)w̃). (5.4) Then y(z) = 1 (1− (1 +A)w̃)2 ∞∑ n=0 cn(−4π(Im z0)w̃) n+α (5.5) is a solution of (5.2) if and only if the series ỹ(x) = ∑∞ n=0 cnx n+α satisfies d3 dx3 ỹ(x) + Q̃(x) d dx ỹ(x) + ( 1 2 d dx Q̃(x) + R̃(x) ) ỹ(x) = 0. (5.6) 32 Z. Chen, C.-S. Lin and Y. Yang Proof. Let γ = ( −4π Im z0 0 0 1 )( 1 0 A 1 )( 1 −z0 1 −z0 ) = ( −4π Im z0 (4π Im z0)z0 1 +A −Az0 − z0 ) with det γ = −4π(Im z0)(z0 − z0). Let x = γz = −4π(Im z0)w̃. Note that if we write γ as γ = ( a b c d ) , then (a− cx)2 det γ = 2πi(1− (1 +A)w̃)2. Thus, by (5.3), y(z) and ỹ(x) are related by y(z) = 2πi ( ỹ ∣∣ −2 γ ) (z). Hence, applying Lemma 5.4 with r = 2, we obtain D3 qy(z) = 1 (2πi)2 d3 dz3 ( ỹ ∣∣ −2 γ ) (z) = (1− (1 +A)w̃)4 d3 dx3 ỹ(x). Also, a direct computation yields, by (5.3), dx dz = dγz dz = det γ (cz + d)2 = (a− cx)2 det γ = 2πi(1− (1 +A)w̃)2, and dw̃ dz = − 1 4π Im z0 dx dz = (1− (1 +A)w̃)2 2i Im z0 . Hence, Dqy(z) = 1 2πi d dz ( 1 (1− (1 +A)w̃)2 ỹ(x) ) = − 1 +A 2π(Im z0)(1− (1 +A)w̃) ỹ(x) + d dx ỹ(x), and DqQ(z) = 1 2πi d dz ( (1− (1 +A)w̃)4Q̃(x) ) = (1− (1 +A)w̃)6 ( 1 +A π(Im z0)(1− (1 +A)w̃) Q̃(x) + d dx Q̃(x) ) . Putting everything together, we find that D3 qy(z) +Q(z)Dqy(z) + ( 1 2 Dq(z) +R(z) ) y(z) = (1− (1 +A)w̃)4 ( d3 dx3 ỹ(x) + Q̃(x) d dx ỹ(x) + ( 1 2 d dx Q̃(x) + R̃(x) ) ỹ(x) ) . Thus, y(z) is a solution of (5.2) if and only if ỹ(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 w̃ rather than z − z0. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 33 Remark 5.6. In practice, the power series Q̃(x) and R̃(x) can be computed using Proposi- tion A.4 of [25]. For example, for the Eisenstein series E4(z) and E6(z) on SL(2,Z), we find that E4(z) = (1− (1 +A)w̃)4 ( B − 1 3 Cu+ 5 72 B2u2 − 5 432 BCu3 + · · · ) , E6(z) = (1− (1 +A)w̃)6 ( C − 1 2 B2u+ 7 48 BCu2 + · · · ) , (5.7) where u = −4π(Im z0)w̃, B = E4(z0), and C = E6(z0). 5.2 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 zj , 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 = zj , i, or ρ, we assume that κ (1) z < κ (2) z < κ (3) z , are rational numbers in 1 3Z such that κ (1) z + κ (2) z + κ (3) z = 3 and κ (2) z − κ (1) z ∈ Z (and hence κ (3) z − κ (1) z ∈ Z). When z = i, we further assume that{ 3κ (1) i , 3κ (2) i , 3κ (3) i } ≡ {0, 0, 1} mod 2. (5.8) 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{ κ(1)ρ , κ(2)ρ , κ(3)ρ } ≡ {0, 1, 2} mod 3. (5.9) Finally, for the cusp ∞ of SL(2,Z), we let κ (1) ∞ ≤ κ (2) ∞ ≤ κ (3) ∞ be three rational numbers in 1 3Z such that κ (1) ∞ + κ (2) ∞ + κ (3) ∞ = 0 and κ (2) ∞ − κ (1) ∞ , κ (3) ∞ − κ (2) ∞ ∈ 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 zj, i, or ρ and are holomorphic at other points and cusps are of the form Q(z) = r∞E4(z) + r (2) i E4(z)∆0(z) E6(z)2 + r(2)ρ ∆0(z) E4(z)2 + m∑ j=1 ( r(2)zj E4(z)∆0(z) 2 Fj(z)2 + r(1)zj E4(z)∆0(z) Fj(z) ) , R(z) = s∞E6(z) + s (3) i ∆0(z) 2 E6(z)3 + s (1) i ∆0(z) E6(z) + s(3)ρ E6(z)∆0(z) E4(z)3 + m∑ j=1 3∑ k=1 s(k)zj E6(z)∆0(z) k Fj(z)k , (5.10) where ∆0(z) = 1728∆(z) = (E4(z) 3 − E6(z) 2) and Fj(z) = E4(z) 3 − tjE6(z) 2 with tj = E4(zj) 3/E6(zj) 2. 34 Z. Chen, C.-S. Lin and Y. Yang 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 (2) zj , r (1) zj such that Q(z)− r (2) i E4(z)∆0(z) E6(z)2 − r(2)ρ ∆0(z) E4(z)2 − m∑ j=1 ( r(2)zj E4(z)∆0(z) 2 Fj(z)2 + r(1)zj E4(z)∆0(z) Fj(z) ) is a holomorphic modular form of weight 4 on SL(2,Z), so it must be a multiple of E4(z). The proof for R(z) is similar. ■ We first determine the indicial equations of (5.2) at ∞, i, ρ, and zj , j = 1, . . . ,m. 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 x3 + r∞x+ s∞ = 0. 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) = qα(1 +O(q)), α ∈ R. We compute that D3 qy(z) +Q(z)Dqy(z) + ( 1 2 DqQ(z) +R(z) ) y(z) = α3qα + r∞αqα + s∞qα +O ( qα+1 ) , from which we see that the indicial equation at ∞ is x3 + r∞x+ s∞ = 0. ■ We now consider the indicial equation of (5.2) at a point in H. Let z0 be one of zj , i, or ρ and w̃ be defined by (5.1) with ϕ∗(z) = E∗ 2(z) := E2(z)+6/(πi(z−z)). Recall that in Lemma 5.5 we have proved that if Q̃(x) and R̃(x) are the Laurent series in x such that (5.4) holds. Then y(z;α) = 1 (1− (1 +A)w̃)2 ∞∑ n=0 cn(α)(−4π(Im z0)w̃) n+α, c0(α) = 1, is a solution of (5.2) if and only if the series ỹ(x;α) = ∑∞ n=0 cn(α)x n+α satisfies d3 dx3 ỹ(x;α) + Q̃(x) d dx ỹ(x;α) + ( 1 2 d dx Q̃(x) + R̃(x) ) ỹ(x;α) = 0. (5.11) Let Q̃(x) = ∞∑ n=−2 anx n, 1 2 d dx Q̃(x) + R̃(x) = ∞∑ n=−3 bnx n, (5.12) where each an and bn 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 Rn(α) := f(α+ n)cn(α) + n−1∑ k=0 [(α+ k)an−k−2 + bn−k−3]ck(α) = 0 (5.13) for all n ≥ 0, where f(t) := t(t − 1)(t − 2) + a−2t + b−3. In particular, R0(α) : f(α) = 0 is the indicial equation at z0, i.e., f(t) = ( t− κ(1)z0 )( t− κ(2)z0 )( t− κ(3)z0 ) . Using (5.7), we can work out the coefficients a−2 and b−3. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 35 Lemma 5.9. Suppose that Q(z) and R(z) are meromorphic modular forms given by (5.10). Then the indicial equations of (5.2) at i, ρ, and zj, j = 1, . . . ,m, are x(x− 1)(x− 2) + 4r (2) i x− 4r (2) i − 8s (3) i = 0, x(x− 1)(x− 2)− 9r(2)ρ x+ 9r(2)ρ + 27s(3)ρ = 0, x(x− 1)(x− 2) + r (2) zj tj x− r (2) zj tj + s (3) zj t2j = 0, respectively. Proof. For the elliptic point i, we know that E6(i) = 0. Hence, using (5.7), we find that Q̃(x) = 4r (2) i x2 + . . . , R̃(x) = − 8s (2) i x3 + · · · . Therefore, we have a−2 = 4r (2) i and b−3 = −4r (2) i −8s (2) i . Similarly, for the elliptic point ρ, using E4(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 zj , letting B = E4(zj) and C = E6(zj), we compute that ∆0(z) (1− (1 +A)w̃)12 = ( B3 − C2 ) +O(w̃) = C2(tj − 1) +O(w̃), Fj(z) (1− (1 +A)w̃)12 = ( B − 1 3 Cu+O ( w̃2 ))3 − tj ( C − 1 2 B2u+O ( w̃2 ))2 = (tj − 1)B2Cu+O ( w̃2 ) , where u = −4π(Im zj)w̃. It follows that Q̃(x) = r(2)zj (tj − 1)2BC4 (tj − 1)2B4C2x2 + · · · = r (2) zj tjx2 + . . . , and R̃(x) = s(3)zj (tj − 1)3C7 (tj − 1)3B6C3 + · · · = s (3) zj t2jx 3 + · · · . Therefore, a−2 = r (2) zj /tj and b−3 = −r (2) zj /tj + s (3) zj /t 2 j . This proves the lemma. ■ The two lemmas show that the parameters r∞, s∞, r (2) i , s (3) i , r (2) ρ , s (3) ρ , r (2) zj , and s (3) zj , j = 1, . . . ,m, solely depend on the local exponents κ’s. The remaining parameters are{ r (1) zj , s (2) zj , s (1) zj , for j = 1, . . . ,m, s (1) i . That is, the number of remaining parameters is 3m+1. We now show that the apparentness con- dition will impose 3m+1 polynomial constraints on the remaining parameters. For convenience, we let r and s denote r (1) z1 , . . . , r (1) zm and s (1) i , s (2) z1 , s (1) z1 , . . . , s (2) zm , s (1) zm , respectively. Let z0 ∈ {i, ρ, z1, . . . , zm}. Assume that α is one of the local exponents κ (k) z0 . We observe that one can recursively determine cn(α) using (5.13) as long as f(α + n) ̸= 0. Thus, there always exists a solution ỹ ( x;κ (3) z0 ) with local exponent κ (3) z0 ; see, e.g., Lemma A.2. For the exponent α = κ (2) z0 , because f ( α+ κ(3)z0 − κ(2)z0 ) = f ( κ(3)z0 ) = 0, 36 Z. Chen, C.-S. Lin and Y. Yang we can only solve for cn(α) up to n = κ (3) z0 − κ (2) z0 − 1. At n′ = κ (3) z0 − κ (2) z0 , the relation Rn′(α) becomes n′−1∑ k=0 [(α+ k)an′−k−2 + bn′−k−3]ck(α) = 0. (5.14) If ck(α), k = 0, . . . , n′−1, do not satisfy this relation, then there is no solution with exponent κ (2) z0 . If ck(α), k = 0, . . . , n′− 1, satisfy this relation, we then can choose cn′(α) to be any number and solve for cn(α) recursively and obtain a solution ỹ ( x;κ (2) z0 ) with local exponent κ (2) z0 . Likewise, for the local exponent α = κ (1) z0 , there will be two conditions (5.14) corresponding to n′ = κ (2) z0 −κ (1) z0 and n′ = κ (3) z0 − κ (1) z0 that ck(α) must satisfy. Thus, there are three polynomial equations Pz0,k1,k2(r, s) = 0, (k1, k2) ∈ {(1, 2), (1, 3), (2, 3)} that r and s need to satisfy. However, we will see in a moment that when z0 is an elliptic point, some or all of the three polynomial equations hold trivially. Lemma 5.10. Assume that z0 is an elliptic point of order e of SL(2,Z). Let (k1, k2) ∈ {(1, 2), (1, 3), (2, 3)}. If κ (k2) z0 − κ (k1) z0 ̸≡ 0 mod e, then the polynomial Pz0,k1,k2(r, s) is identically zero. In particular, under our assumptions (5.8) and (5.9), the differential equation (5.2) is apparent at ρ for any r and s, and also there is at most one pair (k1, k2) such that Pi,k1,k2(r, s) ̸= 0. Proof. By Corollary 5.3, the coefficients an in the expansion of Q̃(x) vanish whenever n ̸≡ −2 mod e. Likewise, the coefficients bn in R̃(x) vanish whenever n ̸≡ −3 mod e. Using these facts, we find that the condition (5.13) reduces to f(α+ n)cn(α) + ∑ k≡n mod e, k≤n−1 [(α+ k)an−k−2 + bn−k−3]ck(α) = 0. (5.15) Then we can easily prove by induction up to n = n′ − 1 that cn(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 an′−k−2 and bn′−k−3 are nonzero only when k ≡ n′ mod e and ck(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 Pi(r, s) denote the only nonzero polynomial Pi,k1,k2(r, s) in the lemma. The discussion above shows that there are 3m + 1 polynomial equations Pi(r, s) = 0, Pzj ,k1,k2(r, s) = 0, j = 1, . . . ,m, (k1, k2) ∈ {(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, z1, . . . , zm} with the given local exponents if and only if the parameters r and s are com- mon roots of the polynomials. We now consider the degree of these polynomials. Proposition 5.11. We have degPzj ,k1,k2(r, s) = { κ (k2) zj − κ (k1) zj , if (k1, k2) = (1, 2) or (2, 3), κ (3) zj − κ (1) zj − 1, if (k1, k2) = (1, 3), and degPi(r, s) = ( κ (k2) i − κ (k1) i ) /2, Modular Ordinary Differential Equations on SL(2,Z) of Third Order 37 where (k1, k2) is the unique pair such that κ (k2) i − κ (k1) i ≡ 0 mod 2. To be more precise, for a polynomial P (r, s) in r and s, we let LT(P ) denote the sum of the terms of highest degree in P . Then, up to nonzero scalars, we have LT(Pzj ,k1,k2(r, s)) = κ (k2) zj −κ (k1) zj∏ k=1 (( κ(k1)zj + k − 3/2 ) d1r (1) zj + d2s (2) zj ) for (k1, k2) = (1, 2) or (2, 3), and LT(Pzj ,1,3(r, s)) = ( d3s (1) zj + d′1r (1) z1 + · · ·+ d′mr(1)zm ) × ∏ k=1, k ̸=κ (2) zj −κ (1) zj ,κ (2) zj −κ (1) zj +1 (( κ(k1)zj + k − 3/2 ) d1r (1) zj + d2s (2) zj ) , where d1, d2, d3, d ′ 1, . . . , d ′ m are complex numbers with d1, d2, d3 ̸= 0. Also, LT(Pi(r, s)) is a product of ( κ (k2) i −κ (k1) i ) /2 linear sums in s (1) i and r with the coefficients of s (1) i being nonzero. Proof. By Lemma 5.7, each of an and bn in (5.12) is a linear combination of r’s and s’s. The key observation here is that only r (2) zj (which has been determined by the local exponents and is regarded as a constant) appears in a−2, only r (2) zj and r (1) zj appear in a−1, only r (2) zj and s (3) zj (which has been determined by the local exponents and is regarded as a constant) appear in b−3, only r (2) zj , r (1) zj , s (3) zj , and s (2) zj appear in b−2, and only r (2) zj , r (1) zj , s (3) zj , s (2) zj , and s (1) zj appear in b−1. In particular, we have LT(a−1) = d1r (1) zj , LT(b−2) = −1 2 d1r (1) zj + d2s (2) zj , (5.16) where d1 and d2 are the leading coefficients in the series expansions of E4(z)∆(z)/Fj(z) and E6(z)∆(z)2/Fj(z) 2 in w̃ (and hence are nonzero). Consider (5.14) for the cases (α, n′) =( κ (2) zj , κ (3) zj − κ (2) zj ) and (α, n′) = ( κ (1) zj , κ (2) zj − κ (1) zj ) first. From (5.13) and (5.16), we can eas- ily show inductively that, up to n = n′ − 1, LT(cn(α)) = n∏ k=1 ( −LT((α+ k − 1)a−1 + b−2) f(α+ k) ) = n∏ k=1 ( − (α+ k − 3/2)d1r (1) zj + d2s (2) zj f(α+ k) ) , (5.17) where d1 and d2 are the two nonzero complex numbers in (5.16), and hence LT(Pzj ,k1,k2(r, s)) = κ (k2) z0 −κ (k1) z0∏ k=1 ( − (α+ k − 3/2)d1r (1) zj + d2s (2) zj f(α+ k) ) for (k1, k2) = (1, 2) or (2, 3). We now consider (5.14) for the remaining case (α, n′) = ( κ (1) zj , κ (3) zj −κ (1) zj ) . Let n′′ = κ (2) zj −κ (1) zj . Up to n = n′′ − 1, the terms of highest degree in cn(α) is given by (5.17). Since Pzj ,1,2(r, s) = 0 is assumed to hold, (5.13) holds for n = n′′ for arbitrary cn′′(α). Here, we simply choose cn′′(α) to be 0. Then we have, by (5.13) cn′′+1(α) = − 1 f(α+ n′′ + 1) n′′−1∑ k=0 [(α+ k)an′′−k−1 + bn′′−k−2]ck(α), 38 Z. Chen, C.-S. Lin and Y. Yang and hence LT(cn′′+1(α)) = − 1 f(α+ n′′ + 1) LT((α+ n′′ − 1)a0 + b−1) LT(cn′′−1(α)), which, by (5.17), is equal to −LT((α+ n′′ − 1)a0 + b−1) f(α+ n′′ + 1) n′′−1∏ k=1 ( − (α+ k − 3/2)d1r (1) zj + d2s (2) zj f(α+ k) ) . Then using (5.13) we can inductively show that for n with n′′ + 1 ≤ n ≤ n′ − 1, LT(cn(α)) = LT(cn′′+1(α)) n∏ k=n′′+2 ( −LT((α+ k − 1)a−1 + b−2) f(α+ k) ) = LT(cn′′+1(α)) n∏ k=n′′+2 ( − (α+ k − 3/2)d1r (1) zj + d2s (2) zj f(α+ i) ) . Thus, for (k1, k2) = (1, 3), we have LT(Pzj ,1,3(r, s)) = LT(cn′′+1(α)) n′∏ k=n′′+2 ( − (α+ k − 3/2)d1r (1) zj + d2s (2) zj f(α+ k) ) = −LT((α+ n′′ − 1)a0 + b−1) f(α+ n′′ + 1) × n′∏ k=1, k ̸=n′′,n′′+1 ( − (α+ k − 3/2)d1r (1) zj + d2s (2) zj f(α+ k) ) . Note that LT((α+ n′′ − 1)a0 + b−1) is of the form d3s (1) zj + d′1r (1) z1 + · · ·+ d′mr(1)zm , where d3, d ′ 1, . . . , d ′ m are complex numbers with d3 ̸= 0. We next consider the case where z0 = i is the elliptic point of order 2. There exists a unique pair of k1 and k2 with k1 < k2 such that κ (k2) i − κ (k1) i ≡ 0 mod 2. We let α = κ (k1) i and n′ = κ (k2) i −κ (k1) z0 . We have seen earlier that cn(α) ̸= 0 only when 2\|n. Also, from (5.15), we can inductively show that LT(cn(α)) = n/2∏ k=1 ( −LT((α+ 2k − 2)a0 + b−1) f(α+ 2k) ) and LT(Pi(r, s)) = n′/2∏ k=1 ( −LT((α+ 2k − 2)a0 + b−1) f(α+ 2k) ) . Noting that LT((α + 2k − 2)a0 + b−1)) = ds (1) i + (a linear sum in r) for some nonzero complex number d (which is the leading coefficient of the expansion of ∆(z)/E6(z) at i in w̃), we conclude that LT(Pi(r, s) is a product of ( κ (k2) i − κ (k1) i ) /2 linear sums in s (1) i and r with all coefficients of s (1) i nonzero. This completes the proof. ■ Modular Ordinary Differential Equations on SL(2,Z) of Third Order 39 Remark 5.12. The proposition suggests that when the local exponents κ’s are fixed, for generic points zj , the number of pairs (Q,R) of modular forms such that (5.2) satisfies the conditions (S1)–(S3) is κ (k2) i − κ (k1) i 2 m∏ j=1 ( κ(2)zj − κ(1)zj )( κ(3)zj − κ(2)zj )( κ(3)zj − κ(1)zj − 1 ) , where k1 and k2 are the integers such that κ (k2) i − κ (k1) i ∈ 2N. (Notice that if ( κ (1) z , κ (2) z , κ (3) z ) = (0, 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. 6 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 Q2(z), Q3(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 ∈ M̃≤r k (SL(2,Z)) is said to be extremal if its vanishing order at ∞ is equal to dim M̃≤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 M̃≤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 D3 qy(z) +Q(z)Dqy(z) + ( 1 2 DqQ(z) +R(z) ) y(z) = 0 (6.1) be the differential equation satisfied by f(z)/ 3 √ Wf (z) as derived in Section 2. (i) If k ≡ 0 mod 4, then Q(z) = −k2 48 E4(z), R(z) = − k3 864 E6(z). (ii) If k ≡ 2 mod 4, then Q(z) = −(k − 2)2 48 E4(z)− 1 3 E4(z) ( E4(z) 3 − E6(z) 2 ) E6(z)2 , and R(z) = −(k − 2)3 864 E6(z) + 5 54 ( E4(z) 3 − E6(z) 2 )2 E6(z)3 + 12− (k − 2)2 144 E4(z) 3 − E6(z) 2 E6(z) . 40 Z. Chen, C.-S. Lin and Y. Yang Remark 6.3. We note that the differential equation in the case k ≡ 0 mod 4 is equivalent to the following equation studied by Kaneko and Koike [18, Theorem 3.1]: D3 qf − k 4 E2D 2 qf + k(k − 1) 4 DqE2Dqf − k(k − 1)(k − 2) 24 D2 qE2f = 0. Indeed, by letting f(z) = ∆(z) k 12 y(z), a direct computation shows that y(z) solves D3 qy − k2 48 E4(z)Dqy − 2k2 123 (3E2(z)E4(z) + (k − 3)E6(z)) y = 0. To prove Theorem 6.2, we need the following general lemma, in which the quasimodular form f(z) is not assumed to be extremal. Lemma 6.4. Assume that f(z) = f0(z) + f1(z)E2(z) + f2(z)E2(z) 2 ∈ M̃≤2 k (SL(2,Z)), fj(z) ∈ Mk−2j(SL(2,Z)). Let g(z) = f1(z) + 2f2(z)E2(z), h(z) = f2(z), m = min(ord∞ f, ord∞ g, ord∞ h). Let κ (1) ∞ ≤ κ (2) ∞ ≤ κ (3) ∞ be the local exponents of (2.4) at ∞. (i) If ord∞ f = m, then ord∞Wf = 3m and κ (j) ∞ = 0 for all j. (ii) If ord∞ g = m and ord∞ f ̸= m, then ord∞Wf = ord∞ f + 2ord∞ g and κ (1) ∞ = κ (2) ∞ = (ord∞ g − ord∞ f)/3 and κ (3) ∞ = 2(ord∞ f − ord∞ g)/3. (iii) If ord∞ h < ord∞ f ≤ ord∞ g, then ord∞Wf = 2ord∞ f + ord∞ h and κ (1) ∞ = 2(ord∞ h− ord∞ f)/3 and κ (2) ∞ = κ (3) ∞ = (ord∞ f − ord∞ h)/3. (iv) If ord∞ h < ord∞ g < ord∞ f , then ord∞Wf = ord∞ f + ord∞ g + ord∞ h and κ (1) ∞ = ord∞ h− 1 3 ord∞Wf , κ (2) ∞ = ord∞ g − 1 3 ord∞Wf , and κ (3) ∞ = ord∞ f − 1 3 ord∞Wf . Proof. Let r = 1 3 ord∞Wf . Since up to scalars, g3(z) = f(z)/ 3 √ Wf (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 (3) ∞ = ord∞ f−r. Likewise, because g2(z) = (2zf(z) + αg(z))/ 3 √ Wf (z) and g1(z) = ( z2f(z) + αzg(z) + α2h(z) ) / 3 √ Wf (z) are the other two linearly independent solutions of (2.4), we have κ(2)∞ = min(ord∞ f, ord∞ g)− r, κ(1)∞ = min(ord∞ f, ord∞ g, ord∞ h)− r. Analyzing case by case, we obtain the claimed conclusions. ■ Proof of Theorem 6.2. First of all, recall that dim M̃≤2 k (SL(2,Z)) = 1 + ⌊ k 4 ⌋ . (6.2) Let f(z) = f0(z) + f1(z)E2(z) + f2(z)E2(z) 2, fj ∈ Mk−2j(SL(2,Z)), be an extremal quasimod- ular form in M̃≤2 k (SL(2,Z)). Note that fj(z) cannot have a common zero on H. To see this, say, assume that fj(z) has a common zero at z0. Let F (z) be a modular form of weight k′ Modular Ordinary Differential Equations on SL(2,Z) of Third Order 41 with a simple zero at z0 and nonvanishing elsewhere. Then f(z)/F (z) ∈ M̃≤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, fj(z) have no common zeros on H. Now according to Pellarin’s argument [28], one has ord∞Wf = ord∞ f = ⌊k/4⌋. Hence, we have Wf (z) = { c∆(z)k/4, if k ≡ 0 mod 4, c∆(z)(k−2)/4E6(z), if k ≡ 2 mod 4, for some nonzero complex number c. Also, by Lemma 6.4, we must have ord∞(f1 + 2f2E2) = 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 x3 − r2 3 x− 2r3 27 = 0. (6.3) 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 E4(z), while R(z) is a multiple of E6(z). In view of (6.3) and Lemma 5.8, we see that Q(z) = −k2 48 E4(z), R(z) = − k3 864 E6(z). We now consider the case k ≡ 2 mod 4. In this case, Wf (z) = c∆(z)(k−2)/4E6(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 x3 − 3x2 + 2 3 x+ 16 27 = 0. (6.4) We will use this information, together with the apparentness property, to determine Q(z) and R(z). First of all, according to Lemma 5.7, Q(z) is of the form Q(z) = r∞E4(z) + r (2) i E4(z) ( E4(z) 3 − E6(z) 2 ) E6(z)2 , while R(z) is of the form R(z) = s∞E6(z) + s (3) i ( E4(z) 3 − E6(z) 2 )2 E6(z)3 + s (1) i E4(z) 3 − E6(z) 2 E6(z) for some complex numbers r∞, r (2) i , s∞, s (3) i , and s (1) i . The parameters r∞ and s∞ are determined by the local exponents at ∞. As in the case k ≡ 0 mod 4, we find that r∞ = − (k−2)2 48 and s∞ = − (k−2)3 864 . We now determine the other parameters. By (6.4) and Lemma 5.9, we have r (2) i = −1 3 and s (3) i = 5 54 . To determine the remaining parameter s (1) i , we let w = (z − i)/(z + i) and recall that, by (5.7), E4(z) = (1− w)4 ( B + 5 72 B2u2 + 5 6912 B3u4 + · · · ) and E6(z) = (1− w)6 ( −1 2 B2u− 7 432 B3u3 − 7 17280 B4u5 + · · · ) , 42 Z. Chen, C.-S. Lin and Y. Yang where u = −4πw and B = E4(i). (Note that E6(z0) and the constant A in Lemma 5.5 are both 0 when z0 = i.) Then the power series Q̃(x) and R̃(x) such that Q(z) = (1−w)4Q̃(−4πw) and R(z) = (1− w)6R̃(−4πw) are Q̃(x) = 4r (2) i x2 + ( r∞ − 4r (2) i 27 ) B + · · · = − 4 3x2 + ( −(k − 2)2 48 + 4 81 ) B + · · · and R̃(x) = − 8s (3) i x3 + ( −2s (1) i + 13 9 s (3) i ) B x + · · · = − 20 27x3 + ( −2s (1) i + 65 486 ) B x + · · · , respectively. By Lemma 5.5, the series (5.5) with c0 = 1 is a solution of (6.1) if and only if the power series ỹ(x) = ∑∞ n=0 cnx n+α satisfies (5.6). Consider ỹ(x) with α = 2/3. The coefficients cn need to satisfy ∞∑ n=0 cn(n+ 2/3)(n− 1/3)(n− 4/3)xn−7/3 + ( − 4 3x2 + ( 4 81 − (k − 2)2 48 ) B + · · · ) ∞∑ n=0 cn(n+ 2/3)xn−1/3 + ( 16 27x3 + ( 65 486 − 2s (1) i ) B x + · · · ) ∞∑ n=0 cnx n+2/3 = 0. Considering the coefficients of x−1/3, we find that s (1) i = 12−(k−2)2 144 . This completes the proof of the theorem. ■ A The solution structure of third order ODE In this appendix, we apply Frobenius’ method to study the solution structure for Ly := d3 dx3 y(x) +Q(x) d dx y(x) +R(x)y(x) = 0. (A.1) See, e.g., [15, 16] for detailed expositions of Frobenius’ method. The following arguments are known to experts in this field. However, since we can not find a suitable reference, we would like to provide all necessary details for later usage. Suppose 0 is a regular singular point of (A.1) with three local exponents κ1, κ2 = κ1 +m1, κ3 = κ2 +m2, where m1,m2 ∈ Z≥0, Since the exponent differences are all integers, there might be logarithmic singularities for solu- tions of (A.1), or more precisely, the local expansion of some solutions at x = 0 might contains lnx terms or even (lnx)2 terms. This leads us to give the following definition. Definition A.1. (1) We say (A.1) is apparent at x = 0 if all solutions have no logarithmic singularities at x = 0. Otherwise (A.1) is called not apparent at x = 0. (2) If (A.1) is not apparent at x = 0 and the local expansion of some solutions contains (lnx)2 terms, we say (A.1) is completely not apparent at x = 0. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 43 Since 0 is a regular singular point of (A.1), both x2Q(x) and x3R(x) are holomorphic at x = 0, so we may write Q(x) = ∞∑ n=−2 anx n, R(x) = ∞∑ n=−3 bnx n. Let y(x;α) = xα ∞∑ n=0 cn(α)x n, where c0(α) = 1. Then Ly(x;α) = ∞∑ n=0 ( f(α+ n)cn(α) + n−1∑ k=0 [(α+ k)an−k−2 + bn−k−3]ck(α) ) xn+α−3 =: xα−3 ∞∑ n=0 Rn(α)x n, (A.2) where f(s) := s(s− 1)(s− 2) + sa−2 + b−3 = 3∏ j=1 (s− κj), i.e., f(s) = 0 is the indicial equation of (A.1) at x = 0. Note R0(α) = f(α). For any α satisfying \|α− κ3\| < 1/2, we have f(α+ n) ̸= 0 for any n ≥ 1, so by letting Rn(α) = f(α+ n)cn(α) + n−1∑ k=0 [(α+ k)an−k−2 + bn−k−3]ck(α) = 0, n ≥ 1, we see that cn(α) can be uniquely solved for any n ≥ 1 such that Ly(x;α) = xα−3f(α), for any \|α− κ3\| < 1/2. (A.3) Note that cn(α) ∈ C(α) is a rational function of α for any n ≥ 1. In particular, letting α = κ3 in (A.3) leads to Ly(x;κ3) = 0, so Lemma A.2. y(x;κ3) = xκ3 ∞∑ n=0 cn(κ3)x n is always a solution of (A.1) with the local exponent κ3. By (A.3), we have L∂y(x;α) ∂α = xα−3f(α) lnx+ xα−3f ′(α), so L∂y(x;α) ∂α ∣∣∣ α=κ3 = xκ3−3f ′(κ3). (A.4) 44 Z. Chen, C.-S. Lin and Y. Yang Similarly, L∂2y(x;α) ∂α2 ∣∣∣ α=κ3 = 2xκ3−3f ′(κ3) lnx+ xκ3−3f ′′(κ3). (A.5) Note that ∂y(x;α) ∂α ∣∣∣ α=κ3 = (lnx)y(x; k3) + xκ3 ∞∑ n=1 c′n(κ3)x n, (A.6) ∂2y(x;α) ∂α2 ∣∣∣ α=κ3 = (lnx)2y(x; k3) + 2(lnx)xκ3 ∞∑ n=1 c′n(κ3)x n + xκ3 ∞∑ n=1 c′′n(κ3)x n. (A.7) Theorem A.3. If κ1 = κ2 = κ3, then ∂y(x;α) ∂α \|α=κ3 and ∂2y(x;α) ∂α2 \|α=κ3 given in (A.6) and (A.7) are the other two linearly independent solutions of (A.1), namely (A.1) is completely not ap- parent at x = 0. Proof. Since f(s) = (s−κ3) 3, we have f ′(κ3) = f ′′(κ3) = 0, so this theorem follows from (A.4) and (A.5). ■ Next we consider the case κ1 < κ2 = κ3, i.e., m1 > 0, m2 = 0 and f(s) = (s− κ1)(s− κ3) 2. Then f ′(κ3) = 0 and f ′′(κ3) ̸= 0, so ∂y(x;α) ∂α \|α=κ3 given in (A.6) is the second solution of (A.1), and (A.5) becomes L∂2y(x;α) ∂α2 ∣∣∣ α=κ3 = xκ3−3f ′′(κ3) ̸= 0. (A.8) On the other hand, f(κ1 + n) ̸= 0 for n ∈ N \ {m1}. Thus by letting cm1(κ1) = 0 and Rn(κ1) = 0 for any n ∈ N \ {m1} in Ly(x;κ1) (see (A.2) with α = κ1), we see that cn(κ1) can be uniquely solved for any n ∈ N \ {m1} such that Ly(x;κ1) = xκ1−3Rm1(κ1)x m1 = Rm1(κ1)x κ3−3, (A.9) where Rm1(κ1) = m1−1∑ k=0 [(κ1 + k)am1−k−2 + bm1−k−3]ck(κ1) is a constant. Thus we obtain Theorem A.4. Suppose κ1 < κ2 = κ3. Then ∂y(x;α) ∂α \|α=κ3 given in (A.6) is the second solution of (A.1). Furthermore, (1) If Rm1(κ1) = 0, then y(x;κ1) = xκ1 ∞∑ n=0 cn(κ1)x n is the third solution of (A.1) that has the local exponent κ1. (2) If Rm1(κ1) ̸= 0, then (A.8) and (A.9) imply that y3(x) := ∂2y(x;α) ∂α2 ∣∣∣ α=κ3 − f ′′(κ3) Rm1(κ1) y(x;κ1) Modular Ordinary Differential Equations on SL(2,Z) of Third Order 45 = (lnx)2y(x; k3) + 2(lnx)xκ3 ∞∑ n=1 c′n(κ3)x n + xκ3 ∞∑ n=1 c′′n(κ3)x n − f ′′(κ3) Rm1(κ1) y(x;κ1) is the third solution of (A.1) that corresponds to the local exponent κ1, namely (A.1) is completely not apparent at x = 0. The remaining case is κ1 ≤ κ2 < κ3, i.e., m2 = κ3 − κ2 > 0. Then for any α satisfying \|α− κ2\| < 1/2, we have f(α+ n) ̸= 0 for any n ∈ N \ {m2}, and f(α+ n) = 0 for n ≥ 1 if and only if α = κ2 and n = m2, so by letting cm2(α) = 0 and Rn(α) = 0 for any n ∈ N \ {m2} in Ly(x;α) (see (A.2)), we see that cn(α) can be uniquely solved for any n ∈ N \ {m2} such that Ly(x;α) = xα−3f(α) + xα+m2−3Rm2(α), for \|α− κ2\| < 1/2, (A.10) where Rm2(α) = m2−1∑ k=0 [(α+ k)am2−k−2 + bm2−k−3]ck(α) ∈ C(α), because ck(α) ∈ C(α) for any n ∈ N \ {m2}. Similarly as before, it follows from (A.10) that Ly(x;κ2) = xκ3−3Rm2(κ2), (A.11) L∂y(x;α) ∂α ∣∣∣ α=κ2 = xκ2−3f ′(κ2) + xκ3−3R′ m2 (κ2) + (lnx)xκ3−3Rm2(κ2), (A.12) where ∂y(x;α) ∂α ∣∣∣ α=κ2 = (lnx)y(x;κ2) + xκ2 ∞∑ n=1 c′n(κ2)x n. Theorem A.5. Suppose κ1 = κ2 < κ3. (1) If Rm2(κ2) = 0, then (A.11) implies that y(x;κ2) is the second solution of (A.1), and y3(x) := ∂y(x;α) ∂α ∣∣∣ α=κ2 − R′ m2 (κ2) f ′(κ3) ∂y(x;α) ∂α ∣∣∣ α=κ3 = (lnx)y(x;κ2) + xκ2 ∞∑ n=1 c′n(κ2)x n − R′ m2 (κ2) f ′(κ3) ( (lnx)y(x;κ3) + xκ3 ∞∑ n=1 c′n(κ3)x n ) is the third solution of (A.1). 46 Z. Chen, C.-S. Lin and Y. Yang (2) If Rm2(κ2) ̸= 0, then (A.4) and (A.11) imply that ∂y(x;α) ∂α ∣∣∣ α=κ3 − f ′(κ3) Rm2(κ2) y(x;κ2) = (lnx)y(x;κ3) + xκ3 ∞∑ n=1 c′n(κ3)x n − f ′(κ3) Rm2(κ2) y(x;κ2) is the second solution of (A.1), and y3(x) := ∂2y(x;α) ∂α2 ∣∣∣ α=κ3 − 2f ′(κ3) Rm2(k2) ∂y(x;α) ∂α ∣∣∣ α=κ2 − f ′′(κ3)Rm2(κ2)− 2f ′(κ3)R ′ m2 (κ2) Rm2(κ2) 2 y(x;κ2) = (lnx)2y(x; k3) + 2(lnx)xκ3 ∞∑ n=1 c′n(κ3)x n + xκ3 ∞∑ n=1 c′′n(κ3)x n − 2f ′(κ3) Rm2(k2) ( (lnx)y(x;κ2) + xκ2 ∞∑ n=1 c′n(κ2)x n ) − f ′′(κ3)Rm2(κ2)− 2f ′(κ3)R ′ m2 (κ2) Rm2(κ2) 2 y(x;κ2). is the third solution of (A.1), namely (A.1) is completely not apparent at x = 0. Proof. Since κ1 = κ2 < κ3, i.e., m1 = 0, m2 > 0 and f(s) = (s − κ2) 2(s − κ3), so f ′(κ2) = 0 and f ′(κ3) ̸= 0. (1) Note that (A.12) becomes L∂y(x;α) ∂α ∣∣∣ α=κ2 = xκ3−3R′ m2 (κ2), we see from (A.4) that Ly3 = 0. (2) Note that (A.12) becomes L∂y(x;α) ∂α ∣∣∣ α=κ2 = xκ3−3R′ m2 (κ2) + (lnx)xκ3−3Rm2(κ2). (A.13) Then (A.5), (A.11) and (A.13) together imply Ly3 = 0. ■ Finally, we consider the last case κ1 < κ2 < κ3, i.e., m1 > 0 and m2 > 0. Then f ′(κj) ̸= 0 for all j. Since f(κ1 + n) = 0 for n ≥ 1 if and only if n ∈ {m1,m1 +m2}, by letting cm1(κ1) = cm1+m2 = 0 and Rn(κ1) = 0 for any n ∈ N\{m1,m1+m2} in Ly(x;κ1) (see (A.2) with α = κ1), we see that cn(κ1) can be uniquely solved for any n ∈ N \ {m1,m1 +m2} such that Ly(x;κ1) = Rm1(κ1)x κ2−3 +Rm1+m2(κ1)x κ3−3, (A.14) where Rm1(κ1) = m1−1∑ k=0 [(κ1 + k)am1−k−2 + bm1−k−3]ck(κ1), Rm1+m2(κ1) = m1+m2−1∑ k=0 [(κ1 + k)am1+m2−k−2 + bm1+m2−k−3]ck(κ1), are constants. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 47 Theorem A.6. Suppose κ1 < κ2 < κ3 and Rm2(κ2) = 0. Then (A.11) implies that y(x;κ2) is the second solution of (A.1). Furthermore, (1) If Rm1(κ1) = Rm1+m2(κ1) = 0, then (A.14) implies that y(x;κ3) is the third solution of (A.1), namely 0 is an apparent singularity of (A.1). (2) If Rm1(κ1) = 0 and Rm1+m2(κ1) ̸= 0, then y3(x) := ∂y(x;α) ∂α ∣∣∣ α=κ3 − f ′(κ3) Rm1+m2(κ1) y(x;κ1) = (lnx)y(x;κ3) + xκ3 ∞∑ n=1 c′n(κ3)x n − f ′(κ3) Rm1+m2(κ1) y(x;κ1) is the third solution of (A.1). (3) If Rm1(κ1) ̸= 0, then y3(x) := ∂y(x;α) ∂α ∣∣∣ α=κ2 − f ′(κ2) Rm1(k1) y(x;κ1) − Rm1(κ1)R ′ m2 (κ2)− f ′(κ2)Rm1+m2(κ1) f ′(κ3)Rm1(κ1) ∂y(x;α) ∂α ∣∣∣ α=κ3 = (lnx)y(x;κ2) + xκ2 ∞∑ n=1 c′n(κ2)x n − f ′(κ2) Rm1(k1) y(x;κ1)− Rm1(κ1)R ′ m2 (κ2)− f ′(κ2)Rm1+m2(κ1) f ′(κ3)Rm1(κ1) ( (lnx)y(x;κ3) + xκ3 ∞∑ n=1 c′n(κ3)x n ) is the third solution of (A.1). Proof. Note that (A.12) becomes L∂y(x;α) ∂α ∣∣∣ α=κ2 = xκ2−3f ′(κ2) + xκ3−3R′ m2 (κ2). (A.15) (2) Note that (A.14) becomes Ly(x;κ1) = Rm1+m2(κ1)x κ3−3 ̸= 0. (A.16) From here and (A.4), we easily obtain Ly3 = 0. (3) Similarly, it is easy see from (A.4), (A.14) and (A.15) that Ly3 = 0. ■ Similarly, we can obtain Theorem A.7. Suppose κ1 < κ2 < κ3 and Rm2(κ2) ̸= 0. Then (A.4) and (A.11) imply that ∂y(x;α) ∂α ∣∣∣ α=κ3 − f ′(κ3) Rm2(κ2) y(x;κ2) = (lnx)y(x;κ3) + xκ3 ∞∑ n=1 c′n(κ3)x n − f ′(κ3) Rm2(κ2) y(x;κ2) is the second solution of (A.1). Furthermore, (1) If Rm1(κ1) = Rm1+m2(κ1) = 0, then (A.14) implies that y(x;κ3) is the third solution of (A.1). (2) If Rm1(κ1) = 0 and Rm1+m2(κ1) ̸= 0, then (A.11) and (A.16) imply that y(x;κ1)− Rm1+m2(κ1) Rm2(κ2) y(x;κ2) is the third solution of (A.1). 48 Z. Chen, C.-S. Lin and Y. Yang (3) If Rm1(κ1) ̸= 0, then (A.5), (A.11), (A.12) and (A.14) imply that y3(x) := ∂2y(x;α) ∂α2 ∣∣∣ α=κ3 − 2f ′(κ3) Rm2(κ2) ∂y(x;α) ∂α ∣∣∣ α=κ2 + C1y(x;κ1)− C2y(x;κ2) = (lnx)2y(x;κ3) + 2(lnx)xκ3 ∞∑ n=1 c′n(κ3)x n + xκ3 ∞∑ n=1 c′′n(κ3)x n − 2f ′(κ3) Rm2(k2) ( (lnx)y(x;κ2) + xκ2 ∞∑ n=1 c′n(κ2)x n ) + C1y(x;κ1)− C2y(x;κ2) is the third solution of (A.1), where C1 := 2f ′(κ3)f ′(κ2) Rm2(κ2)Rm1(κ1) , C2 := 1 Rm2(κ2) [ f ′′(κ3)− 2f ′(κ3)R ′ m2 (κ2) Rm2(κ2) + 2f ′(κ3)f ′(κ2)Rm1+m2(κ1) Rm2(κ2)Rm1(κ1) ] . In particular, (A.1) is completely not apparent at x = 0. Remark A.8. It follows from Theorem A.6(1) that 0 can be apparent only for the case κ1 < κ2 < κ3. Remark A.9. Clearly all the above arguments work when we study whether the regular sin- gularity ∞ is apparent or not for y′′′(z) +Q2(z)y ′(z) +Q3(z)y(z) = 0, z ∈ H, (A.17) when the local exponents κ (1) ∞ ≤ κ (2) ∞ ≤ κ (3) ∞ satisfy κ (j) ∞ − κ (j−1) ∞ ∈ Z. Since Qj(z)’s have Fourier expansions in terms of qN = 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 qN = 0 is apparent or not for ( qN d dqN )3 y + N2 (2πi)2 Q2qN d dqN y + N3 (2πi)3 Q3y = 0. (A.18) All the above statements are true for (A.18) in terms of qN . In particular, (A.17) or equiva- lently (A.18) always has a solution of the form y+(z) := qκ (3) ∞ N ∞∑ n=0 cn(κ (3) ∞ )qnN , c0 = 1, (A.19) and (A.17) is completely not apparent at z = ∞ or equivalently (A.18) is completely not apparent at qN = 0 if and only if the local expansion of some solutions in terms of qN contains the term z2y+(z) because of ln qN = 2πiz/N . More precisely, if (A.17) is completely not apparent at z = ∞, then it follows from Theorems A.3, A.4(2), A.5(2) and Theorem A.7(3) that (A.17) has two solutions of the following form y−(z) := z2y+(z) + zη1(z) + η2(z), y⊥(z) := zy+(z) + η3(z), (A.20) such that (y−, y⊥, y+) t is a basis of solutions, where η1(z) = qκ (2) ∞ N ∞∑ n=0 cn,1q n N , η2(z) = qκ (1) ∞ N ∞∑ n=0 cn,2q n N , η3(z) = qκ (2) ∞ N ∞∑ n=0 cn,3q n N . Note that c0,j = 0 may happen for any j; see Theorem A.3 for example. Modular Ordinary Differential Equations on SL(2,Z) of Third Order 49 Acknowledgements We would like to thank the referees for many valuable comments and pointing out some refer- ences. The research of Z. Chen was supported by NSFC (No. 12071240). References [1] Arike Y., Kaneko M., Nagatomo K., Sakai Y., Affine vertex operator algebras and modular linear differential equations, Lett. Math. Phys. 106 (2016), 693–718. [2] Beukers F., Heckman G., Monodromy for the hypergeometric function nFn−1, Invent. 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Soc. 9 (1996), 237–302, arXiv:math.RT/9912013. https://arxiv.org/abs/2103.04890 https://doi.org/10.1142/S1793042107000973 https://doi.org/10.1016/0370-2693(88)91765-0 https://doi.org/10.2206/kyushujm.74.401 https://arxiv.org/abs/1910.11668 https://doi.org/10.1515/forum-2020-0025 https://arxiv.org/abs/2002.00493 https://doi.org/10.1007/978-1-4684-9884-4 https://doi.org/10.2140/pjm.2001.197.491 https://arxiv.org/abs/math.RT/9912013 https://doi.org/10.1007/978-3-540-74119-0_1 https://doi.org/10.1090/S0894-0347-96-00182-8 https://arxiv.org/abs/math.RT/9912013 1 Introduction 2 Quasimodular forms of depth 2 and its associated 3rd order MODE 3 The MODE on SL(2,Z) 3.1 Proof of Theorem 1.4(1) 3.2 Proofs of Theorems 1.3 and 1.4, and Corollary 1.5 4 Reducibility and SU(3) Toda systems on SL(2,Z) 5 Polynomial systems derived from the conditions (H1)–(H3) 5.1 Solution expansions for MODEs 5.2 Existence of Q(z) and R(z) 6 Extremal quasimodular forms A The solution structure of third order ODE References
id	nasplib_isofts_kiev_ua-123456789-211532
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2026-03-18T10:46:51Z
publishDate	2022
publisher	Інститут математики НАН України
record_format	dspace
spelling	Chen, Zhijie Lin, Chang-Shou Yang, Yifan 2026-01-05T12:27:19Z 2022 Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications. Zhijie Chen, Chang-Shou Lin and Yifan Yang. SIGMA 18 (2022), 013, 50 pages 1815-0659 2020 Mathematics Subject Classification: 11F11; 34M03 arXiv:2106.12438 https://nasplib.isofts.kiev.ua/handle/123456789/211532 https://doi.org/10.3842/SIGMA.2022.013 In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form ′′′ + ₂()′ + ₃() = 0, ∈ ℍ = { ∈ ℂ \| Im > 0}, where ₂() and ₃() − 1/2′₂() are meromorphic modular forms on SL(2, ℤ) of weight 4 and 6, respectively. We show that any quasimodular form of depth 2 on SL(2, ℤ) leads to such a MODE. Conversely, we introduce the so-called Bol representation ^: SL(2, ℤ) → SL(3, 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 SU(3) Toda systems. Note that the SU( + 1) Toda systems are the classical Plücker infinitesimal formulas for holomorphic maps from a Riemann surface to ℂℙᴺ. We would like to thank the referees for their many valuable comments and for pointing out some references. The research of Z. Chen was supported by NSFC (No. 12071240). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications Article published earlier
spellingShingle	Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications Chen, Zhijie Lin, Chang-Shou Yang, Yifan
title	Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications
title_full	Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications
title_fullStr	Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications
title_full_unstemmed	Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications
title_short	Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications
title_sort	modular ordinary differential equations on sl(2, ℤ) of third order and applications
url	https://nasplib.isofts.kiev.ua/handle/123456789/211532
work_keys_str_mv	AT chenzhijie modularordinarydifferentialequationsonsl2zofthirdorderandapplications AT linchangshou modularordinarydifferentialequationsonsl2zofthirdorderandapplications AT yangyifan modularordinarydifferentialequationsonsl2zofthirdorderandapplications

Modular Ordinary Differential Equations on SL(2, ℤ) of Third Order and Applications

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