On rings of differential operators derived from automorphic forms
aa r X i v : . [ m a t h . C V ] J un On rings of differential operators derived from automorphic forms
Atsuhira NaganoOctober 17, 2018
Abstract
We study linear ordinary differential equations which are analytically parametrized on Hermitiansymmetric spaces and invariant under the action of symplectic groups. They are generalizations of theclassical Lam´e equation. Our main result gives a closed relation between such differential equationsand automorphic forms for symplectic groups. Our study is based on techniques concerning withthe monodromy of complex differential equations, the Baker-Akhiezer functions and algebraic curvesattached to rings of differential operators.
Introduction
The main purpose of this paper is to study linear ordinary differential operators of a complex independentvariable which are analytically parametrized on Hermitian symmetric domains and invariant under theaction of symplectic groups. Our main result gives a closed relation between commutative rings of suchdifferential operators and automorphic forms for symplectic groups.Let us start the introduction with a typical example of the differential equations we study: the Lam´edifferential equation (cid:16) − ∂ ∂z + B℘ (Ω , z ) (cid:17) u = Xu, (0.1)where
B, X ∈ C and ℘ (Ω , z ) is the Weierstrass ℘ -function with the double periods 1 and Ω ∈ H = { z ∈ C | Im( z ) > } . The Lam´e differential equation has the regular singular points at every z ∈ Z + Z Ω. If B = ρ ( ρ + 1) , the characteristic exponents at every singular point are ρ + 1 and − ρ. When ρ ∈ Z > , a system of basis of the space of solutions of (0.1) is generated by Λ( z ) and Λ( − z ). Here,Λ( z ) = ρ Y j =1 σ (Ω , z + κ j ) σ (Ω , z ) e − zζ (Ω ,κ j ) , where σ (Ω , z ) and ζ (Ω , z ) are the classical Weierstrass functions and κ j ( j = 1 , · · · , ρ ) can be calculated by X (for detail, see [WW]). We remark that Λ is a single-valuedfunction of z . However, for generic ρ ∈ C , the solutions of (0.1) are multivalued on C − ( Z + Z Ω) . TheLam´e equation is an important topic in mathematics. For example, the periodic solutions of (0.1) isstudied in many body theoretical physics. Also, via the double covering E → P ( C ), where E is anelliptic curve with the double periods 1 and Ω, the equation (0.1) gives a Fuchsian differential equationwith an accessary parameter. Moreover, special types of (0.1) promoted a development of the theory ofintegrable systems and finite zone problems. For example, see [WW], [DMN], [MM] and [T]. In theseresearches, to the best of the author’s knowledge, the double periodicity of the coefficient ℘ (Ω , z ) of (0.1)played an essential role. Keywords: Ordinary Differential Operators ; Automorphic Forms ; Algebraic Curves.Mathematics Subject Classification 2010: Primary 16S32 ; Secondary 47E05, 32N10, 14G35, 14H70, 33E05.Running head: Differential operators derived from automorphic formsNote: This is the corrected version of the published article DOI: 10.1007/s11785-017-0663-7. Typos and misleadingphrases are corrected here. Especially, for simplicity, criteria in Section 1.5 and 2.7 are corrected using arithmetic generaof algebraic curves. ℘ (Ω , z )satisfies the transformation law ℘ (cid:16) a Ω + bc Ω + d , zc
Ω + d (cid:17) = ( c Ω + d ) ℘ (Ω , z ) (0.2)for any (cid:18) a bc d (cid:19) ∈ SL (2 , Z ). Due to (0.2), the Lam´e equation becomes to be invariant under the ac-tion of the elliptic modular group SL (2 , Z ). Namely, via the transformation (Ω , z, X ) (Ω , z , X ) = (cid:16) a Ω + bc Ω + d , zc
Ω + d , ( c Ω+ d ) X (cid:17) , the differential equation (0.1) can be identified with (cid:16) − ∂ ∂z + B℘ (Ω , z ) (cid:17) u = X u. By the way, holomorphic functions on H which are invariant under the action of SL (2 , Z ) are calledelliptic modular forms. The invariance between two differential equations suggests a strong and non-trivial relation between the Lam´e equation and elliptic modular forms. Furthermore, elliptic modularforms are quite important in number theory (see [Sm1]). The author expects that the Lam´e equationmay have some effective applications in number theory.Based on the above observation and expectation, we study a class of ordinary differential equations P u = Xu of a complex variable z (for detail, see Definition 2.2). Here, the differential operator P = ∂ N ∂z N + a (Ω , z ) ∂ N − ∂z N − + a (Ω , z ) ∂ N − ∂z N − + · · · + a N (Ω , z )is parametrized by Ω of a product H gn of the Siegel upper half planes and invariant under the action ofa congruence subgroup Γ of the symplectic group. Such a class contains the Lam´e equation because theaction of the group Γ on H gn is a natural extension of the action of SL (2 , Z ) on H . In this paper, we studycommutative rings of differential operators which commute with P. Here, we recall the importance of commutative rings of differential operators. Commutative ringsof differential operators were firstly studied by Burchnall and Chaundy [BC]. In the later half of the20th century, the relation between commutative rings of differential operators and algebraic curves wasstudied in the celebrated works of Krichever [K] and Mumford [Mm]. Their results are very important inthe theory of integrable systems. Also, they yielded a substantial progress of the geometry of Riemannsurfaces and abelian varieties. In fact, they were used to resolve the classical Riemann-Schottky problemfor Riemann surfaces ([So], [KS]).In this paper, we will give a relation between commutative rings of differential operators and auto-morphic forms. We study the structures of rings of differential operators which are invariant under theaction of Γ and commute with the fixed differential operator P . Such a ring will be denoted by D P inSection 2. Our main result gives an isomorphism χ : D P ≃ S P of rings, where S P is a ring of generatingfunctions for sequences of automorphic forms for Γ (for the definition, see Definition 2.3 and 2.4). Here,we note that automorphic forms are natural extension of elliptic modular forms (see Definition 2.1).These rings are graded by the weight K induced from the action of Γ: D P = ∞ M K =0 D PK , S P = ∞ M K =0 S PK . Theisomorphism χ induces an isomorphisms among three vector spaces: D PK χ −→ S PK −→ W K (see Theorem 2.6). Here, W K is a vector space explicitly parametrized by automorphic forms for Γ.Therefore, the structure of the ring D P is closely related to the structure of the rings of automorphicforms.For our study, we will use the Baker-Akhiezer functions. In [K], the Baker-Akhiezer functions givesolutions of differential equations whose coefficients are smooth functions. However, for our purpose,it is natural to study differential equations whose coefficients have poles (precisely, see Remark 1.2).So, we need to modify the techniques of the Baker-Akhiezer functions for differential equations withsome singularities. Section 1 will be devoted to such techniques. Namely, we will study the multivaluedBaker-Akhiezer functions and its monodromy around singular points of P .2n Section 2, we prove our main result. This is based on an invariance of the multivalued Baker-Akhiezer functions under the action of Γ, which is proved in Theorem 2.1. Moreover, we will see thefollowing results: • For fixed P and an operator Q ∈ D P , there exists an algebraic curve R Ω : X j,k f j,k (Ω) X j Y k = 0such that ( X, Y ) = (
P, Q ) gives a point of R Ω . Here, the coefficients f j,k (Ω) are automorphic formsfor Γ (see Theorem 2.7). Namely, from the differential operators P and Q , we obtain a family ofalgebraic curves {R Ω | Ω ∈ H gn } parametrized on H gn via automorphic forms. • If the coefficients of the fixed operator P have poles in z -plane, the coefficients of Q ∈ D P can bemultivalued functions of z (for detail, see Proposition 1.3 and Theorem 2.2). However, if the genusof the algebraic curve R Ω is small enough, every coefficients of Q ∈ D P must be single-valued. Wewill have a sufficient criterion for Q to be single-valued (see Theorem 2.8).Throughout the paper, the Lam´e differential equation is a prototype of our story. Via our new resultsbetween differential operators and automorphic forms, we have a simple interpretation of classical resultsof the Lam´e equation via elliptic modular forms (Example 2.3, 2.4 and 2.5). This is an important exampleof our story.Our results enable us to study differential equations based on the structures of rings of automorphicforms. In number theory, there are many famous generalizations of elliptic modular forms (for example,Siegel modular forms, Hilbert modular forms, etc.). Our results can be applied to such generalized formsalso. The author expects that this paper may give a first step of the study of differential equations fromthe viewpoint of automorphic forms. In this subsection, we obtain the multivalued Baker-Akhiezer functions for the ordinary differential op-erator P z = d N dz N + a ( z ) d N − dz N − + a ( z ) d N − dz N − + · · · + a N ( z ) (1.1)of the complex variable z . Here, we assume the coefficients a ( z ) , · · · , a N ( z ) are meromorphic functionsof z . More precisely, we assume that a ( z ) , · · · , a N ( z ) are holomorphic on C − N , where N is the unionof the sets of the poles of a j ( z ) ( j = 2 , · · · , N ). Remark 1.1.
If a differential operator P z = d N dz N + a ( z ) d N − dz N − + a ( z ) d N − dz N − + · · · + a N ( z ) is given, bya gauge transformation vP z v − for some unit function v = v ( z ) , P z is transformed to P z . So, in ourstudy, we only consider the differential operator in the form (1.1) without loss of generality.
Let X be the universal covering of C − N . By taking a fixed point w ∈ C − N , any s ∈ X is representedby s = ( z, [ γ ]) , where z ∈ C − N , γ is an arc in C − N from w to z and [ γ ] is the homotopy class of γ .We note that z gives a local coordinate of X . Proposition 1.1.
There exists the unique formal solution
Ψ(( z, [ γ ]) , w, λ ) of the differential equation P z u = λ N u (1.2)3 n the form Ψ(( z, [ γ ]) , w, λ ) = (cid:16) ∞ X s =0 ξ s (( z, [ γ ]) , w ) λ − s (cid:17) e λ ( z − w ) (1.3) such that ( ξ (( z, [ γ ]) , w ) ≡ ,ξ s (( w, [ id ]) , w ) = 0 ( s ≥ . (1.4) Here, ξ s are locally holomorphic functions of ( z, w ) .Proof. In this proof, set a ( z ) ≡ , a ( z ) ≡
0. Putting u = (cid:16) ∞ X s =0 η s ( z ) λ − s (cid:17) e λ ( z − w ) to (1.2), we have N X m =0 a N − m ( z ) m X l =0 (cid:18) ml (cid:19) ∞ X s =0 (cid:16) ∂ m − l ∂z m − l η s ( z ) λ l − s (cid:17) e λ ( z − w ) = (cid:16) ∞ X s =0 η s ( z ) λ N − s (cid:17) e λ ( z − w ) . Comparing the coefficients of λ − s , we have N X m =0 a N − m ( z ) m X l =0 (cid:18) ml (cid:19) ∂ m − l ∂z m − l η l + s ( z ) = η N + s ( z ) . (1.5)Since η N + s ( z ) appears in the left hand side only when m = l = N , the terms of η N + s ( z ) is cancelledfrom the relation (1.5). The function η N + s − ( z ) and its derivation appears in (1.5) only when m = N and l = N −
1. Here, we used a N − ( z ) ≡
0. Then, the equation (1.5) becomes to be
N ∂∂z η N + s − ( z ) = (cid:18) a polynomial in ∂ ν ∂z ν η l ( z ) ( l < N + s − , ν ∈ Z ≥ ) and a j ( z ) defined over Z (cid:19) . (1.6)By the integration of the relation (1.6) on the arc γ ∈ C − N whose start point is w , we can obtain theexpression of η µ ( z ) in terms of η ν ( z ) ( ν < µ ) and a l ( z ). Especially, the condition that η ( z, [ γ ]) ≡ η s ( w, [ id ]) = 0 ( s ≥
1) uniquely determines the sequence { η s ( z ) } s . Such functions η s ( z ) give the requiredfunctions ξ s (( z, [ γ ]) , w ) ( s ≥ ξ s are locallyholomorphic functions of ( z, w ).We call Ψ(( z, [ γ ]) , w, λ ) of (1.3) the multivalued Baker-Akhiezer function for the equation (1.2). Remark 1.2.
Krichever [K] studied ordinary differential equations whose coefficients are smooth func-tions of a real variable. Also, Mumford [Mm] studied differential equations whose coefficients are formalpower series. For the purposes of their research, it is sufficient to study single-valued solutions of differ-ential equations. However, for our main purpose of this paper, it is natural to study differential equationsof a complex independent variables whose coefficients allow some singularities. In Section 2, we willconsider the transformation z z = zj α (Ω) , where j α (Ω) is complex valued. In such cases, even if z isa real variable, z is not always a real variable. Moreover, we will give results for a class of differentialequations containing the Lam´e equation. Since the Lam´e equation has singularities, it is natural to studydifferential equations which admit singularities. This is the reason why we need the multivalued solution z Ψ(( z, [ γ ]) , w, λ ) of (1.3). Remark 1.3. If a ( z ) , · · · , a N ( z ) are holomorphic on the whole z -plane, we do not need to consider theuniversal covering X of C − N . In this case, the function Ψ in the above theorem is given in the form Ψ( z, w, λ ) = (cid:16) ∞ X s =0 ξ s ( z, w ) λ − s (cid:17) e λ ( z − w ) . Here, Ψ and ξ s ( s ≥ are single-valued functions of z ∈ C . P z u = Xu, (1.7)where X ∈ P ( C ) − {∞} . Let λ , · · · , λ N be the solutions of the equation λ N = X . Lemma 1.1.
For fixed w ∈ C − N , the solutions Ψ(( z, [ γ ]) , w, λ j ) ( j = 1 , · · · , N ) of (1.3) are linearindependent for generic X .Proof. For µ , · · · , µ N ∈ C , suppose N X j =1 µ j Ψ(( z, [ γ ]) , w, λ j ) = 0 (1.8)holds for generic X . Since the right hand side of the relation (1.8) is invariant under the permu-tation of λ , · · · , λ N , together with the definition of Ψ of (1.3), we can assume that µ = · · · = µ N = µ . Set E s ( z, w, X ) = N X j =1 λ − sj e λ j ( z − w ) . For fixed z and w , X E s ( z, w, X ) is a formal powerseries in X − and E s ( z, w, X ) ( s = 0 , , · · · ) are linearly independent. The relation (1.8) becomes ∞ X s =0 µξ s (( z, [ γ ]) , w ) E s ( z, w, X ) = 0 for generic X . Therefore, µ = 0 follows. Proposition 1.2.
Let u = u (( z, [ γ ]) , w ) be a series given by the form u (( z, [ γ ]) , w ) = ∞ X s =0 η s ( z, [ γ ]) λ − s ! e λ ( z − w ) , (1.9) where η s ( z, [ γ ]) are analytic on X and λ satisfies λ N = X . Then, u is a formal solution of the differentialequation (1.7) if and only if u is given by u (( z, [ γ ]) , w ) = A ( w, λ )Ψ(( z, [ γ ]) , w, λ ) (1.10) for generic λ , where A ( w, λ ) does not depend on ( z, [ γ ]) .Proof. It is clear that u (( z, [ γ ]) , w ) of (1.10) is a solution of the differential equation (1.7).Conversely, we assume that u (( z, [ γ ]) , w ) in the form (1.9) is a solution of the differential equation(1.7), where X = λ N . From Lemma 1.1, the space of solutions of (1.7) is generated by Ψ(( z, [ γ ]) , w, λ j )( j = 1 , · · · , N ) for generic λ . We can assume λ of (1.9) coincides with λ j for some j ∈ { , · · · , N } .Since the space X is simply connected, ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ j ) ( j = 1 , · · · , N ) are single-valued on X . So, the solution in the form (1.9) must be an element of the 1-dimensional vector space generated by h Ψ(( z, [ γ ]) , w, λ j ) i . Hence, u is given by the form (1.10).We will consider a differential operator Q ( z, [ γ ]) = b ( z, [ γ ]) d M dz M + b ( z, [ γ ]) d M − dz M − + · · · + b M ( z, [ γ ]) . (1.11)Here, we assume that the coefficients b k ( z, [ γ ]) ( k = 0 , · · · , M ) are multivalued analytic functions on C − N . The operator Q ( z, [ γ ]) is defined on X .From now on, we consider the action of the operator Q ( z, [ γ ]) on the function Ψ(( z, [ γ ]) , w, λ ). If P z and Q ( z, [ γ ]) are commutative, we can apply Proposition 1.2 to Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ ). Therefore, it is naturalto consider differential operator (1.11) whose coefficients are multivalued functions of z (for detail, seethe proof of the next proposition). 5 roposition 1.3. Let P z ( Q ( z, [ γ ]) , resp.) be the differential operator of (1.1) ((1.11), resp.). Then, P z and Q ( z, [ γ ]) are commutative if and only if the quotient Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ )Ψ(( z, [ γ ]) , w, λ ) coincides with A ( λ ) = ∞ X s = − M A s λ − s (1.12) for generic λ , where Ψ is given in (1.3) and A ( λ ) does not depend on ( z, [ γ ]) and w .Proof. Suppose that P z and Q ( z, [ γ ]) are commutative. Then, Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ ) gives a solution ofthe differential equation (1.7). Remark that Q ( z, [ γ ]) (Ψ(( z, [ γ ]) , w, λ )) is in the form (1.9) for some { η s } s .So, due to Proposition 1.2, we have Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ ) = A ( w, λ )Ψ(( z, [ γ ]) , w, λ ) , (1.13)for some A ( w, λ ). Take any w ′ ∈ C − N . Then, Ψ(( z, [ γ ]) , w ′ , λ ) e λ ( w ′ − w ) has the form (1.9) and is asolution of (1.7). So, according to Proposition 1 . B ( w, λ ) such thatΨ(( z, [ γ ]) , w ′ , λ ) e λ ( w ′ − w ) = B ( w, λ )Ψ(( z, [ γ ]) , w, λ ) . (1.14)From (1.13) and (1.14), A ( w ′ , λ ) = Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w ′ , λ )Ψ(( z, [ γ ]) , w ′ , λ ) = Q ( z, [ γ ]) ( e λ ( w − w ′ ) B ( w, λ )Ψ(( z, [ γ ]) , w, λ )) B ( w, λ )Ψ(( z, [ γ ]) , w, λ ) e λ ( w − w ′ ) = Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ )Ψ(( z, [ γ ]) , w, λ ) = A ( w, λ ) . This shows that A ( w, λ ) does not depend on the variable w . So, we set A ( λ ) = A ( w, λ ). Hence, therelation (1.13) becomes to be Q ( z, [ γ ]) ∞ X s =0 ξ s (( z, [ γ ]) , w ) λ − s ! e λ ( z − w ) ! = ∞ X s = α A s λ − s ! ∞ X s =0 ξ s (( z, [ γ ]) , w ) λ − s ! e λ ( z − w ) . (1.15)Since Q ( z, [ γ ]) is a differential operator of rank M , a non-zero term which contains λ M appears in the lefthand side of (1.15) as the higher term in λ . Therefore, considering the right hand side of (1.15), the seriesof A ( λ ) must be in the form A ( λ ) = ∞ X s = − M A s λ − s .Conversely, we assume that the relation (1.13) holds. Then, we have P z Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ ) = P z A ( λ )Ψ(( z, [ γ ]) , w, λ ) = λ N A ( λ )Ψ(( z, [ γ ]) , w, λ ). This is clearly equal to Q ( z, [ γ ]) P z Ψ(( z, [ γ ]) , w, λ ).Therefore, we have [ P z , Q ( z, [ γ ]) ]Ψ(( z, [ γ ]) , w, λ ) = 0 . (1.16)Here, the relation (1.16) means that the ordinary differential equation [ P z , Q ( z, [ γ ]) ] u = 0 has solutions { Ψ(( z, [ γ ]) , w, λ ) } λ parametrized by λ . Since Ψ is given by the form of (1.3), it follows that the differentialoperator [ P z , Q ( z, [ γ ]) ] must be 0. Proposition 1.4.
Let P z be the differential operator of (1.1). Let Q (1) z, [ γ ] and Q (2) z, [ γ ] be the differentialoperator given by the form (1.11). If P z commutes with both Q (1) z, [ γ ] and Q (2) z, [ γ ] , then Q (1) z, [ γ ] commutes with Q (2) z, [ γ ] .Proof. By the assumption and Proposition 1.3, there exist series A (1) ( λ ) and A (2) ( λ ) in λ such that Q ( j )( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ ) = A ( j ) ( λ )Ψ(( z, [ γ ]) , w, λ ) ( j = 1 ,
2) for Ψ of (1.3). So, we have[ Q (1)( z, [ γ ]) , Q (2)( z, [ γ ]) ]Ψ(( z, [ γ ]) , w, λ ) = ( A (1) ( λ ) A (2) ( λ ) − A (2) ( λ ) A (1) ( λ ))Ψ(( z, [ γ ]) , w, λ ) = 0 . As in the end of the proof of Proposition 1.3, we have [ Q (1)( z, [ γ ]) , Q (2)( z, [ γ ]) ] = 0 . P z of (1.1), let L ( P z , X ) be the space of solutions of the differentialequation P z u = Xu . Suppose Q ( z, [ γ ]) of (1.11) is a differential operator which commutes with P z . Then, Q ( z, [ γ ]) defines a linear operator Q [ γ ] ,X on the vector space L ( P z , X ) . Let us take two arcs γ and γ ′ from w to z in C − N . Setting δ = γ − · γ ′ , [ δ ] gives an element ofthe fundamental group π ( C − N ) . For λ such that λ N = X , since the coefficients of P z of (1.1) aresingle-valued, each Ψ(( z, [ γ ]) , w, λ ) and Ψ(( z, [ γ ′ ]) , w, λ ) are solutions of the differential equation (1.7) forgeneric X . Based on Lemma 1.1, setting the vectorΨ v (( z, [ γ ]) , w, λ ) = (Ψ(( z, [ γ ]) , w, λ ) , · · · , Ψ(( z, [ γ ]) , w, λ N )) , (1.17)there exists a matrix M ([ δ ] , w, λ ) ∈ GL ( N, C ) such thatΨ v (( z, [ γ ′ ]) , w, λ ) = Ψ v (( z, [ γ ]) , w, λ ) M ([ δ ] , w, λ ) . (1.18)The matrix M ([ δ ] , w, λ ) is called the monodromy matrix of [ δ ] ∈ π ( C −N ) for the system Ψ v (( z, [ γ ]) , w, λ )of (1.17). We note that r : π ( C − N ) → GL ( N, C ) given by [ δ ] M ([ δ ] , w, λ ) (1.19)is a homomorphism of groups.Let γ, γ ′ and δ be as above. Suppose Q ( z, [ γ ]) commutes with P z of (1.1). By considering the analyticcontinuation along the closed arc δ , Q ( z, [ γ ′ ]) also commutes with P z . For the linear operator Q [ γ ] ,X on L ( P z , X ), set [ δ ] ∗ ( Q [ γ ] ,X ) = Q [ γ ′ ] ,X . (1.20) Theorem 1.1.
Let γ, γ ′ , δ, Q [ γ ] ,X be as above.(1) The set of the eigenvalues of the linear operator Q [ γ ] ,X coincides with that of the linear operator [ δ ] ∗ ( Q [ γ ] ,X ) for generic X .(2) There exists N ∈ Z > such that ([ δ ] ∗ ) N ( Q [ γ ] ,X ) = Q [ γ ] ,X for any [ δ ] ∈ π ( C − N ) and generic X .Proof. Set Q ( z, [ γ ]) Ψ v (( z, [ γ ]) , w, λ ) = ( Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ ) , · · · , Q ([ z ] ,γ ) Ψ(( z, [ γ ]) , w, λ N )) . Accordingto Proposition 1.3 together with the notation (1.17), we have Q ( z, [ γ ]) Ψ v (( z, [ γ ]) , w, λ ) = Ψ v (( z, [ γ ]) , w, λ ) A ( λ ) 0 · · · A ( λ N ) . For [ δ ] ∈ π ( C − N ), using the monodromy matrix M ([ δ ] , w, λ ) of (1.18), we have Q ( z, [ γ ′ ]) Ψ v (( z, [ γ ]) , w, λ ) = Q ( z, [ γ ′ ]) Ψ v (( z, [ γ ′ ]) , w, λ ) M ([ δ ] , w, λ ) − = Ψ v (( z, [ γ ′ ]) , w, λ ) A ( λ ) 0 · · · A ( λ N ) M ([ δ ] , w, λ ) − = Ψ v (( z, [ γ ]) , w, λ ) M ([ δ ] , w, λ ) A ( λ ) 0 · · · A ( λ N ) M ([ δ ] , w, λ ) − . (1.21)Here, we used the fact that Q ( z, [ γ ′ ]) Ψ(( z, [ γ ′ ]) , w, λ j ) = A ( λ j )Ψ(( z, [ γ ′ ]) , w, λ j ) due to Proposition 1.3.7n the other hand, since Q ( z, [ γ ′ ]) commutes with P z , we can directly apply Proposition 1.3 to Q ( z, [ γ ′ ]) .Then, there exist A ′ ( λ ) , · · · , A ′ ( λ N ) such that Q ( z, [ γ ′ ]) Ψ v (( z, [ γ ]) , w, X ) = Ψ v (( z, [ γ ]) , w, λ ) A ′ ( λ ) 0 · · · A ′ ( λ N ) . (1.22)So, from (1.21) and (1.22), we have A ′ ( λ ) 0 · · · A ′ ( λ N ) = M ([ δ ] , w, λ ) A ( λ ) 0 · · · A ( λ N ) M ([ δ ] , w, λ ) − . This implies that the set of eigenvalues { A ( λ ) , · · · , A ( λ N ) } for Q [ γ ] ,X coincides with that of eigenvalues { A ′ ( λ ) , · · · , A ′ ( λ N ) } for Q [ γ ′ ] ,X .(2) From the above (1), the correspondence [ δ ] of (1.20) induces a permutation of the eigenvalues.Therefore, setting N = N !, we have(([ δ ] ∗ ) N Q ( z, [ γ ]) )Ψ v (( z, [ γ ]) , w, λ ) = Ψ v (( z, [ γ ]) , w, λ ) A ( λ ) 0 · · · A ( λ N ) = Q ( z, [ γ ]) Ψ v (( z, [ γ ]) , w, λ ) . So, for j ∈ { , · · · , N } , we have (([ δ ] ∗ ) N ( Q ( z, [ γ ]) ) − Q ( z, [ γ ]) )Ψ(( z, [ γ ]) , w, λ j ) = 0 . Therefore, by a similarargument to the end of the proof of Proposition 1.3, we obtain ([ δ ] ∗ ) N ( Q ( z, [ γ ]) ) = Q ( z, [ γ ]) . Corollary 1.1. If N is a finite set of C , the coefficients b k ( z, [ γ ]) ( k = 0 , · · · , M ) of Q ( z, [ γ ]) of (1.11)are at most algebraic functions of z .Proof. By Theorem 1.1 and the assumption of the corollary, the image of the correspondence π ( C −N ) ∋ [ δ ] [ δ ] ∗ of (1.20) is finite. This implies the assertion. R For generic X ∈ C , we have the distinct N values λ , · · · , λ N such that λ Nj = X ( j = 1 , · · · , N ). FromLemma 1.1, { Ψ(( z, [ γ ]) , w, λ j ) | j ∈ { , · · · , N }} gives a system of basis of L ( P z , X ). From Proposition 1.3,Ψ(( z, [ γ ]) , w, λ j ) gives an eigenfunction with the eigenvalue A ( λ j ) of Q [ γ ] ,X . Hence, the characteristicpolynomial of Q [ γ ] ,X on L ( P z , X ) is given by N Y j =1 ( Y − A ( λ j )) . (1.23) Lemma 1.2.
Let { C l (( z, [ γ ]) , w, X ) } l =0 , , ··· ,N − be a system of basis of the vector space L ( P z , X ) satis-fying ∂ r ∂z r C l (( z, [ γ ]) , w, X ) (cid:12)(cid:12)(cid:12) ( z, [ γ ])=( w, [ id ]) = δ l,r . (1.24) (1) For fixed w ∈ C − N and ( z, [ γ ]) ∈ X , the correspondence X C l (( z, [ γ ]) , w, X ) gives a holomor-phic function on C = P ( C ) − {∞} .(2) For Q ( z, [ γ ]) of (1.11), the components of the representation matrix of the linear operator Q [ γ ] ,X for the system of basis { C l (( z, [ γ ]) , w, X ) } l =0 , , ··· ,N − are given by polynomials in X and special valuesof a j ( z ) ( j = 0 , · · · , N ) and b j (( z, [ γ ]) , w ) ( k = 0 , · · · , M ) .Proof. (1) The solutions C l (( z, [ γ ]) , w, X ) are given by solving the initial value problem for the differentialequation (1.7). Hence, the correspondence X C l (( z, [ γ ]) , w, X ) is holomorphic.82) Since C l (( z, [ γ ]) , w, X ) ( l = 0 , · · · , N −
1) are solutions of the equation (1.7), we obtain ∂ N ∂z N C l (( z, [ γ ]) , w, X ) = XC l (( z, [ γ ]) , w, X ) − N − X k =0 a k ( z ) ∂ k ∂z k C l (( z, [ γ ]) , w, X ) . (1.25)By the way, since Q ( z, [ γ ]) C l (( z, [ γ ]) , w, X ) ∈ L ( P z , X ) , there exists constants c l,m ( w, X ) ( l, m ∈ { , · · · , N − } ) for z such that Q ( z, [ γ ]) C l (( z, [ γ ]) , w, X ) = N − X m =0 c l,m C m (( z, [ γ ]) , w, X ). Due to (1.24), we have c l,m ( w, X ) = ∂ m ∂z m Q ( z, [ γ ]) C l (( z, [ γ ]) , w, X ) (cid:12)(cid:12)(cid:12) ( z, [ γ ])=( w, [ id ]) . (1.26)Using the relation (1.25), we can see that ∂ r ∂z r C l (( z, [ γ ]) , w, X ) (cid:12)(cid:12)(cid:12) ( z, [ γ ])=( w, [ id ]) are given by a polynomial in X and the special values a k ( w ) for any r ∈ Z . So, according to (1.26), we can see that c l,m ( w, X ) aregiven by polynomials in X and special values of a j ( z ) and b k (( z, [ γ ]) , w ) .We note that λ j ( j = 1 , · · · , N ) are distinct solutions of the algebraic equation λ N = X . From(1.13), A (Ω , λ j ) is a Laurent series in λ − j . Since the right hand side of (1.23) is symmetric series in λ − j ( j = 1 , · · · , N ), the right hand side of (1.23) gives a Laurent series in X − . Moreover, we have thefollowing. Corollary 1.2.
The characteristic polynomial (1.23) defines a polynomial in X .Proof. We have a representation matrix of Q [ γ ] ,X whose components are polynomial in X from the abovelemma. Therefore, its characteristic polynomial is given by a polynomial in X .In the following, let F ( X, Y ) be the polynomial (1.23) in the variables X and Y . Theorem 1.2.
The differential operators P z of (1.1) and Q ( z, [ γ ]) of (1.11) satisfy F ( P z , Q ( z, [ γ ]) ) = 0 . Proof.
For generic X ∈ C , letting λ be a solution of the equation of λ N = X , we have F ( P z , Q ( z, [ γ ]) )Ψ(( z, [ γ ]) , w, λ ) = F ( X, Q ( z, [ γ ]) )Ψ(( z, [ γ ]) , w, λ ) = 0 . Here, the last equality is due to the Hamilton-Cayley theorem. Then, the ordinary differential equation F ( P z , Q ( z, [ γ ]) ) u = 0 has a family { Ψ(( z, [ γ ]) , w, λ ) } λ of solutions with the parameter λ . By a similarargument to the end of the proof of Proposition 1.3, the operator F ( P z , Q ( z, [ γ ]) ) is equal to 0.The equation F ( X, Y ) = 0 defines an algebraic curve R . This curve should be in the form R : X j,k f j,k X j Y k = 0 . (1.27)Let R be the algebraic curve in Theorem 1.2. Let π : R → P ( C ) be the projection given by( X, Y ) X . Let p ∞ be the point of R corresponding to X = ∞ ∈ P ( C ) . Then, p ∞ is a ramificationpoint of the mapping π. We note that X = X gives a complex coordinate around p ∞ ∈ R . By the procedure of the algebraic curve R and the covering π : R → P ( C ) , Proposition 1.3 andTheorem 1.1 (1) imply that any [ δ ] ∈ π ( C − N ) induces the correspondence σ [ δ ] : R → R given by p j = ( X, A ( λ j )) σ [ δ ] ( p j ) = p k = ( X, A ( λ k )) , (1.28)when ([ δ ] ∗ ( Q [ γ ] ,X ))Ψ(( z, [ γ ]) , w, λ j ) = A ( λ k )Ψ(( z, [ γ ]) , w, λ j ) . (1.29)Hence, letting Aut( π ) be the group of transformations for the covering π , we have the homomorphism π ( C − N ) → Aut( π )of groups given by [ δ ] σ [ δ ] . 9 heorem 1.3. (1) All coefficients of the operator Q ( z, [ γ ]) are single-valued on C − N if and only if σ [ δ ] = id for every [ δ ] ∈ π ( C − N ) .(2) For λ j ( j = 1 , · · · , N ) satisfying λ Nj = X , assume A ( λ ) , · · · , A ( λ N ) are distinct for generic X .Then, σ [ δ ] = id if and only if Ψ(( z, [ γ ′ ]) , w, λ j ) = µ j Ψ(( z, [ γ ]) , w, λ j ) ( j = 1 , · · · , N ) , (1.30) where µ j is a constant function of z .Proof. (1) If all coefficients of Q ( z, [ γ ]) are single-valued, we have [ δ ] ∗ ( Q [ γ ] ,X ) = Q [ γ ] ,X for any [ δ ] ∈ π ( C − N ). Then, by (1.28) and (1.29), we have σ [ δ ] = id. Conversely, if σ [ δ ] = id for any [ δ ] ∈ π ( C − N ), from (1.28) and (1.29), we have[ δ ] ∗ ( Q [ γ ] ,X )Ψ(( z, [ γ ]) , w, λ j ) = A ( λ j )Ψ(( z, [ γ ]) , w, λ j ) = Q [ γ ] ,X Ψ(( z, [ γ ]) , w, λ j )for generic X . So, by a similar argument to the proof of Proposition 1.3, we have Q ( z, [ γ ′ ]) = Q ( z, [ γ ]) , where γ ′ = γ · δ . Hence, the assertion holds.(2) By the assumption, Ψ(( z, [ γ ]) , w, λ j ) spans the 1-dimensional eigenspace for the eigenvalue A ( λ j )of Q [ γ ] ,X . Set γ ′ = γ · δ. If σ [ δ ] = id , from (1), we have ([ δ − ] ∗ ) Q [ γ ′ ] ,X = Q [ γ ′ ] ,X for any [ δ ] ∈ π ( C − N ).This implies that Q z, [ γ ] Ψ(( z, [ γ ′ ]) , w, λ j ) = Q z, [ γ ′ ] Ψ(( z, [ γ ′ ]) , w, λ j ) = A ( λ j )Ψ(( z, [ γ ′ ]) , w, λ j )for generic X , where λ Nj = X . Here, we used Proposition 1.3. Therefore, Ψ(( z, [ γ ′ ]) , w, λ j ) is an eigen-function for the eigenvalue A ( λ j ). So, Ψ(( z, [ γ ′ ]) , w, λ j ) ∈ h Ψ(( z, [ γ ]) , w, λ j ) i C holds.Conversely, if we have (1.30), then, due to Proposition 1.3, Q ( z, [ γ ′ ]) Ψ(( z, [ γ ]) , w, λ j ) = µ − j Q ( z, [ γ ′ ]) Ψ(( z, [ γ ′ ]) , w, λ j )= µ − j A ( λ j )Ψ(( z, [ γ ′ ]) , w, λ j ) = Q ( z, [ γ ]) Ψ(( z, [ γ ]) , w, λ j )holds for generic X . Hence, as in (1), we have σ [ δ ] = id. ψ In this subsection, we use the same notation which we use in the previous subsection. Moreover, wesuppose that there exists s ( s ≥ − M ) , where N and s are coprime , such that A s = 0 (1.31)for { A s } of (1.12). Then, the operator Q [ γ ] ,X on L ( P z , X ) has N distinct eigenvalues A ( λ j ) ( j = 1 , · · · , N )in the sense of Proposition 1.3. Hence, the eigenspace for the eigenvalue A s ( λ j ) is 1-dimensional.Since X is simply connected, for a general X ∈ C and p ∈ π − ( X ) ⊂ R , we can take the uniqueeigenfunction on X : ψ (( z, [ γ ]) , w, p ) = N − X l =0 h l ( w, p ) C l (( z, [ γ ]) , w, X ) , (1.32)where h ( w, p ) ≡
1. Here, C l (( z, [ γ ]) , w, X ) ( l = 0 , · · · , N −
1) are given in Lemma 1.2 and h l ( w, p ) doesnot depend on z . Lemma 1.3.
Let ψ (( z, [ γ ]) , w, p ) be the function of (1.32).(1) For fixed w ∈ C − N , p h l ( w, p ) gives a meromorphic function on R − { p ∞ } .(2) For fixed w ∈ C − N , the poles of R − { p ∞ } ∋ p ψ (( z, [ γ ]) , w, p ) ∈ P ( C ) do not depend on ( z, [ γ ]) ∈ X . (3) Let U ∞ ⊂ P ( C ) be a sufficiently small neighborhood of X = ∞ . Let V ⊂ C − N be a sufficientlysmall and simply connected neighborhood of w . If π ( p ) ∈ U ∞ − {∞} , z ∈ V and γ ⊂ V , then p ψ (( z, [ γ ]) , w, p ) is analytic and has an exponential singularity at p = p ∞ . roof. (1) We had the representation matrix c ( w, X ) = ( c jk ( w, X )) of the linear operator Q [ γ ] ,X on L ( P z , X ) for the system of basis { C l (( z, [ γ ]) , w, X ) } l =0 , ··· ,N − of (1.24). Here, by Lemma 1.2 (2), c l,m ( w, X ) are given by polynomials in X . Let p ∈ R be a point corresponding to X ∈ C and theeigenvalue Y . We can obtain h l ( w, p ) of (1.32) by solving the linear equation c ( w, X ) h ( w, p ) h ( w, p ) · · · h N − ( w, p ) = Y h ( w, p ) h ( w, p ) · · · h N − ( w, p ) , where h ( w, p ) ≡
1. This implies that h l ( w, p ) ( l = 1 , · · · , N −
1) are given by rational functions of X and Y . Therefore, p h l ( w, p ) is meromorphic on R .(2) From Lemma 1.2 (1) and the expression (1.32) of ψ , the poles of R−{ p ∞ } ∋ p ψ (( z, [ γ ]) , w, p ) ∈ P ( C ) are coming only from the poles of p h l ( w, p ) ( l = 1 , · · · , N − z, [ γ ]).(3) From the procedure of the Riemann surface R , we can take sufficiently small neighborhood U ∞ such that the set π − ( X ) consists N distinct points for any X ∈ U ∞ − { p ∞ } . Then, p ∈ R − { p ∞ } such that π ( p ) = X ∈ U ∞ corresponds to ( X, Y ) = (
X, A ( λ j )) for j = 1 , · · · , N , where λ Nj = X . Then, ψ (( z, [ γ ]) , w, p ) corresponds to Ψ(( z, [ γ ]) , w, λ j ) of (1.3) and (1.32). So, from (1.3), h ( w, p ) = ∂∂z Ψ(( z, [ γ ]) , w, λ j ) (cid:12)(cid:12)(cid:12) ( z, [ γ ])=( w, [ id ]) = λ j (1 + O ( λ − j )) . (1.33)By the way, we take a sufficiently small and simply connected neighborhood V ⊂ C − N of w . Let x ∈ V . We have the logarithmic derivative of ψ at w : ∂∂z log ψ (( z, [ γ ]) , w, p ) (cid:12)(cid:12)(cid:12) ( z, [ γ ])=( w, [ id ]) = ∂∂z ψ (( z, [ γ ]) , w, p ) ψ (( z, [ γ ]) , w, p ) (cid:12)(cid:12)(cid:12) ( z, [ γ ])=( w, [ id ]) = h ( w, P ) . By changing the base point w , which defines the universal covering X , to a point x of the simply connectedneighborhood V , we can regard x h ( x, p ) as a single-valued holomorphic function on V . So, if z ∈ V and γ ⊂ V , then we locally have the expression ψ (( z, [ γ ]) , w, p ) = exp (cid:16) Z γ h ( x, p ) dx (cid:17) . (1.34)From (1.33) and (1.34), we have the assertion of (3). Remark 1.4.
The expression (1.34) is valid only for sufficiently close ( x, [ γ ]) to ( w, [ id ]) , because weused the change of the base point from w to x . We note that h ( x, p ) depends on the choice of the basepoint. Generically, x h ( x, p ) can be globally multivalued and the expression (1.34) does not holds. For X ∈ C , π − ( X ) = p + · · · + p N gives a divisor on R . We set G (( z, [ γ ]) , w, X ) = det ψ (( z, [ γ ]) , w, p ) · · · ψ (( z, [ γ ]) , w, p N ) ∂∂z ψ (( z, [ γ ]) , w, p ) · · · ∂∂z ψ (( z, [ γ ]) , w, p N ) · · · · · · · · · ∂ N − ∂z N − ψ (( z, [ γ ]) , w, p ) · · · ∂ N − ∂z N − ψ (( z, [ γ ]) , w, p N ) . (1.35)for fixed ( z, [ γ ]) and w . We note that X G (( z, [ γ ]) , w, X ) is well-defined on X ∈ C . Lemma 1.4.
For ( z, [ γ ]) ∈ X and w ∈ C −N , any poles of the function R−{∞} ∋ p ψ (( z, [ γ ]) , w, p ) ∈ P ( C ) analytically depend on the base point w . They are not independent of w .Proof. For a fixed base point w , if q is a pole of p ψ (( z, [ γ ]) , w, p ), by Lemma 1.3 (2), it holds that ψ (( z, [ γ ]) , w, q ) = ∞ for any ( z, [ γ ]) ∈ X . So, if we can take q which is independent of w , we have11 (( z, [ γ ]) , w, q ) = ∞ for any ( z, [ γ ]) ∈ X and w ∈ C − N . This is a contradiction, because we have ψ (( w, [ id ]) , w, q ) = 1 by the definition (1.32) of ψ . So, any pole of p ψ (( z, [ γ ]) , w, p ) is not independentof w . Moreover, from the proof of Lemma 1.3 (1), such poles are coming from the zeros of the commondenominator of h ( w, p ) , · · · , h N − ( w, p ). We note that the common denominator is given by a polynomialin X and Y analytically parametrized by w . So, poles analytically depend on w . Lemma 1.5.
Take w ∈ C − N and ( z, [ γ ]) ∈ X . Then, the correspondence X G (( z, [ γ ]) , w, X ) of(1.35) gives a rational function of X . Moreover, this rational function has a pole at X = ∞ of degree N − .Proof. Due to Lemma 1.3 (1) and the properties of determinant of (1.35), the correspondence X G (( z, [ γ ]) , w, X ) defines a meromorphic function on C = P ( C ) − {∞} . Now, take a sufficiently small andsimply connected neighborhood V ⊂ C − N of w . Although p ψ (( z, [ γ ]) , w, p ) for z ∈ V and γ ⊂ V has an exponential singularity at p ∞ (Lemma 1.3 (3)), we will see that X G (( z, [ γ ]) , w, X ) is analyticaround X = ∞ . Taking a sufficiently small neighborhood U ∞ , ψ of (1.32) is given by Ψ of (1.3) andholomorphic on U ∞ − { p ∞ } . Then, considering the properties of the determinant of (1.35), and the factthat e λ ( z − w ) · · · e λ N ( z − w ) = 1, we can see that G (( z, [ γ ]) , w, X ) has the form det O ( λ − ) · · · O ( λ − N ) λ (1 + O ( λ − )) · · · λ N (1 + O ( λ − N )) · · · · · · · · · λ N − (1 + O ( λ − )) · · · λ N − N (1 + O ( λ − N )) , (1.36)around X = ∞ . Then, (1.36) is a symmetric series in λ , · · · , λ N with the highest term of degree2(0 + 1 + · · · + ( N − N ( N − X = X , X gives a complex coordinate around X = ∞ and (1.36) gives a Laurent series in X . Due to Lemma 1.3 (1) and the assumption (1.31), applying theRiemann extension theorem, (1.36) is holomorphic at X = 0 and has a zero of degree N ( N − N = N − z ∈ V and γ ⊂ V . By the analytic continuation in terms of ( z, [ γ ]) ∈ X , we have the assertion. Theorem 1.4.
Assume the condition (1.31). Suppose the algebraic curve R given by the defining equation(1.27) is non-singular and of genus g . Then, for w ∈ C − N and ( z, [ γ ]) ∈ X , the function R − { p ∞ } ∋ p ψ (( z, [ γ ]) , w, p ) ∈ P ( C ) (1.37) has g poles.Proof. Let κ be the number of poles of the function of (1.37). Since ψ is given by (1.32), togetherwith Lemma 1.3 (2), we can see that the set of the poles of p ψ (( z, [ γ ]) , w, p ) corresponds to that of p ∂ r ∂z r ψ (( z, [ γ ]) , w, p ) ( r ≥ X G (( z, [ γ ]) , w, X ) of(1.35), the number of the poles of the function P ( C ) − {∞} ∋ X G (( z, [ γ ]) , w, X ) ∈ P ( C )is equal to 2 κ . Together with Lemma 1.5, the number of poles of the function P ( C ) ∋ X G (( z, [ γ ]) , w, X ) ∈ P ( C ) (1.38)is equal to 2 κ + N −
1. Since (1.38) is a rational function of the variable X , this function has 2 κ + N − P ( C ) − {∞} .On the other hand, from Lemma 1.4 and the fact that the ramification points of π are isolated points of R , for generic base point w ∈ C − N , all poles of p ψ (( z, [ γ ]) , w, p ) are out of the set of the ramificationpoints of π . We fix such a base point w . From the definition (1.35), the function X G (( z, [ γ ]) , w, X )vanishes at X ( = ∞ ) if and only if X is a branch point of the covering π. Letting e p be the ramificationindex of π at p ∈ R . From the property of determinants of matrices, the right hand side of (1.35)12as zeros of degree 2(0 + 1 + ( e p − e p ( e p −
1) of a coordinate around p ∈ R . So, at X = π ( p ), X G (( z, [ γ ]) , w, X ) has zeros of degree e p ( e p − e p = e p − . Therefore, the degree of zeros of the functionof (1.38) coincides with X p ∈R−{ p ∞ } ( e p − . So, together with Lemma 1.5, X p ∈R ( e p −
1) = 2 κ + 2 N − . (1.39)By the way, applying the Riemann-Hurwitz formula, we have X p ∈R ( e p −
1) = (2 g −
2) + N (2 − . (1.40)By (1.39) and (1.40), we have κ = g . Therefore, we have proved the assertion for generic w . Sincethe number of poles of p ψ (( z, [ γ ]) , w, p ) on R is analytically dependent on the variable w , this is aconstant function of w . Thus, for every w , the number of poles is equal to g .Next, we consider the case that the algebraic curve R of (1.27) has singular points S ( ⊂ R ). We have aresolution of singularities σ : ˜ R → R . Here, σ is given by a composition ˜ R = R l → R l − → · · · → R = R of blowing ups σ ν : R ν → R ν − for a singular point of multiplicity m ν ∈ Z > ( ν = 1 , · · · , κ ). We havean N to 1 covering π ◦ σ : ˜ R → P ( C ) . By considering the divisor ( π ◦ σ ) − ( X ) for X ∈ P ( C ) − {∞} , we can define the function X G (( z, [ γ ]) , w, X ), also. By a similar argument of the proof of Theorem1.4 and considering properties of the blowing ups (for example, see [G]), X G (( z, [ γ ]) , w, X ) has zeros,not only at the branch points of π ◦ σ , but also the images of S under π , where the sum of the orders ofzeros is at most l X ν =1 m ν ( m ν − . Applying the argument of the proof of Theorem 1.4 to the non-singularcurve ˜ R , we have the following. Corollary 1.3.
Using the above notations and letting g be the genus of R , the function p ψ (( z, [ γ ]) , w, p ) has at most g + l X ν =1 m ν ( m ν − poles. We note that ̟ ( R ) = g + l X ν =1 m ν ( m ν − R . If R is non-singular, g = ̟ ( R ) holds. From Proposition 1.3, operators Q ( z, [ γ ]) of (1.11), which commutes with P z of (1.1), can be multivaluedon C − N . However, they are sometimes single-valued on C − N . In this subsection, we give a criterionfor such single-valued differential operators by applying the results of the eigenfunction ψ in the previoussubsection. Theorem 1.5.
For the differential operators P z and Q ( z, [ γ ]) , assume the condition (1.31). Suppose N is a prime number. Let ̟ ( R ) be the arithmetic genus of R . If ̟ ( R ) < N , every coefficient of Q ( z, [ γ ]) issingle-valued on C − N .Proof. By the assumption (1.31), the eigenvalues A ( λ ) , · · · , A ( λ N ) are distinct. We have the eigenfunc-tion ψ of (1.32). Due to Lemma 1.4, we can take the base point w such that there exist a pole q of thefunction p ψ (( z, [ γ ]) , w, p ) which is not a ramification point of the projection π : R → P ( C ) . We assume that there exists [ δ ] ∈ π ( C − N ) such that σ [ δ ] = id. (1.41)13or generic X ∈ P ( C ) − {∞} where X is not a branch point of π and π − ( X ) consists of N distinctpoints p j = ( X, A ( λ j )) ( j = 1 , · · · , N ), there are k , k ∈ { , · · · , N } such that k = k and σ − δ ] ( p k ) = p k . (1.42)From (1.28) and (1.29), (1.42) means that Q [ γ ] ,X Ψ(( z, [ γ · δ ]) , w, λ k ) = A ( λ k )Ψ(( z, [ γ · δ ]) , w, λ k ) . Since the eigenvalues of Q [ γ ] ,X are distinct, we obtainΨ(( z, [ γ · δ ]) , w, λ k ) = const Ψ(( z, [ γ ]) , w, λ k ) (1.43)for generic ( z, [ γ ]) and X . Since N is a prime number, by fixing the branch λ of N √ X and letting ζ N bethe N -th root of the unity, we can suppose that λ k = λ and λ k = ζ lN λ for some l ∈ { , · · · , N − } .Recalling the form of Ψ of (1.3), the equation (1.43) induces the relation (cid:16) ∞ X s =0 ξ s (( z, [ γ · δ ]) , w ) λ − s (cid:17) e λ ( z − w ) = const (cid:16) ∞ X s =0 ξ s (( z, [ γ ]) , w )(( ζ lN λ ) − s ) (cid:17) e ( ζ lN λ )( z − w ) (1.44)for generic ( z, [ γ ]) and λ . By substituting ζ lN λ for λ , we have (cid:16) ∞ X s =0 ξ s (( z, [ γ · δ ]) , w )( ζ lN λ ) − s (cid:17) e ( ζ lN λ )( z − w ) = const (cid:16) ∞ X s =0 ξ s (( z, [ γ ]) , w )( ζ lN λ ) − s (cid:17) e ( ζ lN λ )( z − w ) from (1.44). This means that it holds Ψ(( z, [ γ · δ ]) , w, λ k ) = const Ψ(( z, [ γ ]) , w, λ k ) for generic ( z, [ γ ])and λ , where λ = ζ lN λ . Setting p k = ( X, A ( λ )), we have σ − δ ] ( p k ) = p k because Ψ(( z, [ γ · δ ]) , w, λ k ) = const Ψ(( z, [ γ · δ ]) , w, λ k ) = const Ψ(( z, [ γ ]) , w, λ k ) holds. This implies that σ − δ ] ( p k ) = p k . Repeating this argument, putting p m = ( X, A ( λ k m )) for λ k m = ζ mlN λ , we have σ − δ m ] ( p k ) = p k m ( m = 0 , · · · , N − . (1.45)Since N is a prime number and A ( λ ) is given by the form (1.12), p k , · · · , p k N − are distinct and π − ( X ) = { p k , · · · , p k N − } . Namely, (1.45) means that the action of the group h σ [ δ ] i , which is generated by σ [ δ ] ,is transitive on the fibre π − ( X ) for generic X .Recalling the eigenfunction ψ , (1.45) implies that ψ (( z, [ γ · δ m ]) , w, p ) = const ψ (( z, [ γ ]) , w, σ − δ m ] ( p )) (1.46)for m = 0 , · · · , N −
1, if π ( p ) is not a branch point of π . At the beginning of the proof, we tookthe pole q of the function p ψ (( z, [ γ ]) , w, p ) such that π ( q ) is not a branch point. Since the polesof p ψ (( z, [ γ ]) , w, p ) do not depend on ( z, [ γ ]) (see Lemma 1.3 (2)), (1.46) yields that σ [ δ m ] ( q ) for m ∈ { , · · · , N − } are also poles. So, we have at least N distinct poles of p ψ (( z, [ γ ]) , w, p ).However, due to the assumption, Theorem 1.4 and its corollary, we have at most ̟ ( R )( < N ) polesof p ψ (( z, [ γ ]) , w, p ). This is a contradiction. So, the assumption (1.41) is false. Therefore, σ [ δ ] = id for any [ δ ] ∈ π ( C − N ). According to Theorem 1.3, this means that all of the coefficients of Q ( z, [ γ ]) aresingle-valued. For a commutative algebra A , we set Sp ( n, A ) = { α ∈ GL (2 n, A ) | t αJα = J } , where J = (cid:18) − I n I n (cid:19) .The Siegel upper half plane H n is given by H n = { Ω ∈ M n ( C ) | t Ω = Ω , Im(Ω) > } . Here, Im(Ω) > n = 1, H is the ordinary upper half plane14 = { z ∈ C | Im( z ) > } . For α ∈ Sp ( n, R ) given by α = (cid:18) A BC D (cid:19) , where
A, B, C, D ∈ M n ( R ), we havethe point α (Ω) = ( A Ω + B )( C Ω + D ) − ∈ H n . Set j ( α, Ω) = det( C Ω + D ) . We note that j α (Ω) = 0 . To define automorphic forms, we will consider the case that the commutative ring A is given by atotally real field F such that [ F : Q ] = g. Let ϕ , · · · , ϕ g : F ֒ → R be distinct g embeddings. Set a = { ϕ , · · · , ϕ g } . For any α ∈ Sp ( n, F ), let α ϕ j be the matrix whose components are given by theimage of the components of α under ϕ j . So, a embeds Sp ( n, F ) to Sp ( n, R ) g by α ( α ϕ , · · · , α ϕ g ) . From now on, we will identify Sp ( n, F ) with its image in Sp ( n, R ) g via this embedding. Then, for α = ( α ϕ , · · · , α ϕ g ) ∈ Sp ( n, F ) g and Ω = (Ω , · · · , Ω g ) ∈ H gn , we set α (Ω) = ( α ϕ (Ω ) , · · · , α ϕ g (Ω g )) ∈ H gn . (2.1)For any C -valued function f on H gn and K ∈ Z , we set f | [ α ] K (Ω) = j α (Ω) − K f ( α (Ω)) , (2.2)where j α (Ω) = g Y ν =1 j ( α ϕ j , Ω j ) . Throughout this paper, we use these notations.Let O F be the ring of integers of F . For an ideal c ⊂ O F , we set Γ( c ) = { α ∈ Sp ( n, O F ) | α − I n ∈ c M (2 n, O F ) } . For a group Γ ⊂ Sp ( n, F ), if there exists an ideal c such that Γ contains Γ( c ) as a finiteindex subset, Γ is called a congruence subgroup of Sp ( n, F ) . Definition 2.1.
Let Γ ⊂ Sp ( n, F ) be a congruence subgroup. If a function f on H gn satisfies the followingconditions (i), (ii) and (iii), we call f an automorphic form for Γ of weight K .(i) f is holomorphic on H gn .(ii) f satisfies f | [ α ] K = f for any α ∈ Γ .(iii) When F = Q and n = 1 , f | [ α ] K (Ω) has a holomorphic Fourier expansion at cusps for any α ∈ SL (2 , Z ) . Namely, f | [ α ] K (Ω) = ∞ X k =0 ˜ f α,k exp (cid:16) π √− k Ω N α (cid:17) , holds, where ˜ f α,k ∈ C and N α ∈ Z > . Here, ‘holomorphic’ means that the Fourier expansion does not have any terms for k < . Remark 2.1.
The case of F = Q and n = 1 is an exceptional case. The condition (iii) is a growthcondition for the cusps of Γ . When F = Q or n ≥ , such a condition follows from the conditions (i) and(ii) (Koecher’s principle). We note that automorphic forms of several variables are defined in various literature. Our definitionabove of automorphic forms is due to [Sm2]. This definition seems general enough for applicationsbecause we can obtain important modular functions as reductions. For example, if Γ = Sp ( n, Q ), thenthe corresponding automorphic forms are well-known Siegel modular forms. If F = Q and n = 1, thenthe corresponding automorphic forms are Hilbert modular forms. Let a l (Ω , z ) ( l = 2 , · · · , N ) be a function of Ω = (Ω , · · · , Ω g ) ∈ H gn and z ∈ C . We suppose thatΩ a l (Ω , z ) is holomorphic for generic z . Moreover, for fixed Ω, let z a l (Ω , z ) be an analytic functionwith at most poles. We consider the cases that a l (Ω , z ) satisfies the transformation law a l (cid:18) α (Ω) , zj α (Ω) (cid:19) = j α (Ω) l a l (Ω , z ) , (2.3)for α ∈ Γ. For fixed Ω ∈ H gn , let N Ω ⊂ C = ( z -plane) be the union of the sets of poles of the function a j (Ω , z ) ( j = 2 , · · · , n ). Namely, for a fixed Ω ∈ H gn , a (Ω , z ) , · · · , a N (Ω , z ) are holomorphic functions of z ∈ C − N Ω . 15 emark 2.2. If n = 1 , the action (Ω , z ) (cid:16) α (Ω) , zj α (Ω) (cid:17) is equal to the action which defines the Jacobiforms of degree (see [EZ]). However, if n ≥ , our action is different from the action for Jacobi formsof higher degrees studied in [Z]. Lemma 2.1.
For any α ∈ Γ and Ω ∈ H gn , z ∈ C − N Ω if and only if zj α (Ω) ∈ C − N α (Ω) . Proof.
Due to the transformation law (2.3), it holds that z ∈ C − N Ω ⇐⇒ a l (Ω , z ) = ∞ ( l = 2 , · · · , N ) ⇐⇒ a l (cid:16) α (Ω) , zj α (Ω) (cid:17) = j α (Ω) l a l (Ω , z ) = ∞ . ⇐⇒ zj α (Ω) ∈ C − N α (Ω) . Let X Ω be the universal covering of C − N Ω . For a fixed point w ∈ C − N Ω , any s ∈ X Ω is representedby s = ( z, [ γ ]) , where z ∈ C − N Ω , γ is an arc in C − N Ω from w to z and [ γ ] is the homotopy class of γ .We note that z gives a local coordinate of X Ω .Let us consider the following ordinary differential operator of the independent variable z : P Ω ,z = ∂ N ∂z N + a (Ω , z ) ∂ N − ∂z N − + a (Ω , z ) ∂ N − ∂z N − + · · · + a N (Ω , z ) . (2.4)Set (Ω , z ) = ( α (Ω) , zj α (Ω) ). Throughout this paper, we assume that Ω a l (Ω , z ) are holomorphic forgeneric z . Since ∂∂z = j α (Ω) ∂∂z and a l (Ω , z ) = j α (Ω) l a l (Ω , z ), we have P Ω ,z = j α (Ω) N P Ω ,z . (2.5) Definition 2.2.
Let D Ω ,z be a linear differential operator of z holomorphically parametrized by Ω ∈ H gn .If D Ω ,z = j α (Ω) K D Ω ,z (2.6) holds for α ∈ Γ , we call D Ω ,z a differential operator of weight K with respect to the action of Γ . We call D Ω ,z u = 0 a linear differential equation of weight K with respect to the action of Γ . There exist many important examples which satisfy the transformation law (2.3).
Example 2.1.
Let Γ be a congruence subgroup of SL ( n, F ) . If f (Ω) be an automorphic form of weight j , then a j + k (Ω , z ) = z − k f (Ω) satisfies the transformation law (2.3) for l = j + k for any k ∈ Z ≥ . If k > , then N Ω = { } holds. Example 2.2.
For a congruence subgroup Γ ⊂ SL (2 , Z ) , a weak Jacobi form H × C ∋ (Ω , z ) f (Ω , z ) ∈ C for Γ ⊂ SL (2 , Z ) of weight K and level m is a holomorphic function with the following properties(i) for any α = (cid:18) a bc d (cid:19) ∈ Γ , f ( α (Ω) , zj α (Ω) ) = j α (Ω) K exp( − πi mcz j α (Ω) ) f (Ω , z ) ,(ii) for any n , n ∈ Z f (Ω , z + n Ω + n ) = exp( − πi ( n Ω + 2 n z )) f (Ω , z ) ,(iii) f has a Fourier expansion f (Ω , z ) = X n,l ∈ Z c n,l exp (cid:16) π √− n Ω N (cid:17) exp(2 π √− nz ) for some N ∈ Z .Weak Jacobi forms are very important in number theory (see [EZ]). If f (Ω , z ) ( g (Ω , z ) , resp.) is a weakJacobi form for Γ of weight K ( K , resp.) and level m , then a l (Ω , z ) = f (Ω ,z ) g (Ω ,z ) satisfies the transformationlaw of (2.3) for n = g = 1 and l = K − K . xample 2.3. As a special case of 2.2, we consider the Lam´e differential operator P Ω ,z = ∂ ∂z − B℘ (Ω , z ) , (2.7) where Ω ∈ H and ℘ (Ω , z ) is the Weierstrass ℘ -function ℘ (Ω , z ) = 1 z + X ( n ,n ) ∈ Z −{ (0 , } (cid:16) z − n − n Ω) − n + n Ω) (cid:17) . We note that z ℘ (Ω , z ) has poles of degree at every z ∈ N Ω := Z + Z Ω .Let Γ be the elliptic full-modular group SL (2 , Z ) . For any α ∈ Γ , we have ℘ (cid:16) α (Ω) , zj α (Ω) (cid:17) = j α (Ω) ℘ (Ω , z ) . Especially, ℘ (Ω + 1 , z ) = ℘ (Ω , z ) holds and ℘ has the Fourier expansion at cusps: ℘ (Ω , z ) = π (cid:16) − ∞ X n =1 nq n − q n (cid:17) + π sin ( πz ) − π ∞ X n =1 cos(2 nπz ) nq n − q n , where q = exp(2 π √− (for detail, see [EMOF]). Therefore, in terms of Definition 2.2, P Ω ,z of (2.7)is a differential operator of weight with respect to the action of Γ = SL (2 , Z ) . When γ : [0 , → C − N Ω is an arc with γ (0) = w and γ (1) = z , let γ = γj α (Ω) be the arc given by γ ( t ) = γ ( t ) j α (Ω) . By virtue of Lemma 2.1, γ is an arc in C − N α (Ω) . Theorem 2.1.
Let P Ω ,z be the differential operator of (2.4).(1) There exists the unique formal solution Ψ(Ω , ( z, [ γ ]) , w, λ ) of the differential equation P Ω ,z u = λ N u (2.8) in the form Ψ(Ω , ( z, [ γ ]) , w, λ ) = (cid:16) ∞ X s =0 ξ s (Ω , ( z, [ γ ]) , w ) λ − s (cid:17) e λ ( z − w ) (2.9) such that ( ξ (Ω , ( z, [ γ ]) , w ) ≡ ,ξ s (Ω , ( w, [ id ]) , w ) = 0 ( s ≥ . (2.10) Here, Ω ξ s (Ω , ( z, [ γ ]) , w ) are holomorphic for generic (( z, [ γ ]) , w ) . Moreover, for a fixed Ω ∈ H gn , (( z, [ γ ]) , w ) ξ s (Ω , ( z, [ γ ])) are locally holomorphic.(2) For any α ∈ Γ , it holds Ψ (cid:16) α (Ω) , (cid:16) zj α (Ω) , h γj α (Ω) i(cid:17) , wj α (Ω) , j α (Ω) λ (cid:17) = Ψ(Ω , ( z, [ γ ]) , w, λ ) . (2.11) The function ξ s (Ω , ( z, [ γ ]) , w ) in (2.9) satisfies the transformation law ξ s (cid:18) α (Ω) , (cid:18) zj α (Ω) , (cid:20) γj α (Ω) (cid:21)(cid:19) , wj α (Ω) (cid:19) = j α (Ω) s ξ s (Ω , ( z, [ γ ]) , w ) . (2.12) Proof. (1) For fixed Ω ∈ H g , putting u = (cid:16) ∞ X s =0 η s (Ω , z ) λ − s (cid:17) e λ ( z − w ) to (2.8), by the same argument as inthe proof of Proposition 1.1, we can obtain N ∂∂z η N + s − (Ω , z )= (cid:18) a polynomial in ∂ ν ∂z ν η l (Ω , z ) ( l < N + s − , ν ∈ Z ≥ ) and a j (Ω , z ) defined over Z (cid:19) (2.13)17or any s . By the integration of the relation (2.13) on arc γ ∈ C − N Ω whose start point is w , we canobtain the expression of η µ (Ω , z ) in terms of η ν (Ω , z ) ( ν < µ ) and a l (Ω , z ). From the conditions that η ≡ η s (Ω , ( w, [ id ])) = 0 ( s ≥ { η s (Ω , z ) } s uniquely. Such η s (Ω , ( z, [ γ ])) give the required functions ξ s (Ω , ( z, [ γ ]) , w ).Moreover, since the coefficients a l (Ω , z ) of (2.4) are holomorphic functions of Ω for generic z and ξ s (Ω , ( z, [ γ ]) , w ) are determined by the construction via (2.13), ξ s (Ω , ( z, [ γ ]) , w ) are holomorphic functionsof Ω ∈ H gn for generic (( z, [ γ ]) , w ). Also, for fixed Ω, ξ s (Ω , ( z, [ γ ]) , w ) are locally holomorphic functionsof (( z, [ γ ]) , w ) ∈ X Ω × ( C − N Ω ).(2) We consider the transformation(Ω , ( z, [ γ ]) , w, λ ) (Ω , ( z , [ γ ]) , w , λ ) = (cid:18) α (Ω) , (cid:18) zj α (Ω) , (cid:20) γj α (Ω) (cid:21)(cid:19) , wj α (Ω) , j α (Ω) λ (cid:19) . (2.14)By virtue of (2.5), the differential equation P Ω ,z u = λ N u gives the same equation with P Ω ,z u = λ N u under the correspondence (2.14). Since we have the uniqueness of the solution Ψ in the form of (2.9) andthe condition (2.10), we obtain (2.11). Then, we have (cid:16) ∞ X s =0 ξ s (Ω , ( z, [ γ ]) , w ) λ − s (cid:17) e λ ( z − w ) = (cid:16) ∞ X s =0 ξ s (Ω , ( z , [ γ ]) , w ) λ − s (cid:17) e λ ( z − w ) . (2.15)By cancelling e λ ( z − w ) = e λ ( z − w ) and comparing the coefficient of λ − s = λ − s j α (Ω) s , we have thetransformation law (2.12). F = Q or n ≥ ) We consider the differential operator Q Ω , ( z, [ γ ]) = b (Ω , ( z, [ γ ])) ∂ M ∂z M + b (Ω , ( z, [ γ ])) ∂ M − ∂z M − + · · · + b M (Ω , ( z, [ γ ])) , (2.16)which commutes with the differential operator P Ω ,z of (2.4). Here, we assume that the coefficients b k (Ω , ( z, [ γ ])) ( k = 0 , · · · , M ) are locally analytic functions of ( z, [ γ ]) ∈ X Ω . Theorem 2.2. (1) Let P Ω ,z ( Q Ω , ( z, [ γ ]) , resp.) be the differential operator of (2.4) ((2.16), resp.). Then, P Ω ,z and Q Ω , ( z, [ γ ]) are commutative if and only if the quotient Q Ω , ( z, [ γ ]) Ψ(Ω , ( z, [ γ ]) , w, λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) for Ψ of (2.9)coincides with A (Ω , λ ) = ∞ X s = − M A s (Ω) λ − s (2.17) for generic λ , where A (Ω , λ ) does not depend on the variables z and w .(2) If P Ω ,z commutes with both Q (1)Ω , ( z, [ γ ]) and Q (2)Ω , ( z, [ γ ]) , then Q (1)Ω , ( z, [ γ ]) commutes with Q (2)Ω , ( z, [ γ ]) .(3) If the differential operator Q Ω , ( z, [ γ ]) is of weight K with respect to the action of Γ , then the membersof the sequence { A s (Ω) } s satisfy A s ( α (Ω)) = j α (Ω) K + s A s (Ω) . (2.18) Proof. (1) (2) These are proved by a similar argument to the proof of Propositon 1.3 and Proposition1.4.(3) We recall that Ψ in (2.9) satisfies (2.11). Since Q Ω , ( z, [ γ ]) is of weight K , we have A (Ω , λ ) = j α (Ω) K A (Ω , λ ) . Namely, we have ∞ X s = − M A s (Ω ) λ − s = j α (Ω) K ∞ X s = − M A s (Ω) λ − s . By comparing the coefficients of λ − s , the assertion follows.18 heorem 2.3. For any j ∈ { , − , · · · , − M } , let A j (Ω) satisfy the transformation law A j (Ω ) = j α (Ω) K + j A j (Ω) (2.19) for any α ∈ Γ . If there exists a differential operator Q Ω , ( z, [ γ ]) of rank M of weight K with respect to Γ satisfying Q Ω , ( z, [ γ ]) Ψ(Ω , ( z, [ γ ]) , w, λ ) = A (Ω , λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) , (2.20) where A (Ω , λ ) = ∞ X s = − M A s (Ω) λ − s , then Q Ω , ( z, [ γ ]) is uniquely determined only by given operator P Ω ,z of(2.4) and the functions A j (Ω) ( j = 0 , · · · , − M ) . Here, A s (Ω) ( s ≥ are also uniquely determined onlyby P Ω ,z and A j (Ω) ( j = 0 , · · · , − M ) .Proof. Let Ψ(Ω , ( z, [ γ ]) , w, λ ) be the solution of (2.9) for the differential equation P Ω ,z u = Xu , where X = λ N . Let { A s (Ω) } s be the sequence satisfying the relation (2.18) and set A (Ω , λ ) = ∞ X s = − M A s (Ω) λ − s . If there exists a differential operator Q Ω , ( z, [ γ ]) ,w satisfying (2.20), then Q Ω , ( z, [ γ ]) ,w is a differential operatorof weight K with respect to Γ and commutes with P Ω ,z . Next, taking w ′ ∈ C − N Ω and another solutionΨ(Ω , ( z, [ γ ]) , w ′ , λ ) , we suppose that there is an operator Q Ω , ( z, [ γ ] ,w ′ ) such that Q Ω , ( z, [ γ ]) ,w ′ Ψ(Ω , ( z, [ γ ]) , w ′ , λ ) = A (Ω , λ )Ψ(Ω , ( z, [ γ ]) , w ′ , λ ) . (2.21)As in Proposition 1.2 and Proposition 1.3, there exists B (Ω , λ ) such that Ψ(Ω , ( z, [ γ ]) , w ′ , λ ) e λ ( w ′ − w ) = B (Ω , λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) . Therefore, by (2.20) and (2.21), Q Ω , ( z, [ γ ]) ,w ′ Ψ(Ω , ( z, [ γ ]) , w, λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) = Q Ω , ( z, [ γ ]) ,w ′ B (Ω , λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) e λ ( w − w ′ ) B (Ω , λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) e λ ( w − w ′ ) = Q Ω , ( z, [ γ ]) ,w ′ Ψ(Ω , ( z, [ γ ]) , w ′ , λ )Ψ(Ω , ( z, [ γ ]) , w ′ , λ ) = A (Ω , λ ) = Q Ω , ( z, [ γ ]) ,w Ψ(Ω , ( z, [ γ ]) , w, λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) . Therefore, we obtain ( Q Ω , ( z, [ γ ]) ,w − Q Ω , ( z, [ γ ]) ,w ′ )Ψ(Ω , ( z, [ γ ]) , w, λ ) = 0 . By a similar argument to the endof the proof of Proposition 1.3, we can see that Q Ω , ( z, [ γ ]) ,w = Q Ω , ( z, [ γ ]) ,w ′ . Hence, a differential operator Q Ω , ( z, [ γ ]) ,w satisfying (2.20) does not depend on the base point w. So, we use the notation Q Ω , ( z, [ γ ]) instead of Q Ω , ( z, [ γ ]) ,w .Now, we see that the differential operator Q Ω , ( z, [ γ ]) and the series A (Ω , λ ) satisfying (2.20) are uniquelydetermined by P Ω ,z and A (Ω) , · · · , A − M (Ω). The relation (2.20) is equal to ∞ X s =0 M X k =0 k X α =0 (cid:18) kα (cid:19) b M − k (Ω , ( z, [ γ ])) ∂ α ∂z α ξ s (Ω , ( z, [ γ ]) , w ) λ k − α − s = ∞ X t = − M A t (Ω) λ − t ! ∞ X s =0 ξ s (Ω , ( z, [ γ ]) , w ) λ − s ! . (2.22)We note that { ξ s } s is determined only by the given differential operator P Ω ,z by Theorem 2.1 (1). Forany j ∈ { , , · · · , M } , recalling that ξ ≡ λ M − j in theequation (2.22), we have b j (Ω , ( z, [ γ ])) + (cid:16) a polynomial in b k (Ω , ( z, [ γ ])) ( k ≤ j −
1) and ∂ ν ∂z ν ξ s (Ω , ( z, [ γ ]) , w ) ( ν ∈ Z ≥ ) (cid:17) = (cid:16) a polynomial in A t (Ω) ( t ≤ j − M ) and ξ s (Ω , ( z, [ γ ]) , w ) (cid:17) . (2.23)From (2.23), we can obtain b j (Ω , ( z, [ γ ])) ( j = 0 , , · · · , M ) inductively. This shows that A − M (Ω) , · · · , A ( M )and the differential operator P Ω ,z uniquely determine Q Ω , ( z, [ γ ]) . Moreover, since ξ ≡ λ − s in the relation (2.22), we have A s (Ω) + (cid:16) a polynomial in A t (Ω) ( t < s ) and ∂ ν ∂z ν ξ s (Ω , ( z, [ γ ]) , w ) ( ν ∈ Z ≥ ) (cid:17) = (cid:16) a polynomial in b k (Ω , ( z, [ γ ])) (0 ≤ k ≤ M ) and ξ s (Ω , ( z, [ γ ]) , w ) (cid:17) , (2.24)for any s ≥ − M . Especially, A s (Ω) ( s ≥
1) are inductively determined by (2.24). Here, we note thatsuch A s (Ω) ( s ≥
1) do not depend on z and w by virtue of Theorem 2.2 (1). Therefore, the assertionfollows.The following theorem gives a correspondence between automorphic forms and differential operatorswhich commutes with P Ω ,z for generic cases of F = Q or n ≥ Theorem 2.4.
Suppose F = Q or n ≥ . Let P Ω ,z of (2.4) ( Q Ω , ( z, [ γ ]) of (2.16), resp.) be differentialoperators studied in Theorem 2.3.(1) If A j (Ω) ( j = 0 , · · · , − M ) are automorphic forms of weight K + j for Γ , then any coefficients b j (Ω , ( z, [ γ ])) of Q Ω , ( z, [ γ ]) ( A s (Ω) ( s ≥ ) of A (Ω , λ ) , resp.), which is derived from P Ω ,z and A j (Ω)( j = 0 , − , · · · , − M ) , give holomorphic functions Ω b j (Ω , ( z, [ γ ])) for generic ( z, [ γ ]) (automorphicforms of weight K + s for Γ , resp.).(2) Conversely, if every coefficient b j (Ω , ( z, [ γ ])) of Q Ω , ( z, [ γ ]) gives a holomorphic function Ω b j (Ω , ( z, [ γ ])) for generic ( z, [ γ ]) , then A s (Ω) ( s ≥ − M ) , which are determined in the sense of Theo-rem 2.2, are automorphic forms for Γ .Proof. (1) Recall Definition 2.1. From the assumption, A j (Ω) ( k ∈ { , · · · , − M } ) are holomorphicfunction of Ω ∈ H gn satisfying the transformation law (2.19). From Theorem 2.1, ξ s (Ω , ( z, [ γ ]) , w ) areholomorphic of Ω ∈ H gn for generic (( z, [ γ ]) , w ). So, due to the construction of b j (Ω , ( z, [ γ ])) via therelation (2.23), b j (Ω , ( z, [ γ ])) are holomorphic of Ω for generic ( z, [ γ ]). Also, from (2.24), A s (Ω) ( s ≥
1) arealso holomorphic in Ω. Moreover, by Theorem 2.2, we obtain A s (Ω) ( s ≥
1) satisfying the transformationlaw (2.18). So, from Definition 2.1, A s (Ω) ( s ≥
1) are automorphic forms for Γ of weight K + s .(2) From Theorem 2.1 and Theorem 2.2, we only need to see that Ω A s (Ω) are holomorphic underour assumption. However, we can see this property, because A (Ω) , · · · , A − M (Ω) are determined by b j (Ω , ( z, [ γ ])) via (2.24) and do not depend on ( z, [ γ ]) and w . F = Q and n = 1 ) In this subsection, we consider exceptional cases of F = Q and n = 1 carefully. In such cases, we needto consider the action of α ∈ SL (2 , Z ), because automorphic forms for such exceptional cases requireholomorphic Fourier expansion at cusps in the sense of Definition 2.1 (iii).Recall that the set of poles of P Ω ,z of (2.4) is given by N Ω . If the coefficients a l (Ω , z ) ( l = 2 , · · · , N ) of P Ω ,z satisfy the transformation law (2.3) for α ∈ SL (2 , Z ), we have N Ω = N α (Ω) for any α ∈ SL (2 , Z ) byvirtue of Lemma 2.1. However, the transformation law (2.3) for α ∈ SL (2 , Z ) generically does not hold.So, we need a bit delicate argument for holomorphic Fourier expansions at cusps. Let j α (Ω) · ( C − N α (Ω) )be the set { j α (Ω) z | z ∈ C − N α (Ω) } . If an arc γ is in j α (Ω) · ( C − N α (Ω) ), then γ = γj α (Ω) is in C − N α (Ω) .We have the following lemma. Lemma 2.2.
Suppose F = Q and n = 1 . For any α ∈ SL (2 , Z ) , we suppose that the coefficients a l (Ω , z )( l = 2 , · · · , N ) have the holomorphic Fourier expansion at cusps: j α (Ω) − l a l (cid:16) α (Ω) , zj α (Ω) (cid:17) = X k ≥ ˜ a l,α,k ( z ) exp (cid:18) π √− k Ω N l,α (cid:19) . (2.25)20 here ˜ a l,α,k ( z ) are holomorphic functions of z ∈ C −N α (Ω) and N l,k ∈ Z > . Here ‘holomorphic’ means thatthe expression (2.25) does not contain any terms for k < . Then, the coefficients ξ s of the multivaluedBaker-Akhiezer function Ψ of (2.9) has a holomorphic Fourier expansion at cusps: j α (Ω) − s ξ s (cid:18) α (Ω) , (cid:18) zj α (Ω) , (cid:20) γj α (Ω) (cid:21)(cid:19) , wj α (Ω) (cid:19) = X k ≥ ˜ ξ s,α,k (( z, [ γ ]) , w ) exp (cid:18) π √− k Ω N s,α (cid:19) , (2.26) where ˜ ξ s,α,k (( z, [ γ ]) , w ) are multivalued function on j α (Ω) · ( C − N α (Ω) ) .Proof. For α ∈ SL (2 , Z ), we set (Ω , ( z , [ γ ]) , w ) = (cid:16) α (Ω) , (cid:16) zj α (Ω) , h γj α (Ω) i(cid:17) , wj α (Ω) (cid:17) . We prove theexistence of the holomorphic Fourier expansions (2.26) of ξ s by a induction for s .If s = 0, it is trivial. If s = 1, ξ (Ω , ( z , [ γ ]) , w ) is determined by the integration of the relation N ∂∂z η (Ω , z ) = − a (Ω , z ) (2.27)on the arc γ ⊂ C − N Ω (recall the proof of Theorem 2.1). Dividing (2.27) by j α (Ω) , considering therelation ∂∂z = j α (Ω) ∂∂z and using the assumption (2.25), we have N ∂∂z j α (Ω) − η (cid:16) α (Ω) , zj α (Ω) (cid:17) = − a (Ω , z ) j α (Ω) = − X k ≥ ˜ a ,α,k ( z ) exp (cid:18) π √− k Ω N ,α (cid:19) . (2.28)By integrating (2.28) on the arc γ ⊂ j α (Ω) · ( C − N α (Ω) ), we have the holomorphic Fourier expansion atcusps for ξ of (2.26).Next, assume that we have the holomorphic Fourier expansion (2.26) of ξ s for s = 0 , , · · · , s − s ≥ ξ s . By the proof of Theorem 2.1 (1),especially the relation (2.13), ξ s (Ω , ( z , [ γ ]) , w ) is given by the integration of the relation N ∂∂z η s (Ω , z ) = H s (cid:16) ∂ ν ∂z ν ξ m (Ω , ( z , [ γ ]) , w ) , a l (Ω , z ) (cid:17) . (2.29)Here, H s (cid:16) ∂ ν ∂z ν ξ m (Ω , ( z , [ γ ]) , w ) , a l (Ω , z ) (cid:17) is a polynomial in ∂ ν ∂z ν ξ m (Ω , ( z , [ γ ]) , w ) ( m ≤ s − , ν ∈ Z ≥ ) and a l (Ω , z ) ( l = 2 , · · · , N ). By Theorem 2.1 (3), the polynomial H s is homogeneous ofweight s + 1 with respect to the action of Γ. This implies that, by dividing (2.29) by j α (Ω) s +1 for α ∈ SL (2 , Z ), the relation N ∂∂z j α (Ω) − s η s (cid:16) α (Ω) , zj α (Ω) (cid:17) = H s (cid:16) ∂ ν ∂z ν j α (Ω) − m ξ m (Ω , ( z , [ γ ]) , w ) , j α (Ω) − l a l (Ω , z ) (cid:17) , (2.30)holds similarly to the (2.28). By the assumption, the right hand side of (2.30) has the holomorphicFourier expansion at cusps. So, by the integration of (2.30) on the arc γ ⊂ j α (Ω) · ( C − N α (Ω) ), we havethe holomorphic Fourier expansion (2.26) at cusps for s .Hence, the assertion is proved. Remark 2.3. If Γ = SL (2 , Z ) , the relation (2.3) holds for any α ∈ SL (2 , Z ) . Then, from Lemma 2.1, j α (Ω) · ( C − N α (Ω) ) = ( C − N Ω ) holds. So, in this case, we only need to consider multivalued functionson C − N Ω . However, if Γ = SL (2 , Z ) , we need a detailed condition as we saw in Lemma 2.2. Theorem 2.5.
Suppose F = Q and n = 1 . Let P Ω ,z of (2.4) ( Q Ω , ( z, [ γ ]) of (2.16), resp.) be differentialoperators studied in Theorem 2.3. Moreover, assume that every coefficient a l (Ω , z ) ( l = 2 , · · · , N ) of thedifferential operator P Ω ,z of (2.4) has a holomorphic Fourier expansion at cusps in the form (2.25).(1) If A j (Ω) ( j = 0 , − , · · · , − M ) are automorphic forms of weight K + j for Γ , then any coefficients b j (Ω , ( z, [ γ ])) of Q Ω , ( z, [ γ ]) , which are derived from A j (Ω) ( j = 0 , − , · · · , − M ) in the sense of Theorem .3, give holomorphic functions of Ω b j (Ω , ( z, [ γ ])) for generic ( z, [ γ ]) . Moreover, for α ∈ SL (2 , Z ) ,every coefficient b j (Ω , ( z, [ γ ])) of Q Ω , ( z, [ γ ]) has a holomorphic Fourier expansion j α (Ω) − K − j + M b j (cid:16) α (Ω) , (cid:16) zj α (Ω) , h γj α (Ω) i(cid:17)(cid:17) = X k ≥ ˜ b j,α,k ( z, [ γ ]) exp (cid:16) π √− k Ω N j,α (cid:17) , (2.31) where ˜ b j,α,k (Ω , ( z, [ γ ])) are multivalued analytic function on j α (Ω) · ( C − N α (Ω) ) . Furthermore, A s (Ω)( s ≥ are automorphic forms of weight K + s for Γ .(2) Conversely, we suppose that every coefficient b j (Ω , ( z, [ γ ])) of Q Ω , ( z, [ γ ]) is a holomorphic functionof Ω and has a holomorphic Fourier expansion (2.31) at cusps. Then, A s (Ω) ( s ≥ − M ) , which aredetermined in the sense of Theorem 2.2, are automorphic forms for Γ .Proof. (1) Under the assumption, as in the proof of Theorem 2.4, we can see that Ω b j (Ω , ( z, [ γ ])) areholomorphic for generic ( z, [ γ ]). We prove that b j (Ω , ( z, [ γ ])) have holomorphic Fourier expansions (2.31)for any α ∈ SL (2 , Z ). The coefficients b j (Ω , ( z, [ γ ])) are determined by the relation (2.23) inductively.For α ∈ SL (2 , Z ) and t = 0 , · · · , − M , we have the holomorphic Fourier expansion j α (Ω) − t A t (Ω) = X k ≥ ˜ A t,α,k exp (cid:16) π √− N t,α (cid:17) (2.32)by the assumption. We set (Ω , ( z , [ γ ]) , w ) = (cid:16) α (Ω) , (cid:16) zj α (Ω) , h γj α (Ω) i(cid:17) , wj α (Ω) (cid:17) . From (2.23), it holdsthat b j (Ω , ( z , [ γ ])) = H bj (cid:16) b m (Ω , ( z , [ γ ])) , ∂ ν ∂z ν ξ s (Ω , ( z , [ γ ]) , w ) , A t (Ω ) (cid:17) , (2.33)where H bj ( b m (Ω , ( z , [ γ ])) , ∂ ν ∂z ν ξ s (Ω , ( z , [ γ ]) , w ) , A t (Ω )) is a polynomial in b m (Ω , ( z , [ γ ])) ( m < j ), ∂ ν ∂z ν ξ s (Ω , ( z , [ γ ]) , w ) ( s, ν ∈ Z ≥ ) and A t (Ω ) ( t = − M, · · · , K + j − M with respect to the action of Γ. This implies that, by dividing (2.33)by j α (Ω) K + j − M for α ∈ SL (2 , Z ) and considering ∂∂z = j α (Ω) ∂∂z , we obtain j α (Ω) − K − j + M b j (Ω , ( z , [ γ ]))= H bj (cid:16) j α (Ω) − K − m + M b m (Ω , ( z , [ γ ])) , ∂ ν ∂z ν j α (Ω) − s ξ s (Ω , ( z , [ γ ]) , w ) , j α (Ω) − t A t (Ω ) (cid:17) . (2.34)By virtue of the assumption and Lemma 2.2, ∂ ν ∂z ν j α (Ω) − s ξ s (Ω , ( z , [ γ ]) , w ) ( j α (Ω) − t A t (Ω ) , resp.)have Fourier expansions (2.26) ((2.32), resp.). So, we can inductively obtain the Fourier expansions (2.31)of j α (Ω) − K − j + M b j (Ω , ( z , [ γ ])).Next, we will consider the Fourier expansion of A s (Ω) for s ≥
1. By (2.24), we obtain A s (Ω ) = H as (cid:16) b j (Ω , ( z , [ γ ])) , ∂ ν ∂z ν ξ s (Ω , ( z , [ γ ]) , w ) , A t (Ω ) (cid:17) , (2.35)where H as (cid:16) b j (Ω , ( z , [ γ ])) , ∂ ν ∂z ν ξ s (Ω , ( z , [ γ ]) , w ) , j α (Ω) − t A t (Ω ) (cid:17) is a polynomial in b j (Ω , ( z , [ γ ]))(0 ≤ j ≤ M ), ∂ ν ∂z ν ξ s (Ω , ( z , [ γ ]) , w ) ( s, ν ∈ Z ≥ ) and A t (Ω ) ( t < s ). We can see that the polynomialis homogeneous of weight s under the action of SL (2 , Z ) also. Therefore, dividing (2.35) by j α (Ω) s ( α ∈ SL (2 , Z )) and using ∂∂z = j α (Ω) ∂∂z , we have j α (Ω) − s A s (Ω )= H as (cid:16) j α (Ω) − K − j + M b j (Ω , ( z , [ γ ])) , ∂ ν ∂z ν j α (Ω) − s ξ s (Ω , ( z , [ γ ]) , w ) , j α (Ω) − t A t (Ω ) (cid:17) . (2.36)So, we can also obtain the Fourier expansions of j α (Ω) − s A s (Ω) inductively.(2) We only need to obtain the holomorphic Fourier expansions of A s (Ω) ( s ≥ − M ) at cusps. By thesame argument with the latter of the proof of (1), we can obtain the Fourier expansion of j α (Ω) − s A s (Ω)inductively from the Fourier expansion (2.31). 22 .5 A formulation via a ring of generating functions In this subsection, we will give an interpretation of Theorem 2.2, 2.3, 2.4 and 2.5 using a ring of generatingfunctions for sequences of automorphic forms.Let Γ ⊂ Sp ( n, F ) be a congruence subgroup and M K (Γ) be the vector space of automorphic formsfor Γ of weight K . It is well-known that M K (Γ) = { } if K < . Let M (Γ) = ∞ M K =0 M K (Γ) be thegraded ring of automorphic forms for Γ. Let V be the ring of formal Laurent series in λ − over M (Γ): V = n ∞ X s = − M A s (Ω) λ − s (cid:12)(cid:12)(cid:12) M ∈ Z ≥ , A s (Ω) ∈ M (Γ) o . We take a subspace R K of V defined by R K = n ∞ X s = − M A s (Ω) λ − s ∈ V (cid:12)(cid:12)(cid:12) A s (Ω) ∈ M s + K (Γ) o . Then, R = ∞ M K =0 R K is a graded ring. The ring R can beregarded as a ring of generating functions for sequences { A s (Ω) } ( A s (Ω) ∈ M s + K (Γ)) of automorphicforms. We note that R gives a subring of R .Now, we define a vector space D PK and a ring D P of differential operators. Let P = P Ω ,z of (2.4) bea differential operator of weight N with respect to Γ. However, if F = Q and n = 1, we additionallyassume that the coefficients a l (Ω , z ) ( l = 0 , · · · , N ) have holomorphic Fourier expansions (2.25) at cusps. Definition 2.3.
For a fixed congruence subgroup Γ( ⊂ Sp ( n, F )) , set ˜ D PK = { Q = Q Ω , ( z, [ γ ]) | Q is given by (2 . Q commutes with P ; Q is of weight K with respect to Γ } . Then,(i) if F = Q or n ≥ , set D PK = { Q ∈ ˜ D PK | Ω b j (Ω , ( z, [ γ ])) are holomorphic for generic ( z, [ γ ]) } . (ii) if F = Q and n = 1 , set D PK = { Q ∈ ˜ D PK | Ω b j (Ω , ( z, [ γ ])) are holomorphic for generic ( z, [ γ ]); b j (Ω , ( z, [ γ ])) have holomorphic Fourier expansions (2 . at cusps } . Set D P = ∞ M K =0 D PK . This is a commutative graded ring (see Theorem 2.2 (2)).
Let χ : D P → R be a mapping given by Q Ω , ( z, [ γ ]) = Q A = A (Ω , λ ) (2.37)if Q Ψ = A Ψ for Ψ = Ψ(Ω , ( z, [ γ ]) , w, λ ) of (2.9) for generic λ . Theorem 2.3 implies that χ is an injectivemapping. Moreover, if Q , Q ∈ D P and χ ( Q j ) = A j ( j = 1 , Q + Q )Ψ = ( A + A )Ψ and( Q Q )Ψ = A A Ψ . So, by a similar argument to the end of the proof of Proposition 1.3, we can see that χ ( Q + Q ) = A + A , χ ( Q Q ) = A A and χ (1) = 1 hold. Namely, χ gives an embedding D P ֒ → R of rings. Definition 2.4.
Let S PK be the vector space χ ( D PK )( ⊂ R ) over C . Let S P be the graded ring χ ( D P ) = ∞ M K =0 S PK . For any A = ∞ X s = − M A s (Ω) λ − s ∈ S PK ( A − M (Ω) P rin ( A ) = A − M (Ω) λ M + · · · + A (Ω) , ( A − M (Ω) . (2.38)23e call P rin ( A ) the principal part of A . Set W K = P rin ( S PK ) = K M s =0 M ′ K − s (Γ) λ s . Here, M ′ K − s (Γ) is asubspace of M K − s (Γ). We have the following reformulation of our main results. Theorem 2.6.
The mapping χ : D P → S P given by (2.37) is an isomorphism of graded rings. For fixed K , the mapping P rin : S PK → W K of (2.38) is an isomorphism of vector spaces over C . Especially, thesequence D PK χ −→ S PK P rin −−−→ W K gives isomorphisms of three vector spaces.Proof. Due to Theorem 2.4 and 2.5,
P rin of (2.38) is a bijective mapping. So,
P rin is also an isomorphismof vector spaces over C . Remark 2.4.
We can naturally define
P rin on S P = ∞ M K =0 S PK . However, this does not give a homomor-phism of rings.
From the proof of Theorem 2.3 and 2.4, we can see the following.
Corollary 2.1.
An operator Q ∈ D PK is of rank M if and only if P rin ( χ ( Q )) is given by the form(2.38). Moreover, the leading coefficient of Q ∈ D PK is a constant number c if and only if M = K and A − K (Ω) ∈ M (Γ) is given by A − K (Ω) ≡ c. Let D PK,M be the subspace consisting of differential operators Q whose ranks are at most M . We have C = D PK, ⊂ D PK, ⊂ · · · ⊂ D PK,K = D PK . From Theorem 2.6, any element of Q ∈ D PK,M is parametrized by the elements of the vector space M ′ K − M (Γ) ⊕ · · · ⊕ M ′ K (Γ) . Then, D PK,M has M X s =0 dim M ′ K − s (Γ) complex parameters. Corollary 2.2. dim C D PK,M = M M s =0 dim C M ′ K − s (Γ) . Especially, dim C D PK = K M s =0 dim C M ′ K − s (Γ) . Anyway, if a differential operator P Ω ,z of weight N with respect to the action of Γ and automorphicforms A j (Ω) ∈ M ′ K + j (Γ) ( j = 0 , · · · , − M ) are given, there is the unique differential operator Q Ω , ( z, [ γ ]) = Q ∈ D PK,M which commutes with P = P Ω ,z . We set Q Ω , ( z, [ γ ]) ( P Ω ,z ; A − M (Ω) λ M + · · · + A (Ω)) = χ − ◦ P rin − ( A − M (Ω) λ M + · · · + A (Ω)) . Example 2.4.
In this example, we consider the special case for B = 2 of the Lam´e operator of (2.7): P Ω ,z = ∂ ∂z − ℘ (Ω , z ) . (2.39) As we saw in Example 2.3, P Ω ,z is of weight for SL (2 , Z ) . Automorphic forms for
Γ = SL (2 , Z ) are called elliptic modular forms. Let M k (Γ) be the vectorspace of elliptic modular forms of weight k . According to Theorem 2.4 and 2.5, elliptic modular forms A j (Ω) ∈ M K + j (Γ) ( j ∈ { , · · · , − M } ) determine a differential operator Q Ω , ( z, [ γ ]) of rank M of weight K with respect to Γ , where Q Ω , ( z, [ γ ]) commutes with P Ω ,z . n the following, we shall consider a simple case of K = 3 and M = 3 . In this case, modular formsmust be quite simple: A − (Ω) ≡ const ∈ M (Γ) and A − j (Ω) ≡ ∈ M j (Γ) ( j = 0 , , , because wehave ( M (Γ) = C , M k (Γ) = { } ( k = 1 , ,
3) (2.40) (for detail, see [Sm1]). So, let us obtain the differential operator Q Ω , ( z, [ γ ]) = Q Ω , ( z, [ γ ]) ( P Ω ,z ; λ ) = Q Ω , ( z, [ γ ]) ( P Ω ,z ; λ + 0 λ + 0 λ + 0) .First, we calculate Ψ of (2.9). As we saw in the proof of Proposition 1.1 and Theorem 2.1, we candetermine { ξ s (Ω , ( z, [ γ ]) , w ) } inductively. Taking w = Z + Z Ω , we have in fact ξ (Ω , ( z, [ γ ]) , ) = 1 ,ξ (Ω , ( z, [ γ ]) , ) = − ζ (Ω , z ) + ζ (Ω) ,ξ (Ω , ( z, [ γ ]) , ) = ζ (Ω , z ) + ζ (Ω) − ζ (Ω) ζ (Ω , z ) − ℘ (Ω , z ) + ℘ (Ω) , · · · , (2.41) where ζ (Ω , z ) is the Weierstrass ζ -function ζ (Ω , z ) = 1 z + X ( n ,n ) ∈ Z −{ (0 , } (cid:16) z − n − n Ω + 1 n + n Ω + z ( n + n Ω) (cid:17) and ζ (Ω) = ζ (Ω , ) and ℘ (Ω) = ℘ (Ω , ) . (We note that the right hand side of (2.41) satisfy thetransformation law (2.12).)From our data, set A (Ω , λ ) = P rin − ( λ ) = ∞ X s = − A s (Ω) λ − s = λ + 0 λ + 0 λ + 0 + A (Ω) λ − + · · · . For Ψ(Ω , ( z, [ γ ]) , w, λ ) given by (2.41), we can uniquely find the differential operator Q Ω , ( z, [ γ ]) = X j =0 b j (Ω , ( z, [ γ ])) ∂ − j ∂z − j satisfying Q Ω , ( z, [ γ ]) Ψ(Ω , ( z, [ γ ]) , w, λ ) = A (Ω , λ )Ψ(Ω , ( z, [ γ ]) , w, λ ) . In fact, bya direct calculation as in the proof of Theorem 2.3, we can obtain b (Ω , ( z, [ γ ])) = 1 , b (Ω , ( z, [ γ ])) =0 , b (Ω , ( z, [ γ ])) = − ℘ (Ω , z ) and b (Ω , ( z, [ γ ])) = − ∂∂z ℘ (Ω , z ) . Therefore, Q Ω , ( z, [ γ ]) = Q Ω , ( z, [ γ ]) ( P Ω ,z ; λ ) = ∂ ∂z − ℘ (Ω , z ) ∂∂z − (cid:16) ∂∂z ℘ (Ω , z ) (cid:17) (2.42) is the differential operator we want. This is of weight with respect to Γ = SL (2 , Z ) .From Theorem 2.6, Corollary 2.2 and the fact (2.40), Q Ω , ( z, [ γ ]) of (2.42) is the unique element of D P for P = P Ω ,z of (2.39) up to a constant factor.We remark that the relation between such a Lam´e operator P Ω ,z of (2.39) and Q Ω , ( z, [ γ ]) of (2.42) wereprecisely studied from the viewpoint of integrable systems or physics (see [W], [DMN] or [Ml]). Our resultgives a new interpretation on this topic from the viewpoint of elliptic modular forms. R Ω For X ∈ C = P ( C ) − {∞} and P Ω ,z of (2.4), we consider the differential equation P Ω ,z u = Xu and itsspace of solutions L ( P Ω ,z , X ). Suppose Q Ω , ( z, [ γ ]) of (2.16) commutes with P Ω ,z . Letting λ , · · · , λ N bethe disjoint solutions of λ N = X , Ψ(Ω , ( z, [ γ ]) , w, λ j ) gives an eigenvector for the eigenvalue A (Ω , λ j ).Let Q Ω , [ γ ] ,X be the linear operator derived from Q Ω , ( z, [ γ ]) on L ( P Ω ,z , X ). The characteristic polynomialof Q Ω , [ γ ] ,X is given by N Y j =1 ( Y − A (Ω , λ j )) . (2.43)25ue to Corollary 1.2, (2.43) gives a polynomial F Ω ( X, Y ) in X and Y . From Theorem 1.2, it follows that F Ω ( P Ω ,z , Q Ω , ( z, [ γ ]) ) = 0. We set F Ω ( X, Y ) = X j,k f j,k (Ω) X j Y k . (2.44) Theorem 2.7.
Let P = P Ω ,z be the differential operator of (2.4). Take Q = Q Ω , ( z, [ γ ]) ∈ D PK . Then, thecoefficient f j,k (Ω) in (2.44) is an automorphic form of weight N K − N j − Kk for Γ .Proof. From Theorem 2.6, we suppose that Q ∈ D PK is given by Q = Q Ω , ( z, [ γ ]) ( P Ω ,z ; A − M (Ω) λ M + · · · + A (Ω)). Then, χ ( Q ) = P rin − ( A − M (Ω) λ M + · · · + A (Ω)) is given by a series A (Ω , λ ) = ∞ X s = − M A s (Ω) λ − s .Due to Theorem 2.4 and 2.5, A s (Ω) ( s ≥
1) are automorphic forms of weight s + K . Since the set ofautomorphic forms is a ring, together with the argument in Section 1.3, the coefficients of the polynomial N Y j =1 ( Y − A (Ω , λ j )) in X and Y , where λ Nj = X , are automorphic forms for Γ.Since P Ω ,z ( Q Ω , ( z, [ γ ]) ) is of weight N ( K , resp.) for the action of Γ, we have the action of α ∈ Γ givenby (Ω , X, Y ) ( α (Ω) , j α (Ω) N X, j α (Ω) K Y ) = ( α (Ω) , X , Y ) . When we describe F Ω ( X, Y ) as in (2.44),we have f j,k ( α (Ω)) = j α (Ω) NK − Nj − Kk f j,k (Ω) . Hence, by comparing the coefficients, f j,k (Ω) is of weight N K − N j − Kk.
By Theorem 2.6 and Theorem 2.7, the family { F Ω ( X, Y ) = 0 | Ω ∈ H gn } of algebraic curves is uniquelydetermined by the differential operator P Ω ,z and given principal part A − M (Ω) λ M + · · · + A (Ω) of a Laurentseries P rin − ( A − M (Ω) λ M + · · · + A (Ω)) ∈ S PK . We denote such a family by F ( P Ω ,z ; A − M (Ω) λ M + · · · + A (Ω)) whose members are R Ω = R Ω ( P Ω ,z ; A − M (Ω) λ M + · · · + A (Ω)) . Example 2.5.
Let P Ω ,z ( Q Ω , ( z, [ γ ]) = Q Ω , ( z, [ γ ]) ( P Ω ,z : λ ) , resp.) be the operator of (2.39) ((2.42), resp.),as we saw in Example 2.4. We note that the ℘ -function satisfies the Weierstrass equation (cid:16) ∂∂z ℘ (Ω , z ) (cid:17) = 4 ℘ (Ω , z ) − g (Ω) ℘ (Ω , z ) − g (Ω) , (2.45) where g (Ω) = X ( n ,n ) ∈ Z −{ (0 , } n + n Ω) and g (Ω) = X ( n ,n ) ∈ Z −{ (0 , } n + n Ω) . It is well-knownthat g (Ω) ∈ M (Γ) and g (Ω) ∈ M (Ω) (for detail, see [Sm1]). Using the relation (2.45), we can seethat the defining equation F Ω ( X, Y ) = 0 of R Ω = R Ω ( P Ω ,z ; λ ) is given by F Ω ( X, Y ) = Y − X − g (Ω)4 X − g (Ω)4 . So, f , = f , (Ω) = 1 , f , (Ω) = g (Ω)4 and f , (Ω) = g (Ω)4 . In this case, the family F ( P Ω ,z ; λ ) consistsof non-singular algebraic curves of genus . Let π Ω : R Ω → P ( C ) be the canonical projection given by ( X, Y ) X . Γ In this subsection, we use the same notation with that of Section 2.4 and Section 2.5. Moreover, weassume there exists s ( s ≥ − M ) , where N and s are coprime , such that A s (Ω) { A s (Ω) } of (2.17). 26 emark 2.5. There are so many cases that the condition (2.46) holds. For example, if N and M arecoprime and Q Ω , ( z, [ γ ]) is given by Q Ω , ( z, [ γ ]) ( P Ω ,z ; λ M + A − M +1 (Ω) λ M − + · · · + A (Ω)) (namely the caseof A − M (Ω) ≡ ), the condition (2.46) is satisfied. By a similar argument as in Section 1.4, we have the eigenfunction ψ (Ω , ( z, [ γ ]) , w, p ) = N − X l =0 h l (Ω , w, p ) C l (Ω , ( z, [ γ ]) , w, p ) (2.47)of the operator Q Ω , [ γ ] ,X on L ( P Ω ,z , X ). Here, C l (Ω , ( z, [ γ ]) , w, X ) ∈ L ( P Ω ,z , X ) such that ∂ r ∂z r C l (Ω , ( z, [ γ ]) , w, X ) (cid:12)(cid:12)(cid:12) ( z, [ γ ])=( w, [ id ]) = δ r,l . (2.48)We note that the function C l (Ω , ( z, [ γ ]) , w, X ) of (2.48) satisfies the transformation law C l (Ω , ( z , [ γ ]) , w , X ) = 1 j α (Ω) l C l (Ω , ( z, [ γ ]) , w, X ) , (2.49)where α ∈ Γ and (Ω , ( z , [ γ ]) , w , X ) is given in (2.14), because the equation P Ω ,z u = Xu coincideswith P Ω ,z u = X u under the transformation (Ω , ( z, [ γ ]) , w, X ) (Ω , ( z , [ γ ]) , w , X ) and it holdsthat ∂ r ∂z r C l (Ω , ( z , [ γ ]) , w , X ) = (cid:16) dz dz (cid:17) r ∂ r ∂z r C l (Ω , ( z , [ γ ]) , w , X ) = 1 j α (Ω) l δ r,l . If p = ( X, Y ) ∈ R Ω , set p = ( X , Y ) = ( j α (Ω) N X, j α (Ω) M Y ). From (2.44), ( X , Y ) ∈ R Ω . The vector t ( h (Ω , w, p ) , · · · , h N − (Ω , w, p )) admits a transformation law h (Ω , w , p ) h (Ω , w , p ) · · · h N − (Ω , w , p ) = h (Ω , w, p ) j α (Ω) h (Ω , w, p ) · · · j α (Ω) N − h N − (Ω , w, p ) . (2.50) Theorem 2.8.
Assume that N is a prime number and the differential operators P Ω ,z of (2.4) and Q Ω , ( z, [ γ ]) of (2.16) satisfy the condition (2.46). Suppose the arithmetic genus ̟ ( R Ω ) is smaller than N for any Ω ∈ H gn . Then all coefficients of Q Ω , ( z, [ γ ]) are single-valued functions of z .Proof. As in Theorem 1.4 and Corollary 1.3, the function R Ω ∋ p ψ (Ω , ( z, [ γ ]) , w, p ) ∈ P ( C ) has atmost ̟ ( R Ω ) poles. By our assumption, we have ̟ ( R Ω ) < N for any Ω. By a similar argument to theproof of Theorem 1.5, we can prove that every coefficient Q (Ω , ( z, [ γ ])) are single-valued in z .The phenomenon that we saw in Example 2.3, 2.4 and 2.5 gives a typical example of the criterion ofTheorem 2.8. Namely, the rank of the Lam´e operator P Ω ,z of (2.39) is the prime number N = 2, thecommutative operator Q Ω , ( z, [ γ ]) of (2.42) of rank M = 3 is single-valued, where P Ω ,z and Q Ω , ( z, [ γ ]) givea point of the non-singular curve R Ω ∈ F ( P Ω ,z ; λ ) of genus 1 < N . Acknowledgment
This work is supported by The JSPS Program for Advancing Strategic International Networks to Acceler-ate the Circulation of Talented Researchers “Mathematical Science of Symmetry, Topology and Moduli,Evolution of International Research Network based on OCAMI” and The Sumitomo Foundation Grantfor Basic Science Research Project (No.150108). 27 eferences [BC] J. L. Burchnall and T. W. Chaundy,
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