Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2

In this paper, we construct some examples of commuting differential operators $L_1$ and $L_2$ with rational coefficients of rank 3 corresponding to a curve of genus 2.


Introduction
The study of the commutation equation [L 1 , L 2 ] = 0 of two scalar differential operators is one of the classical problems of the theory of ordinary differential equations. Burchnall and Chaundy in [1,2,3] have shown that "each pair of commuting operators L 1 and L 2 is connected by a nontrivial polynomial algebraic relation Q(L 1 , L 2 ) = 0". The equation Q(z, w) = 0 determines a smooth compact algebraic curve Ξ of finite genus g. For a generic point P ∈ Ξ, there exist common eigenfunctions ψ(x, P ) on Ξ such that L 1 ψ = λψ and L 2 ψ = µψ. The dimension l of the space of these functions corresponding to P ∈ Ξ is called the rank of the commuting pair (L 1 , L 2 ). For simplicity, in this paper we denote "the commuting differential operators of rank l corresponding to a curve of genus g" by "(l, g)-operators".
Burchnall and Chaundy also made significant progress in solving the commutation equation for relatively prime orders m and n. In this case, the rank l equals to 1. The study of this case was completed by Krichever [11,12], who also obtained explicit formulas of the function ψ and the coefficients of L 1 and L 2 in terms of the Riemann Θ-function. Let us remark that there are several papers related to this case, for instance [5,6,23,25,28,29].
But for high rank case i.e. l > 1, it is much more complicated. In [10], the problem of classifying (l, g)-operators was solved by reducing the computation of the coefficients to a Riemann problem. In [13,14] I.M. Krichever and S.P. Novikov developed a method of deforming the Tyurin parameters on the moduli space of framed holomorphic bundles over algebraic curves. By using this method, in certain cases the Riemann problem can be avoided and they found 2 The commuting operators of rank 3 and genus 2 In this section we want to construct (3,2)-operators. The first step is to use a σ-invariance, due to A.E. Mironov [17], to simplify the Krichever-Novikov system (2). The second step is to solve the simplified system by making a crucial hypothesis The last step is to construct the commuting differential operators L 1 and L 2 .

The general principle
Let Γ be a curve of genus 2 defined in C 2 by the equation On the curve Γ, there is a holomorphic involution which has six fixed ramification points. It induces an action on the space of function by (σf )(x, P ) = f (x, σ(P )). Let us take q = (0, √ c 0 ) ∈ Γ. For a generic point P ∈ Γ there exist common eigenfunctions ψ j (x, P ), j = 0, 1, 2 with an essential singularity at q, of the operators L 1 and L 2 . Without loss of generality, we assume that ψ j (x, P ) are normalized by where x 0 is a fixed point. Notice that on Γ − {q}, ψ j (x, P ) are meromorphic and have six simple poles at P 1 , . . . , P 6 independent of x. Let us consider the Wronskian matrix of the vector-valued function Ψ(x, P ; x 0 ), and where χ j = χ j (x, P ) are independent of x 0 and meromorphic functions on Γ with six poles at P 1 (x), . . . , P 6 (x) coinciding with the poles of ψ j (x, P ) at x = x 0 . In a neighborhood of q, the functions χ j (x, P ) have the form where k −1 is a local parameter near q. The expansion of χ j in a neighborhood of the pole P i (x) has the form where k − γ i (x) is a local parameter near P i (x) for 1 ≤ i ≤ 6 and 0 ≤ j ≤ 2.
In this subsection, we discuss explicit forms of χ j (x, P ) corresponding to the curve Γ defined by In order to do this, we assume that and Theorem 2.2. Let γ be a solution of then functions χ 0 , χ 1 , χ 2 are given by the formulas with G s , H s , τ 0 , τ 1 defined in (9)- (14).
Proof . By using the σ-invariance of χ 2 (x, P ), we know According to the properties of χ j (x, P ) in (2), (3) and (5), we could assume that the functions χ j (x, P ) are of the form in (8) with unknown functions G s = G s (x), H s = H s (x), τ r = τ r (x) and h r = h r (x) for s = 1, 2, 3 and r = 0, 1. Substituting (6) into (8), we have which yields that For simplicity we use the following notations It follows from (3) that By substituting α ij and d ij into (4), we get twelve equations We now try to solve these equations. Firstly, it follows from By using (9) and Eq[s + 3, 0] − Eq[s, 0] = 0, we get Furthermore, by solving Let us remark that we have reduced twelve equations to six equations From Neq[1, 1] = 0, we get By using (13) By using (14) which is exactly the equation (7). Thus we complete the proof of the theorem.

D. Zuo
Generally, solutions of (7) are not useful for us to construct (3, 2)-operators with "good" coefficients. But when we choose c 3 = 2 or −2, there are rational solutions. In what follows let us suppose The equation (7) is rewritten as It is easy to check that when (x + s 0 ) 3 + 2 > 0, is a solution of (15). Without loss of generality, we set s 0 = 0. In this case we would like to write γ = γ(x; ). As a corollary of Theorem 2.2, we have be a solution of (15). Then we have where κ = ( 2 + x 3 )z 3 − x 3 and w(z) = 1 − 2z 3 − 2 3888 z 4 + z 6 .
By using (16), let us expand χ j (x, P ) in a neighborhood of z = 0

Commuting dif ferential operators of rank 3
Let Γ be a smooth curve of genus 2 defined by the equation on the (z, w)-plane.
Theorem 2.4. The operator L 1 corresponding to the meromorphic function on Γ with the unique pole at q = (0, 1) and L 1 ψ = λψ has the form Proof . By using (1), we have It follows from (21) that the equation L 1 ψ j = λ(z)ψ j can be rewritten as 8 D. Zuo By substituting ζ 1 and ζ 2 in (17) into the above formula, we obtain explicit expressions of f j in (20).
Next we want to look for a 12 th -order differential operator such that [L 1 , L 2 ] = 0. Let us sketch out our ideas and omit tedious computations. The commutation equation [L 1 , L 2 ] = 0 is written as which yields that W k (f, g) = 0, k = 0, . . . , 18.
By using eleven equations W k (f, g) = 0, k = 8, . . . , 18, we could obtain explicit forms of g m = h m (x; ρ 0 , . . . , ρ 10−m ) + ρ 11−m with integral constants ρ 11−m . The last eight equations will determine some integral constants. For simplicity, we take all arbitrary parameters to be zero, and then obtain all coefficients g j as follows Remark 2.5. By analogy with the process of getting f j in (20), we could obtain the above g j in (25) by choosing another meromorphic function with a unique pole of order 4 at z = 0 on Γ Remark 2.6. One could find another operator L 3 of order 15 from [L 1 , L 3 ] = 0. Furthermore as in [17], the commutative ring of differential operators generated by L 1 , L 2 and L 3 is isomorphic to the ring of meromorphic functions on Γ with the pole at q = (0, 1).
The inverse image of the cuspidal point is the point σ(q), where q = (0, 1) ∈ Γ. In order to make π to be a morphism, we must complementΓ at infinity by a cuspidal point of the type (3,4), then its inverse image is the point q.

Concluding remarks
In summary by using a σ-invariance to simplify the Krichever-Novikov system, we have constructed a pair of commuting differential operators L 1 in (19) and L 2 in (23) of rank 3 with rational coefficients corresponding to the singular curveΓ, which is birationally equivalent to the smooth curve Γ of genus 2.
Let us remark that all of coefficients of L 1 and L 2 are polynomials with respect to the parameter . So if we take More precisely, we have So, when = 0 this is a trivial example. How about the case = 0? Let us comment that in this case, by a direct verification there is not such kind of L of order 3 commuting with L 1 and L 2 . Furthermore, according to the result in [29], any rank one operator with rational coefficients whose second highest coefficient is zero has the property that the limit as x goes to ∞ of the coefficients is zero. So, for example, the absence of a d 11 dx 11 term in L 2 and the x 6 in the coefficient of its d 9 dx 9 term which means that L 2 is not a rank 1 operator.