Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2
In this paper, we construct some examples of commuting differential operators L₁ and L₂ with rational coefficients of rank 3 corresponding to a curve of genus 2.
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irk-123456789-1484572019-02-19T01:23:24Z Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 Zuo, D. In this paper, we construct some examples of commuting differential operators L₁ and L₂ with rational coefficients of rank 3 corresponding to a curve of genus 2. 2012 Article Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 / D. Zuo // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 13N10; 14H45; 34L99; 37K20 DOI: http://dx.doi.org/10.3842/SIGMA.2012.044 http://dspace.nbuv.gov.ua/handle/123456789/148457 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this paper, we construct some examples of commuting differential operators L₁ and L₂ with rational coefficients of rank 3 corresponding to a curve of genus 2. |
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Article |
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Zuo, D. |
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Zuo, D. Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 Symmetry, Integrability and Geometry: Methods and Applications |
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Zuo, D. |
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Zuo, D. |
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Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 |
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Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 |
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Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 |
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Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 |
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Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 |
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commuting differential operators of rank 3 associated to a curve of genus 2 |
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Інститут математики НАН України |
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2012 |
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Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 / D. Zuo // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 29 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT zuod commutingdifferentialoperatorsofrank3associatedtoacurveofgenus2 |
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2025-07-12T19:31:18Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 044, 11 pages
Commuting Differential Operators
of Rank 3 Associated to a Curve of Genus 2
Dafeng ZUO †‡
† School of Mathematical Science, University of Science and Technology of China,
Hefei 230026, P.R. China
E-mail: dfzuo@ustc.edu.cn
‡ Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences,
P.R. China
Received March 12, 2012, in final form July 12, 2012; Published online July 15, 2012
http://dx.doi.org/10.3842/SIGMA.2012.044
Abstract. In this paper, we construct some examples of commuting differential opera-
tors L1 and L2 with rational coefficients of rank 3 corresponding to a curve of genus 2.
Key words: commuting differential operators; rank 3; genus 2
2010 Mathematics Subject Classification: 13N10; 14H45; 34L99; 37K20
1 Introduction
The study of the commutation equation
[L1, L2] = 0
of two scalar differential operators
L1 =
dn
dxn
+
n−1∑
i=0
fi(x)
di
dxi
and L2 =
dm
dxm
+
m−1∑
j=0
gj(x)
dj
dxj
, n < m,
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 L1
and L2 is connected by a nontrivial polynomial algebraic relation Q(L1, L2) = 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 L1ψ = λψ and L2ψ = µψ.
The dimension l of the space of these functions corresponding to P ∈ Ξ is called the rank of the
commuting pair (L1, L2). 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 L1 and L2 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 classi-
fying (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
dfzuo@ustc.edu.cn
http://dx.doi.org/10.3842/SIGMA.2012.044
2 D. Zuo
all (2, 1)-operators. Let us remark that J. Dixmier in [4] also discovered an example of (2, 1)-
operators with polynomial coefficients. Furthermore, P.G. Grinevich found the condition of
(2, 1)-operators with rational coefficients [7]. S.P. Novikov and P.G. Grinevich [24] clarified the
spectral data related to formally self-adjoint (2, 1)-operators. In [21] O.I. Mokhov obtained all
(3, 1)-operators. A.E. Mironov in [17, 19] introduced a σ-invariance to simplify the Krichever–
Novikov system [14] and constructed some examples of (2, 2)-operators, (2, 4)-operators with
polynomial coefficients and also in [18, 20] formally self-adjoint (2, g)-operators and (3, g)-
operators. Recently, an interesting paper is due to O.I. Mokhov in [22] who constructed examples
of (2k, g)-operators and (3k, g)-operators with polynomial coefficients for arbitrary genus g. For
more related results, please see [8, 9, 13, 15, 16, 25, 26, 27] and references therein.
The aim of this paper is to construct examples of commuting differential operators L1 and L2
with rational coefficients of rank 3 corresponding to a curve of genus 2, which is different from
those in [22].
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
γ1 = γ, γ2 = aγ, γ3 = āγ, a =
−1 +
√
3i
2
.
The last step is to construct the commuting differential operators L1 and L2.
2.1 The general principle
Let Γ be a curve of genus 2 defined in C2 by the equation
w2 = z6 + c5z
5 + c4z
4 + c3z
3 + c2z
2 + c1z + c0.
On the curve Γ, there is a holomorphic involution
σ : Γ→ Γ by σ(z, w) = (z,−w),
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,
√
c0) ∈ Γ. For a generic point P ∈ Γ there
exist common eigenfunctions ψj(x, P ), j = 0, 1, 2 with an essential singularity at q, of the opera-
tors L1 and L2. Without loss of generality, we assume that ψj(x, P ) are normalized by
di
dxi
ψj(x0, P ) = δij ,
where x0 is a fixed point. Notice that on Γ−{q}, ψj(x, P ) are meromorphic and have six simple
poles at P1, . . . , P6 independent of x. Let us consider the Wronskian matrix
~Ψ(x, P ;x0) =
ψ0 ψ1 ψ2
ψ′0 ψ′1 ψ′2
ψ′′0 ψ′′1 ψ′′2
of the vector-valued function ~Ψ(x, P ;x0), and
~Ψx
~Ψ−1 =
0 1 0
0 0 1
χ0 χ1 χ2
, (1)
Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 3
where χj = χj(x, P ) are independent of x0 and meromorphic functions on Γ with six poles at
P1(x), . . . , P6(x) coinciding with the poles of ψj(x, P ) at x = x0. In a neighborhood of q, the
functions χj(x, P ) have the form
χ0(x, P ) = k + w0(x) +O
(
k−1
)
, χ1(x, P ) = w1(x) +O
(
k−1
)
,
χ2(x, P ) = O
(
k−1
)
, (2)
where k−1 is a local parameter near q. The expansion of χj in a neighborhood of the pole Pi(x)
has the form
χj(x, P ) = −γ
′
i(x)αij(x)
k − γi(x)
+ dij(x) +O(k − γi(x)), αi2 = 1, (3)
where k − γi(x) is a local parameter near Pi(x) for 1 ≤ i ≤ 6 and 0 ≤ j ≤ 2.
Lemma 2.1 ([11]). The parameters γi(x), αij(x) and dij(x), 1 ≤ i ≤ 6, 0 ≤ j ≤ 2 satisfy the
system
Eq[i, 0] := αi0(x)αi1(x) + αi0(x)di2(x)− α′i0(x)− di0(x) = 0,
Eq[i, 1] := αi1(x)2 − αi0(x) + αi1(x)di2(x)− α′i1(x)− di1(x) = 0. (4)
2.2 Explicit forms of χj(x, P )
In this subsection, we discuss explicit forms of χj(x, P ) corresponding to the curve Γ defined by
w2 = 1 + c3z
3 + c4z
4 + z6. In order to do this, we assume that
σχ2(x, P ) = χ2(x, P ), σPs(x) = Ps+3(x), s = 1, 2, 3, (5)
and
γ1 = γ, γ2 = aγ, γ3 = āγ, a =
−1 +
√
3 i
2
. (6)
Theorem 2.2. Let γ be a solution of
1 + c3γ
3 + γ6 − 6(−3)
1
4 c
1
4
4 γ
′ 3
2 = 0, (7)
then functions χ0, χ1, χ2 are given by the formulas
χ2(x, P ) = −
3∑
s=1
γ′s
z − γs
−
3∑
s=1
γ′s
γs
=
3z3γ′
γ4 − z3γ
,
χ1(x, P ) = τ1 −
3∑
s=1
Gsγ
′
s
z − γs
+
w(z)h1
2(z − γ1)(z − γ2)(z − γ3)
, (8)
χ0(x, P ) =
τ0
2
+
1
2z
−
3∑
s=1
Hsγ
′
s
z − γs
− w(z)(γ1γ2γ3 + zh0)
2z(z − γ1)(z − γ2)(z − γ3)
,
with Gs, Hs, τ0, τ1 defined in (9)–(14).
Proof. By using the σ-invariance of χ2(x, P ), we know
γs(x) = γs+3(x), ds2(x) = ds+3,2(x), s = 1, 2, 3.
4 D. Zuo
According to the properties of χj(x, P ) in (2), (3) and (5), we could assume that the functions
χj(x, P ) are of the form in (8) with unknown functions Gs = Gs(x), Hs = Hs(x), τr = τr(x)
and hr = hr(x) for s = 1, 2, 3 and r = 0, 1.
Substituting (6) into (8), we have
χ2(x, P ) =
3z3γ′
γ4 − z3γ
,
which yields that
di2 = −2γ′
γ
, i = 1, . . . , 6.
For simplicity we use the following notations
a1 = 1, a2 = a, a3 = ā,
as+3 = as, Gs+3 = Gs, Hs+3 = Hs, s = 1, 2, 3.
It follows from (3) that
αs0 = Hs +
w(asγ)h0
6γ2γ′
+
a2sw(asγ)
6γ′
, αs1 = Gs −
w(asγ)h1
6γ2γ′
,
ds0 =
τ0
2
+
a2s
2γ
+
γ′
2∑
m=1
(1− a2sas+m)Hs+m
3γ
+
(h0 + 2(asγ)2)w(asγ)− (h0 + (asγ)2)asγw
′(asγ)
6γ3
,
ds1 = τ1 −
Gs+1γ
′
(as − as+1)γ
− Gs+2γ
′
(as − as+2)γ
+
(asγw
′(asγ)− w(asγ))h1
6γ3
,
αs+3,r = σαsr, ds+3,1 = σds1, r = 0, 1, s = 1, 2, 3.
By substituting αij and dij into (4), we get twelve equations
Eq[i, 0] = 0, Eq[i, 1] = 0, i = 1, . . . , 6.
We now try to solve these equations. Firstly, it follows from
Eq[s+ 3, 1]− Eq[s, 1] = 0, s = 1, 2, 3
that
Gs =
h′1 − h0 − (asγ)2
2h1
+
γ′
2γ
− γ′′
2γ′
, s = 1, 2, 3. (9)
By using (9) and Eq[s+ 3, 0]− Eq[s, 0] = 0, we get
Hs =
(h0 + (asγ)2)h′1 − 2h′0 − 7a2sγγ
′
2h1
− (h0 + (asγ)2)2
2h21
(10)
− h0γ
′
2h1γ
+
(h0 + (asγ)2)γ′′
2h1γ′
, s = 1, 2, 3. (11)
Furthermore, by solving
Eq[s+ 3, 1] + Eq[s, 1] = 0, Eq[s+ 3, 0] + Eq[s, 0] = 0,
Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 5
we have
Neq[s, 1] := −τ1 +
h20 + 6h0(asγ)2 + 6asγ
4 − 6h0h
′
1 − 6(asγ)2h′1 + 3h′21
4h21
+
3h′0 − h′′1 + 9a2sγγ
′
2h1
+
3h0γ
′ − 2h′1γ
′
2h1γ
+
γ′′
2γ
+
γ′′′
2γ′
− 3γ′2
4γ2
− γ′′2
4γ′2
− h1γ
′′
2h1γ′
+
h21w(asγ)
36γ4γ′2
= 0, s = 1, 2, 3,
and
Neq[s, 0] := −τ0 −
a2s
γ
+
4h′′0 + 16a2sγγ
′ + 21(asγ)2
2h1
+
(3h′0 + 9asγγ
′ − h′′1)(h0 + (asγ)2)− 4h′0h
′
1 − 13a2sγγ
′h′1
h21
+
(h0 + (asγ)2)3 − 6h′1(h0 + (asγ)2)2 + 5h′21 (h0 + (asγ)2)
2h31
+
6h′0γ
′ − h0γ′′
h1γ
+
(3h20 − 4h0h
′
1)γ
′
h21γ
− (h0 + (asγ)2)γ′′′
h1γ′
+
(h0 + (asγ)2)h′1γ
′′
h21γ
′ +
(h0 + (asγ)2)γ′′2
2h1γ′2
+
3h0γ
′2
2h1γ2
− (h0 + (asγ)2)h1w(asγ)
18γ4γ′2
, s = 1, 2, 3.
Let us remark that we have reduced twelve equations to six equations
Neq[s, 0] = 0, Neq[s, 1] = 0, s = 1, 2, 3,
with four unknown functions τ1, τ0, h1 and h0.
Let us take
h1 = i(−3)
3
4 c
− 1
4
4 γ
√
γ′, h0 =
i(−3)
3
4 (γγ′′ − 4γ′2)
2c
1
4
4
√
γ′
. (12)
From Neq[1, 1] = 0, we get
τ1 =
4γ′2 − 9γγ′′
2γ2
+
4γ′γ′′′ − 3γ′′2
4γ′2
+
i(γ3 − 1)2
4
√
3c4γ2γ′
. (13)
By using (13), we conclude that Neq[2, 1] = 0 and Neq[3, 1] = 0 always hold true.
From the equation Neq[1, 0] = 0, we obtain
τ0 =
i(γ3 − 1)2√
3c4γ3
− 1
γ
− i(γ3 − 1)2γ′′
4
√
3c4γ2γ′2
− 2i(−3)
3
4 c
3
4
4 γ
3
27γ′
3
2
− i(−3)
3
4 (γ3 − 1)2
18c
1
4
4 γγ
′ 3
2
− 3γ′′′
γ
+
10γ′γ′′
γ2
− 4γ′3
γ3
+
γ(4)
γ′
− 5γ′′γ′′′
2γ′2
+
3γ′′3
2γ′3
− 3γ′′2
γγ′
. (14)
By using (14), both Neq[2, 0] = 0 and Neq[3, 0] = 0 reduce to the same equation
1 + c3γ
3 + γ6 − 6(−3)
1
4 c
1
4
4 γ
′ 3
2 = 0,
which is exactly the equation (7). Thus we complete the proof of the theorem. �
6 D. Zuo
Generally, solutions of (7) are not useful for us to construct (3, 2)-operators with “good”
coefficients. But when we choose c3 = 2 or −2, there are rational solutions. In what follows let
us suppose
c3 = −2, c4 = − ε4
3888
, ε < 0.
The equation (7) is rewritten as
1− 2γ3 + γ6 + εγ′
3
2 = 0. (15)
It is easy to check that when (x+ s0)
3 + ε2 > 0,
γ =
x+ s0
((x+ s0)3 + ε2)
1
3
, s0 ∈ C
is a solution of (15). Without loss of generality, we set s0 = 0. In this case we would like to
write γ = γ(x; ε). As a corollary of Theorem 2.2, we have
Corollary 2.3. Let γ(x; ε) = x
(x3+ε2)
1
3
be a solution of (15). Then we have
χ0(x, P ) =
1
2z
− x3(ε2 + x3)
5832
+
10(z3 − 1)
κ
+
ε2x3z
216κ
− 108w(z) + ε2z2
6κ
− x3w(z)
2κz
+
16ε2z3
κx3
,
χ1(x, P ) =
132ε2z3 − x3[204− 204z3 + 108w(z) + ε2z2]
12x2κ
, χ2(x, P ) = −3ε2z3
xκ
, (16)
where κ = (ε2 + x3)z3 − x3 and w(z) =
√
1− 2z3 − ε2
3888z
4 + z6.
By using (16), let us expand χj(x, P ) in a neighborhood of z = 0
χ0(x, P ) =
1
z
+ ζ1 −
ε2
216
z +
2ε2
3x2
z2 +O
(
z3
)
,
χ1(x, P ) = ζ2 +
ε2
12x2
z2 +O
(
z3
)
, χ2(x, P ) =
3ε2
x4
+O
(
z4
)
,
where
ζ1 =
28
x2
− ε2x3 + x6
5832
and ζ2 =
26
x2
. (17)
2.3 Commuting differential operators of rank 3
Let Γ be a smooth curve of genus 2 defined by the equation
w2 = 1− 2z3 − ε4
3888
z4 + z6 (18)
on the (z, w)-plane.
Theorem 2.4. The operator L1 corresponding to the meromorphic function
λ =
1 + w(z)
2z3
− 1
2
Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 7
on Γ with the unique pole at q = (0, 1) and L1ψ = λψ has the form
L1 =
d9
dx9
+
7∑
n=0
fn
dn
dxn
, (19)
where
f0 =
152
243
− 58240
x9
− 55ε2
243x3
− 37ε4x3
11337408
+
115ε2x6
11337408
+
37x9
1417176
+
ε6x9
198359290368
+
ε4x12
66119763456
+
ε2x15
66119763456
+
x18
198359290368
,
f1 =
58240
x8
+
55ε2
243x2
− 152x
243
+
5ε4x4
5668704
+
2ε2x7
177147
+
17x10
1417176
,
f2 = −43200
x7
+
26 ε2
243x
− 73x2
243
+
ε4x5
1259712
+
ε2x8
419904
+
x11
629856
,
f3 = −143ε2
1944
+
19120
x6
+
79x3
486
+
ε4x6
11337408
+
ε2x9
5668704
+
x12
11337408
,
f4 = −4800
x5
− 2ε2x
243
+
16x4
243
, f5 = −24
x4
+
ε2x2
216
+
x5
108
,
f6 =
384
x3
+
ε2x3
1944
+
x6
1944
, f7 = −78
x2
. (20)
Proof. By using (1), we have
ψ′′′j (x, P ) = χ2(x, P )ψ′′j (x, P ) + χ1(x, P )ψ′j(x, P ) + χ0(x, P )ψj(x, P ). (21)
It follows from (21) that the equation L1ψj = λ(z)ψj can be rewritten as
Q0(x, z)ψj(x, P ) +Q1(x, z)ψ
′
j(x, P ) +Q2(x, z)ψ
′′
j (x, P ) = λ(z)ψj . (22)
According to the independence of χ0(x, P ), χ1(x, P ) and χ2(x, P ) at x = x0, we conclude that
the system (22) is equivalent to three equations
Q0(x, z) = λ(z), Q1(x, z) = 0, Q2(x, z) = 0.
By expanding Qj(x, z) at z = 0, we have
0 = Qj(x, z)− δ0jλ(z) = Qj,−2
1
z2
+Qj,−1
1
z
+Qj0 +O(z).
Then by solving Qj,−s = 0 for s, j = 0, 1, 2, we get the coefficients of L1 given by
f0 = −1− ζ31 −
4ε2 + 3
2x3
− ζ2ζ ′1ζ ′2 −−ζ22ζ ′′1 + 6ζ ′1ζ
′′
1 + 3ζ ′′1 ζ
′′
2 + 3ζ ′2ζ
′′′
1
+ ζ1
(
− ε
2
72
− 3ζ2ζ
′
1 + 3ζ ′′′1
)
+ ζ ′1ζ
′′′
2 + 2ζ2ζ
(4)
1 − ζ
(6)
1 ,
f1 = − 1
4x2
+ 6ζ ′21 + 9ζ1ζ
′′
1 + 12ζ ′2ζ
′′
1 + 9ζ ′1ζ
′′
2 + 3ζ ′′22 − ζ22 (3ζ ′1 + ζ ′′2 ) + 3ζ1ζ
′′′
2
+ 4ζ ′2ζ
′′′
2 + ζ2
(
− ε
2
72
− 3ζ21 − 3ζ1ζ
′
2 − ζ ′22 + 9ζ ′′′1 + 2ζ
(4)
2
)
− 6ζ
(5)
1 − ζ
(6)
2 ,
f2 = 3
[
−ζ22ζ ′2 + 5ζ ′1ζ
′
2 + 5ζ ′2ζ
′′
2 − ζ1ζ22 + 3ζ1(ζ
′
1 + ζ ′′2 ) + ζ2(5ζ
′′
1 + 3ζ ′′′2 )− 5ζ
(4)
1 − 2ζ
(5)
2
]
,
f3 =
ε2
72
+ 3ζ21 − ζ32 + 9ζ1ζ
′
2 + 9ζ ′22 + 3ζ2(4ζ
′
1 + 5ζ ′′2 )− 21ζ ′′′1 − 15ζ
(4)
2 ,
8 D. Zuo
f4 = 15ζ2ζ
′
2 − ζ2(2(−3ζ1 − 9ζ ′2) + 21ζ ′2)− 18ζ ′′1 − 21ζ ′′′2 ,
f5 = 3ζ22 − 9ζ ′1 − 18ζ ′′2 , f6 = −3ζ1 − 9ζ ′2, f7 = −3ζ2.
By substituting ζ1 and ζ2 in (17) into the above formula, we obtain explicit expressions of fj
in (20). �
Next we want to look for a 12th-order differential operator
L2 =
d12
dx12
+
10∑
m=0
gm
dm
dxm
, (23)
such that [L1, L2] = 0. Let us sketch out our ideas and omit tedious computations. The
commutation equation [L1, L2] = 0 is written as
0 =
[
d9
dx9
+
7∑
n=0
fn
dn
dxn
,
d12
dx12
+
10∑
m=0
gm
dm
dxm
]
=
18∑
k=0
Wk(f, g)
dk
dxk
, (24)
which yields that
Wk(f, g) = 0, k = 0, . . . , 18.
By using eleven equations Wk(f, g) = 0, k = 8, . . . , 18, we could obtain explicit forms of
gm = hm(x; ρ0, . . . , ρ10−m) + ρ11−m with integral constants ρ11−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 gj as follows
g0 =
45660160
x12
− 4928ε2
729x6
− 20048
729x3
− 605ε2x3
708588
+
4553x6
708588
+
79ε6x6
99179645184
+
269ε4x9
16529940864
+
683ε2x12
16529940864
+
ε8x12
1156831381426176
+
661x15
24794911296
+
ε6x15
289207845356544
+
ε4x18
192805230237696
+
ε2x21
289207845356544
+
x24
1156831381426176
,
g1 = −45660160
x11
+
4928ε2
729x5
+
20048
729x2
− 203ε4x
2834352
+
1691ε2x4
2834352
+
7111x7
708588
+
55ε6x7
49589822592
+
127ε4x10
16529940864
+
217ε2x13
16529940864
+
325x16
49589822592
,
g2 =
27758080
x10
− 182ε2
27x4
+
296
9x
− 413ε4x2
5668704
+
4339ε2x5
2834352
+
6595x8
1417176
+
ε6x8
3673320192
+
ε4x11
918330048
+
5ε2x14
3673320192
+
x17
1836660096
,
g3 = −5992
729
− 11567360
x9
+
1028ε2
729x3
+
25ε4x3
1417176
+
457ε2x6
708588
+
1393x9
1417176
+
ε6x9
49589822592
+
ε4x12
16529940864
+
ε2x15
16529940864
+
x18
49589822592
,
g4 =
3395840
x8
+
271ε2
243x2
− 2834x
243
+
193ε4x4
11337408
+
317ε2x7
2834352
+
307x10
2834352
,
g5 = −693504
x7
− 13ε2
243x
+
221x2
243
+
ε4x5
314928
+
ε2x8
104976
+
x11
157464
,
g6 = −167ε2
972
+
86464
x6
+
316x3
243
+
ε4x6
5668704
+
ε2x9
2834352
+
x12
5668704
,
Commuting Differential Operators of Rank 3 Associated to a Curve of Genus 2 9
g7 = −672
x5
+
ε2x
486
+
109x4
486
, g8 = −2856
x4
+
ε2x2
108
+
x5
54
,
g9 =
824
x3
+
ε2x3
1458
+
x6
1458
, g10 = −104
x2
. (25)
Remark 2.5. By analogy with the process of getting fj in (20), we could obtain the above gj
in (25) by choosing another meromorphic function with a unique pole of order 4 at z = 0 on Γ
µ(z) =
1 + w(z)
2z4
− 1
2z
.
Remark 2.6. One could find another operator L3 of order 15 from [L1, L3] = 0. Furthermore as
in [17], the commutative ring of differential operators generated by L1, L2 and L3 is isomorphic
to the ring of meromorphic functions on Γ with the pole at q = (0, 1).
2.4 The corresponding Burchnall–Chaundy curve
According to the Burchnall–Chaundy’s correspondence in [1, 2, 3], for each pair of commuting
operators L1 and L2 there is a Burchnall–Chaundy curve defined by a minimal nontrivial poly-
nomial Q(z, w) = 0 such that Q(L1, L2) = 0 (or Q(L2, L1) = 0). Obviously, the above curve Γ
defined by (18) is not the Burchnall–Chaundy curve for L1 and L2 given in (19) and (24).
Actually the corresponding Burchnall–Chaundy curve Γ̃ is given by
w3 − ε4
15552
w2 = z4 + z3,
that is to say,
L3
2 −
ε4
15552
L2
2 = L4
1 + L3
1.
The curve Γ̃ has a cuspidal singularity at (0, 0). The operators L1 and L2 correspond to those
meromorphic functions on Γ
λ =
1 + w(z)
2z3
− 1
2
, µ =
1 + w(z)
2z4
− 1
2z
defining a birational equivalence
π : Γ→ Γ̃, π(z, w) = (λ, µ).
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 complement Γ̃ at infinity by a cuspidal point of the type
(3, 4), then its inverse image is the point q.
3 Concluding remarks
In summary by using a σ-invariance to simplify the Krichever–Novikov system, we have con-
structed a pair of commuting differential operators L1 in (19) and L2 in (23) of rank 3 with
rational coefficients corresponding to the singular curve Γ̃, which is birationally equivalent to
the smooth curve Γ of genus 2.
Let us remark that all of coefficients of L1 and L2 are polynomials with respect to the
parameter ε. So if we take
L1 = lim
ε→0
L1, L2 = lim
ε→0
L2,
10 D. Zuo
then
[L1,L2] = 0, L32 = L41 + L31.
More precisely, we have
L1 = L3 − 1, L2 = L4 − L,
where
L =
d3
dx3
− 26
x2
d
dx
− 28
x3
+
x6
5832
.
So, when ε = 0 this is a trivial example.
How about the case ε 6= 0? Let us comment that in this case, by a direct verification there is
not such kind of L of order 3 commuting with L1 and L2. 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 d11
dx11
term in L2 and the x6 in the coefficient of its d9
dx9
term which means that L2
is not a rank 1 operator.
Acknowledgments
The author is grateful to Andrey E. Mironov for bringing the attention to this project and
helpful discussions. The author also thanks referees’ suggestions and Alex Kasman for pointing
some errors in the first version of this paper, Qing Chen and Youjin Zhang for their constant
supports. This work is supported by “PCSIRT” and the Fundamental Research Funds for the
Central Universities (WK0010000024) and NSFC (No. 10971209) and SRF for ROCS, SEM.
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1 Introduction
2 The commuting operators of rank 3 and genus 2
2.1 The general principle
2.2 Explicit forms of j(x,P)
2.3 Commuting differential operators of rank 3
2.4 The corresponding Burchnall-Chaundy curve
3 Concluding remarks
References
|