Funkcialaj Ekvacioj, 44
(2001) 377-389
Differential Equations for Modular Forms of Level Three
By
Yousuke OHYAMA
(Osaka University, Japan)
1.
Introduction
It is known that elliptic modular forms satisfy nonlinear differential equations of third order. Historically, it was Jacobi who first studied a differential
equation whose solutions are written in terms of theta constants. He showed
that the equation
$(y^{2}y^{¥prime¥prime¥prime}-15yy^{¥prime}y^{¥prime¥prime}+30y^{¥prime 3})^{2}+32(yy^{¥prime¥prime}-3y^{¥prime 2})^{3}=-¥pi^{2}y^{10}(yy^{¥prime¥prime}-3y^{¥prime^{2}})^{2}$
. The study of Jacobi’s equation is complicated,
has solutions
, ,
but Halphen found a simple system of differential equations ([5])
$y=¥theta_{2}$
$¥theta_{3}$
$¥theta_{4}([7])$
$¥left¥{¥begin{array}{l}X^{¥prime}+¥mathrm{Y}^{¥prime}=2X¥mathrm{Y},¥¥¥mathrm{Y}^{¥prime}+Z,=2YZ,¥¥Z,+X^{¥prime}=2ZX,¥end{array}¥right.$
whose solutions are written in terms of logarithmic derivatives of theta constants ([5], [8]). Although Halphen’ equations are not integrable in the classical sence (there exists no algebraic Hamiltonian), Halphen’ equations can be
solved exactly. Halphen’ equations may be considered as fundamental equations for modular forms of level two. J. Chazy also found a differential equation for modular forms of level one ([2]):
$¥mathrm{s}$
$¥mathrm{s}$
$¥mathrm{s}$
(1)
$y^{¥prime/¥prime}=2yy^{¥prime¥prime}-3(y^{¥prime})^{2}$
.
The aim of this paper is to deduce Halphen type equations for modular
forms of level three:
(2)
$¥left¥{¥begin{array}{l}W^{¥prime}+X/+¥mathrm{Y}^{¥prime}=WX+X¥mathrm{Y}+¥mathrm{Y}W,¥¥W/+¥mathrm{Y}^{¥prime}+Z/=WY+¥mathrm{Y}Z+ZW)¥¥W^{¥prime}+X^{¥prime}+Z^{¥prime}=WX+XZ+ZW,¥¥X^{¥prime}+Y,+Z^{¥prime}=XY+¥mathrm{Y}Z+ZX,¥¥e^{4¥pi i}/3(XZ+¥mathrm{Y}W)+e^{2¥pi i}/3(XW+YZ)+(XY+ZW)=0.¥end{array}¥right.$
378
Yousuke OHYAMA
A special solution of
(2) is given by
$W=3¥frac{¥partial}{¥partial¥tau}$
$¥log$
$¥eta(¥frac{¥tau}{3})-¥frac{¥partial}{¥partial¥tau}¥log¥eta(¥tau)$
$X=3¥frac{¥partial}{¥partial¥tau}$
$¥log$
$¥eta(3¥tau)-¥frac{¥partial}{¥partial¥tau}¥log¥eta(¥tau)$
,
,
(3)
,
$¥mathrm{Y}=3¥frac{¥partial}{¥partial¥tau}¥log¥eta(¥frac{¥tau+2}{3})-¥frac{¥partial}{¥partial¥tau}¥log¥eta(¥tau)$
.
$Z=3¥frac{¥partial}{¥partial¥tau}¥log¥eta(¥frac{¥tau+1}{3})-¥frac{¥partial}{¥partial¥tau}¥log¥eta(¥tau)$
The system
(2) is invariant under the following
$SL(2, ¥mathrm{C})$
$¥overline{X}_{j}(¥tau)=¥frac{1}{(c¥tau+d)^{2}}X_{j}(¥frac{a¥tau+b}{c¥tau+d})-¥frac{c}{c¥tau+d}$
for
$¥left(¥begin{array}{ll}a & b¥¥c & d¥end{array}¥right)¥in SL(2, C)$
.
-action.
,
Since (2) is equivalent to an equation of third order,
generic solutions are given by the
-orbit of (3). In section 4, we will
study the whole solution space of (2). The solution space of (2) decomposes
into one generic orbit and several special orbits. Any solution in special orbits
is a rational function.
In section 2, we will study the Hesse pencil, following [1] and [3]. Since
the Hesse pencil is an elliptic modular surface for the modular group of level
three, the period of the Hesse pencil is a modular form of level three. In
section 3, we will deduce Halphen type equations of level three from the PicardFuchs equation of the Hesse pencil. A method for constructing Halphen
type equations was found by Jacobi ([7]) and is generalized in [9]. Recently
this method has been applied to replicable functions by Harnad and McKay
$SL(2, ¥mathrm{C})$
([6]).
Halphen’s equation is a specialization of self-dual Einstein equation ([4])
and other Halphen type equations also have many applications in mathematical
physics ([9]). It would be interesting to find an application of (2) in mathematical physics. It seems that nonlinear differential equations defining modular
forms are closely related to self-dual equations.
Chazy considered the equation (1) in his study on classification of Painleve
type equations of third order ([2]). Although there is no singularity in (1),
solutions of (1) have natural boundaries. Other Halphen type equations also
do not have the Painleve property. Halphen type equations may be significant
examples for considering the meaning of the Painleve property.
Differential
2.
Equations
for
379
Modular Forms
Theta functions and the Hesse pencil
In this section we will review the classical work ([1],
(4)
[3]) on the Hesse pencil
.
$x_{0}^{3}+x_{1}^{3}+x_{2}^{3}-3ax_{0}x_{1}x_{2}=0$
We take Jacobi’s elliptic theta function
$¥theta_{1}(z)=¥sum_{n=-¥infty}^{¥infty}e^{¥pi i(n-1/2)^{2}¥tau+2¥pi i(n-1/2)(z-1/2)}$
for
$¥propto s¥tau>0$
.
The Hesse pencil
,
(4) is uniformized by
$x_{j}(z)=(-¥omega)^{j}¥theta_{1}(z-¥frac{j}{3})¥theta_{1}(z-¥frac{j+¥tau}{3})¥theta_{1}(z-¥frac{j+2¥tau}{3})$
where
$¥omega=e^{2¥pi i/3}$
.
Proposition 2.1.
(1) Consider the complex
torus
$E_{¥tau}=C/(Z+Z¥tau)$ .
The image
of
the embedding
$¥varphi:E_{¥tau}¥rightarrow P^{2}(C)$
$¥varphi(z)=(x_{0}(z):x_{1}(z):x_{2}(z))$
is the Hesse pencil (4).
(2) The modulus
$a$
satisfies
(5)
$a-1=9¥frac{¥eta(3¥tau)^{3}}{¥eta(¥frac{¥tau}{3})^{3}}$
(6)
,
$a-¥omega=¥sqrt{3}e^{-¥pi i/3}¥frac{¥eta(¥frac{¥tau+2}{3})^{3}}{¥eta(¥frac{¥tau}{3})^{3}}$
(7)
Hesse
,
$a-¥omega^{2}=¥sqrt{3}e^{¥pi i/12}¥frac{¥eta(¥frac{¥tau+1}{3})^{3}}{¥eta(¥frac{¥tau}{3})^{3}}$
$¥eta(¥tau)$
is the Dedekind
$¥eta$
function
.
,
380
Yousuke OHYAMA
$¥eta(¥tau)=q^{1/12}¥prod_{n=1}^{¥infty}(1-q^{2n})$
We take inhomogeneous coordinates
tic integrals
,
$q=e^{¥pi i¥tau}$
$x=x_{0}/x_{2}$
$¥kappa=¥int_{0}^{-1}¥frac{dx}{y^{2}-ax}$
The ellip-
,
We have
$¥kappa=-¥frac{2¥pi}{3¥sqrt{3}}¥frac{¥eta(¥frac{¥tau}{3})^{3}}{¥eta(¥tau)}$
(8)
.
The analytic parameter
is
$¥tau$
$¥tau=¥frac{¥omega¥kappa^{¥prime}-¥kappa}{¥kappa}$
$¥kappa$
and
(9)
By
$y=x_{1}/x_{2}$ .
,
$¥kappa^{¥prime}=¥int_{0}^{-1}¥frac{dx}{y^{2}-a¥omega x}$
give periods of the Hesse pencil.
and
.
$¥kappa^{¥prime}$
.
satisfy the Picard-Fuchs equation
$(1 - a^{3})¥frac{d^{2}¥kappa}{da^{2}}-3a^{2}¥frac{d¥kappa}{da}-a¥kappa=0$
$t=a^{3}$
,
.
(9) is changed into
$t(1-t)¥frac{d^{2}¥kappa}{dt^{2}}+(¥frac{2}{3}-¥frac{5}{3}t)¥frac{d¥kappa}{dt}-¥frac{1}{9}¥kappa=0$
Therefore the solutions of (9) near
$2¥mathrm{F}1$
$(¥frac{1}{3},¥frac{1}{3},1;1-t)$
$a=1$
.
are given by
,
,
$¥sum_{n=0}^{¥infty}¥{¥frac{¥Gamma(n+¥frac{1}{3})}{¥Gamma(¥frac{1}{3})n!}¥}^{2}(2¥psi(n+1)-2¥psi(n+¥frac{1}{3})-¥log(1-t))(1-t)^{n}$
where $¥psi(x)=2¥Gamma^{¥prime}(x)/¥Gamma(x)$ .
Since
$¥kappa(1)=¥int_{0}^{-1}¥frac{dx}{y^{2}-x}=-¥frac{2¥pi}{3¥sqrt{3}}$
,
Differential
Equations
for
381
Modular Forms
we have
$¥kappa(a)=-¥frac{2¥pi}{3¥sqrt{3}}2F_{1}(¥frac{1}{3},¥frac{1}{3},1;1-a^{3})$
near
3.
$a=1$ .
The Picard-Fuchs equation
In the following, we will deduce (2) by projecting the Picard-Fuchs equation to the period domain of the Hesse pencil. The independent variable of the
Picard-Fuchs equation is an algebraic parameter . We will obtain a nonlinear
differential equation whose independent variable is an analytic parameter .
This method was found by Jacobi ([7], [9]).
At first we will show the logarithmic derivatives of elliptic periods satisfy
(2). We set $¥delta=(a^{3}-1)d/da$ . Then (9) is
$a$
$¥tau$
$¥delta^{2}¥kappa+a(a^{3}-1)¥kappa=0$
We will change the independent variable
Since
and
are solutions of (9),
$a$
.
into
$¥tau=(¥omega¥kappa^{¥prime}-¥kappa)/¥kappa$
.
$¥kappa^{¥prime}$
$¥kappa$
$¥delta(¥kappa¥delta(¥kappa^{¥prime})-¥kappa^{¥prime}¥delta(¥kappa))=0$
.
Therefore
$¥delta(¥tau)=¥omega¥frac{¥kappa¥delta(¥kappa^{¥prime})-¥kappa^{¥prime}¥delta(¥kappa)}{¥kappa^{2}}=¥frac{c}{¥kappa^{2}}$
for a constant .
Hence we have
$c$
$¥frac{da}{d¥tau}=¥frac{(a^{3}-1)¥kappa^{2}}{c}$
Changing the independent variable
(10)
$a$
for ,
$¥tau$
.
(9)
turns into
$¥frac{c}{¥kappa^{2}}¥frac{d}{d¥tau}(¥frac{c}{¥kappa^{2}}¥frac{d}{d¥tau}¥kappa)+a(a^{3}-1)¥kappa=0$
.
We set
A
$=¥kappa$
,
$ B=(a-1)¥kappa$
,
$ C=(a-¥omega)¥kappa$
,
$ D=(a-¥omega^{2})¥kappa$
and
$W=¥frac{¥mathrm{A}_{¥tau}}{¥mathrm{A}}$
where A
$¥tau=d¥mathrm{A}/d¥tau$
.
,
$X=¥frac{B_{¥tau}}{B}$
,
From (10), we have
$¥mathrm{Y}=¥frac{C_{¥tau}}{C}$
,
$Z=¥frac{D_{¥tau}}{D}$
,
,
382
Yousuke OHYAMA
(12)
$(¥frac{A_{¥tau}}{A})_{¥tau}=¥frac{A_{¥tau}}{¥mathrm{A}}-¥frac{1}{3c^{2}}(B+C+D)BCD$
.
We have
(12)
$X-W=¥frac{A_{¥tau}}{¥mathrm{A}}-¥frac{B_{¥tau}}{B}=¥frac{a_{¥tau}}{a-1}$
$=¥frac{1}{c}(a-¥omega)(a-¥omega^{2})¥kappa^{2}$
$=¥frac{1}{c}CD$
.
In the same way,
(13)
$¥mathrm{Y}-W=¥frac{1}{c}DB$
(14)
$Z-W=¥frac{1}{c}BC$ .
We can eliminate
(12)
$¥mathrm{A}$
,
$B$
,
$C$
and
$D$
,
from (11).
$W_{¥tau}=W^{2}-¥frac{1}{3}((X-W)(¥mathrm{Y}-W)+(¥mathrm{Y}-W)(Z-W)+(Z-W)(X-W))$
$=¥frac{1}{3}(-X¥mathrm{Y}-¥mathrm{Y}Z-ZX+2XW+2¥mathrm{Y}W+2ZW)$
.
From (12), we have
(16)
$X_{¥tau}=W_{¥tau}+¥frac{1}{c}(CD)_{¥tau}$
$=¥frac{1}{3}(2X¥mathrm{Y}-¥mathrm{Y}Z+2ZX+2XW-¥mathrm{Y}W-ZW)$ .
In the same way, we have
(17)
(13)
$¥mathrm{Y}_{¥tau}=¥frac{1}{3}(2X¥mathrm{Y}+2¥mathrm{Y}Z-ZX-XW+2¥mathrm{Y}W-ZW)$
,
$Z_{¥tau}=¥frac{1}{3}(/-X¥mathrm{Y}+2¥mathrm{Y}Z+2ZX-XW-¥mathrm{Y}W+2ZW)$ .
The equations (15), (16),
(17) and (18) are equivalent
$¥left¥{¥begin{array}{l}W_{¥tau}+X_{¥tau}+¥mathrm{Y}_{¥tau}=WX+X¥mathrm{Y}+¥mathrm{Y}W,¥¥W_{¥tau}+¥mathrm{Y}_{¥tau}+Z_{¥tau}=W¥mathrm{Y}+¥mathrm{Y}Z+ZW,¥¥W_{¥tau}+X_{¥tau}+Z_{¥tau}=WX+XZ+ZW,¥¥X_{¥tau}+¥mathrm{Y}_{¥tau}+Z_{¥tau}=X¥mathrm{Y}+¥mathrm{Y}Z+ZX.¥end{array}¥right.$
to the system
Differential
Equations
for
383
Modular Forms
Since $W$ , $X$ ,
and $Z$ are functions of ,
and , there should be an
algebraic relation between them. By (12) and (14), we have
$¥mathrm{Y}$
$¥kappa$
$a$
$¥kappa_{¥tau}$
$X-Z=¥frac{1}{c}(D-B)C$
$=¥frac{¥kappa}{c}(1-¥omega^{2})C$
.
Therefore
$(X -Z)(¥mathrm{Y}-W)=¥frac{¥kappa}{c^{2}}(1-¥omega^{2})BCD$
.
In the same way
$(X -¥mathrm{Y})(Z-W)=¥frac{¥kappa}{c^{2}}(1-¥omega)BCD$
.
Therefore we have
$F(W, X, ¥mathrm{Y}, Z):=(X-Z)(¥mathrm{Y}-W)+¥omega^{2}(X-¥mathrm{Y})(Z-W)$
$=¥omega^{2}(XZ+¥mathrm{Y}W)+¥omega(XW+¥mathrm{Y}Z)+(X¥mathrm{Y}+ZW)=0$.
$F(W, X, ¥mathrm{Y}, Z)$
and
$Z$
is a relative invariant under even permutations of
,
$W$ $X$
,
$¥mathrm{Y}$
.
By direct calculation
$¥frac{d}{d¥tau}F(W, X, ¥mathrm{Y}, Z)=3(W+X+¥mathrm{Y}+Z)F(W, X, ¥mathrm{Y}, Z)$ .
We obtain the main theorem:
Theorem 3.1.
cial solution (3).
Proof.
The system
By (5), (6),
of nonlinear differential
(2) has a
spe-
(7) and (8), we have
$=¥frac{2¥pi}{3¥sqrt{3}}e^{5¥pi i/4}¥frac{¥eta(¥frac{¥tau}{3})^{3}}{¥eta(¥tau)}$
A
equations
,
$C=¥frac{2¥pi}{3}e^{11¥pi i/12}¥frac{¥eta(¥frac{¥tau+2}{3})^{3}}{¥eta(¥tau)}$
$B=2¥sqrt{3}¥pi e^{5¥pi i/4}¥frac{¥eta(3¥tau)^{3}}{¥eta(¥tau)}$
,
,
$D=¥frac{2¥pi}{3}e^{4¥pi i/3}¥frac{¥eta(¥frac{¥tau+1}{3})^{3}}{¥eta(¥tau)}$
.
Taking logarithmic derivatives of the above functions, we obtain Theorem
3.1.
$¥square $
384
Yousuke OHYAMA
From now on, we will study the solution space of (2). (2) is essentially
a system of differential equations of third order. We will show that (2) has
solutions with three parameters, which give generic solutions of (2).
For any function
$f(¥tau)$
$f(¥tau)$
, we define an action of A
$=¥left(¥begin{array}{ll}p & q¥¥r & s¥end{array}¥right)$
$¥in SL(2, C)$
on
:
(19)
$(¥rho(¥mathrm{A})¥cdot f)(¥tau)=¥frac{1}{(r¥tau+s)^{2}}f(¥mathrm{A}¥cdot¥tau)-¥frac{r}{r¥tau+s}$
.
Here
$A$
The action
$p$
Proposition 3.2.
,
$p(¥mathrm{A})$
Proof.
$¥frac{d}{d¥tau}$
$p¥tau+q$
$¥tau=_{¥overline{r¥tau+s}}$
.
is a right action:
$¥rho(¥mathrm{A}B)$
$p(¥mathrm{A})¥cdot X$
.
.
$¥mathrm{Y}$
If
$W(¥tau)$
and $¥rho(A)$ .
,
$Z$
. $f=p(B)$ .
$X(¥tau)$
,
$(p(¥mathrm{A})¥cdot f)$
and
$¥mathrm{Y}(¥tau)$
also satisfy
$Z(¥tau)$
(2) for any
.
satisfy (2), then
A
$=¥left(¥begin{array}{ll}p & q¥¥r & s¥end{array}¥right)$
$¥rho(¥mathrm{A})¥cdot W$
$¥in SL(2, C)$ .
We have
$(¥rho(¥mathrm{A}).
.
f)=¥frac{1}{(r¥tau+s)^{4}}¥frac{df}{d¥tau}(¥frac{p¥tau+q}{r¥tau+s})-¥frac{2r}{(r¥tau+s)^{3}}f(¥mathrm{A}¥cdot¥tau)+¥frac{r^{2}}{(r¥tau+s)^{2}}$
Using the equation above, we can easily prove this proposition.
4.
,
$¥square $
The Space of initial values
We will determine the solution space of the equations
tions of (2) are given by Proposition 3.2. The group
$¥Gamma(3)=¥{$
$¥left(¥begin{array}{ll}p & q¥¥r & s¥end{array}¥right)¥in SL(2, Z);p¥equiv 1$
,
$q¥equiv 0$
,
(2). Generic solu-
$r¥equiv 0,s¥equiv 1(¥mathrm{m}¥mathrm{o}¥mathrm{d} 3)¥}$
is called the modular group of level three.
By the transformation law of the //-function, the special solution (3) is
invariant under the action for any A
. Therefore effective actions on
(3) are
$¥in¥Gamma(3)$
$p$
$¥Gamma(3)¥backslash S¥grave{L}(2, C)$
Moreover the action
and
.
$¥mathrm{Y}_{0}$
$Z_{0}$
$¥Gamma(3)¥backslash SL(2, Z)¥simeq$
$¥mathrm{A}_{4}$
.
induces even permutations of
$W_{0}$
,
$X_{0}$
,
Differential
Equations
for
385
Modular Forms
We will study the solution space of the equations (2) using the solution
space of Chazy’s equation ([8]). At first, we will show that (2) is a Galois
extension of Chazy’ equation. Chazy’ equation (1) is equivalent to
$¥mathrm{s}$
$¥mathrm{s}$
(20)
$¥left¥{¥begin{array}{l}¥frac{d}{d¥tau}h_{1}=h_{2},¥¥¥frac{d}{d¥tau}h_{2}=6h_{3},¥¥¥frac{d}{d¥tau}h_{3}=4h_{1}h_{3}-h_{2}^{2},¥end{array}¥right.$
where
(20) has a special solution
$h_{1}=2y$ .
$h_{1}=6¥frac{d}{d¥tau}¥log¥eta(¥tau)$
,
-symmetry as (2) ([8]).
and has the same
We take the elementary symmetric polynomials of
$SL(2, ¥mathrm{C})$
,
$W$ $X$
,
$¥mathrm{F}$
, $Z$ :
$P_{1}=W+X+¥mathrm{Y}+Z$ ,
$P_{¥mathit{2}}=WX+WY+WZ+XY+YZ+XZ$ ,
$P_{¥mathit{3}}=WXY+WXZ+WYZ+XYZ$ ,
$P_{4}=$
Let
$¥overline{F}$
WXYZ.
be the complex conjugate of
$F$
:
$¥overline{F}(W, X, ¥mathrm{Y}, Z)=¥omega(XZ+¥mathrm{Y}W)+¥omega^{2}(XW+¥mathrm{Y}Z)+(X¥mathrm{Y}+ZW)$
Since
$F¥overline{F}=12P_{4}+P_{2}^{2}-3P_{1}P_{3}$
,
we have
$P_{4}=¥frac{3P_{1}P_{3}-P_{2}^{2}}{12}$
By (2),
(21)
$P_{1}$
,
$P_{2}$
and
$P_{3}$
.
satisfy the following equations
$¥left¥{¥begin{array}{l}¥frac{d}{d¥tau}P_{¥mathrm{l}}=¥frac{2}{3}P_{2},¥¥¥frac{d}{d¥tau}P_{2}=3P_{3},¥¥¥frac{d}{d¥tau}P_{3}=4P_{4}+2P_{1}P_{3}-¥frac{2}{3}P_{2}^{2}=3P_{1}P_{3}-P_{2}^{2}.¥end{array}¥right.$
.
386
Yousuke OHYAMA
If we set
$h_{1}=¥frac{3}{4}P_{1}$
,
$h_{2}=¥frac{1}{2}P_{2}$
,
$h_{3}=¥frac{1}{4}P_{3}$
,
(20) is changed
to
(22)
$¥lambda^{4}-P_{1}¥lambda^{3}+P_{2}¥lambda^{2}-P_{3}¥lambda+(¥frac{3P_{1}P_{3}-P_{2}^{2}}{12})=0$
(21). Hence (21) is equivalent to Chazy’ equation.
$¥mathrm{s}$
The Galois group of the algebraic equation
¥
. A ¥ ¥
is the alternating group
and the action of
defined by (19) induce the Galois action on (22).
$¥mathrm{A}_{4}$
$4 simeq Gamma(3) backslash SL(2, Z)$
$SL(2, Z)$
We will solve (2) for distinct initial values, which gives the generic part of
the solution space.
Theorem 4.1. Assume that holomorphic functions
near $t¥in C$ and have initial values
$W(t)=k_{0}$
,
where the complex numbers
$F(k_{0},k_{1},k_{2},k_{3})=0$
.
,
$X(t)=k_{1}$
$k_{0},k_{1},k_{2},k_{3}$
$¥mathrm{Y}(t)=k_{2}$
,
$W$ $X$
,
$¥mathrm{Y}$
and
$Z(t)=k_{3}$
from
are distinct
Then there exists a matrix A
,
$Z$
(2)
satisfy
,
each other and
$=¥left(¥begin{array}{ll}p & q¥¥r & s¥end{array}¥right)$
$¥in SL(2, C)$
such
that
$W=p(¥mathrm{A})¥cdot W_{0}$
,
$X=p(¥mathrm{A})¥cdot X_{0}$
,
$W=p(¥mathrm{A})¥cdot ¥mathrm{Y}_{0}$
Proof. At first, we
,
will reduce (2) to Chazy’ equation.
initial value problem of (20), whose initial value is
$h_{1}(t)=¥frac{3}{4}P_{1}(t)$
$¥mathrm{s}$
,
$h_{2}(t)=¥frac{1}{2}P_{2}(t)$
,
$W=p(¥mathrm{A})¥cdot Z_{0}$
.
We consider the
$h_{3}(t)=¥frac{1}{4}P_{3}(t)$
.
By the proposition 4.1 in [8], if the discriminant of the equation
(23)
$¥lambda^{3}-h_{1}(t)¥lambda^{2}+h_{2}(t)¥lambda-h_{3}(t)=0$
is not zero, the solution of Chazy’ equation is written in terms of the
function. By direct calculation, the discriminant of (23) is
$¥mathrm{s}$
$¥eta-$
$-¥frac{1}{64}(9P_{1}(t)^{2}P_{2}(t)^{2}-32P_{2}(t)^{3}-27P_{1}(t)^{3}P_{3}(t)+108P_{1}(t)P_{2}(t)P_{3}(t)-108P_{3}(t)^{2})$
which is the same as the discriminant of (22) up to a constant factor.
are distinct, the discriminant of (22) is not zero.
$k_{j}’ ¥mathrm{s}$
,
Since the
Differential
for
Equations
Hence there exists a matrix
$A$
387
Modular Forms
$=¥left(¥begin{array}{ll}p & q¥¥r & s¥end{array}¥right)$
$¥in SL(2, C)$
$P_{1}(¥tau)=¥frac{8}{(r¥tau+s)^{2}}h(¥mathrm{A}¥cdot¥tau)-¥frac{4r}{r¥tau+s}$
such that
,
where
$h(¥tau)=¥frac{d}{d¥tau}$
We can also write
$P_{2}$
and
$P_{3}$
$¥log$
$¥eta(¥tau)$
.
in terms of the //-function by (21):
$P_{2}(¥tau)=¥frac{12}{(r¥tau+s)^{4}}h^{¥prime}(A¥cdot¥tau)-¥frac{24r}{(r¥tau+s)^{3}}h(¥mathrm{A}¥cdot¥tau)+¥frac{6r^{2}}{(r¥tau+s)^{2}}$
,
$P_{3}(¥tau)=¥frac{4}{(r¥tau+s)^{6}}h^{¥prime¥prime}(A¥cdot¥tau)-¥frac{24r}{(r¥tau+s)^{5}}h^{¥prime}(A¥cdot¥tau)+¥frac{24r^{2}}{(r¥tau+s)^{4}}h(¥mathrm{A}¥cdot¥tau)-¥frac{4r^{3}}{(r¥tau+s)^{3}}$
We will find solutions of (22), which give solutions of
consider the case $p=1$ , $q=0$ , $r=0$ , $s=1$ . Since
.
(2). At first we will
$W_{0}+X_{0}+¥mathrm{Y}_{0}+Z_{0}=8h(¥tau)$
by (13),
$s=1$ .
, ,
We will set
$W_{0}$
,
$X_{0}$
$¥mathrm{Y}_{0}$
$Z_{0}$
are the solutions of (22) in the case
$P_{1}^{(0)}=8h(¥tau)$
$p=1$ , $q=0$ , $r=0$ ,
,
$P_{2}^{(0)}=¥frac{12}{(r¥tau+s)^{4}}h^{¥prime}(¥tau)$
,
$P_{3}^{(0)}=¥frac{4}{(r¥tau+s)^{6}}h^{¥prime¥prime}(¥tau)$
,
and
$¥alpha=¥frac{1}{(r¥tau+s)^{2}}$
$¥beta=¥frac{r}{r¥tau+s}$
,
.
Then
$ P_{1}=aP_{1}^{(0)}(¥mathrm{A}¥cdot¥tau)-4¥beta$
,
$P_{2}=¥alpha^{2}P_{2}^{(0)}(¥mathrm{A}¥cdot¥tau)-3¥alpha¥beta P_{1}^{(0)}(A¥cdot¥tau)+6¥beta^{2}$
,
.
$P_{3}=¥alpha^{3}P_{3}^{(0)}(¥mathrm{A}¥cdot¥tau)-2a^{2}¥beta P_{2}^{(0)}(¥mathrm{A}¥cdot¥tau)+3¥alpha¥beta^{2}P_{1}^{(0)}(¥mathrm{A}¥cdot¥tau)-4¥beta^{3}$
388
Yousuke OHYAMA
By the relation between solutions and coefficients, the solution of
written in terms of
and
:
,
,
$W_{0}$
$X_{0}$
$¥mathrm{Y}_{0}$
(22) can be
$Z_{0}$
,
$¥alpha W_{0}(¥mathrm{A}¥cdot¥tau)-¥beta=p(¥mathrm{A})¥cdot W_{0}(¥tau)$
$¥alpha X_{0}(A¥cdot¥tau)-¥beta=p(¥mathrm{A})¥cdot X_{0}(¥tau)$
,
$¥alpha ¥mathrm{Y}_{0}(A¥cdot¥tau)-¥beta=p(¥mathrm{A})¥cdot ¥mathrm{Y}_{0}(¥tau)$
,
$¥alpha Z_{0}(¥mathrm{A}¥cdot¥tau)-¥beta=p(¥mathrm{A})¥cdot Z_{0}(¥tau)$
.
As a set, these four functions have the initial value
Therefore we should take a suitable Galois action
$g¥in ¥mathrm{A}_{4}$
$k_{0}=p(¥mathrm{A})¥cdot W_{0}(t)$
$k_{1}=p(¥mathrm{A})¥cdot X_{0}(t)$
,
$k_{2}=p(¥mathrm{A})¥cdot ¥mathrm{Y}_{0}(t)$
,
$k_{3}=p(¥mathrm{A})¥cdot Z_{0}(t)$
.
at
in order that
$¥{k_{0}, k_{1}, k_{2}, k_{3}¥}$
$¥tau=t$
.
,
Since the Galois action is deduced from the action
$p$
by the isomorphism
A $4¥simeq¥Gamma(3)¥backslash SL(2, Z)$ ,
we can make the initial values coincide with the corresponding functions by
¥
changing
into $gA$ for ¥ ¥
.
$A$
$g in Gamma(3) backslash SL(2, Z)$
$¥square $
, then
or $X=W$ because $F=(¥mathrm{Y}-Z)(X-W)$
$¥omega(X-¥mathrm{Y})(Z-W)$ .
In general, if any two of $W$ , $X$ , , $Z$ coincide, some three
of $W$ , $X$ , , $Z$ coincide. When $X=¥mathrm{Y}=Z$, (2) is reduced to the following
equations:
If
$X=¥mathrm{Y}$
$¥mathrm{Y}=Z$
$-$
$¥mathrm{Y}$
$¥mathrm{Y}$
$¥left¥{¥begin{array}{l}X_{¥tau}=X^{2},¥¥W_{¥tau}+2X_{¥tau}=2WX+X^{2}.¥end{array}¥right.$
Hence
$X=-¥frac{p}{p¥tau+q}$
,
$W=-¥frac{p}{p¥tau+q}+¥frac{r}{(p¥tau+q)^{2}}$
.
We may consider $(p:q)¥in P^{1}(C)$ and $r¥in O(2)$ .
Thus we can solve the initial value problem of (2) for all cases. This gives
the whole space of meromorphic solutions of (2). Summing up our result, we
obtain the following theorem:
Differential
Equations
for
Modular Forms
389
Theorem 4.2. The generic part of the solution space of (2) is isomorphic
to $¥Gamma(3)¥backslash SL(2, C)$ , and the degenerate part is a union of four line bundles $O(2)$ ,
which intersect along the zero section. This intersection corresponds to the case
$k_{0}=k_{1}=k_{2}=k_{3}$
.
References
[1] Bianchi, L., Ueber die Normalformen dritter und Fiunfter Stufe des elliptishen Integrals erster
Gattung.,
Math. Ann., 17 (1880), 234-262.
[2] Chazy, J., Sur les equations differentielles du trousieme ordre et d’ordre superieur dont
l’integrale generale a ses points critiques fixes, Acta Math., 34 (1911), 317-385.
[3] Dixon, A. C., On the doubly periodic functions arising out of the curve $x^{3}+y^{3}-3¥alpha xy=1.$ ,
[4]
[5]
Quart. J., 24 (1890), 167-233.
Gibbons, G. W. and Pope, C. N., The positive action conjecture and asymptotically
Euclidean metrics in quantum gravity, Commun. Math. Phys., 66 (1979), 267-290.
Halphen, G., Sur une system d’equations differentielles, C. R. Acad. Sci., Paris, 92 (1881),
1101-1103.
[6] Harnad, J. and McKay, J., Modular solutions to equations of generalized Halphen type.
[7]
R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 456 (2000), 261-294.
Jacobi, C. G. J., Uber die Differentialgleichung, welcher die Reihen $1¥pm 2q+2q^{4}¥pm 2q^{9}+$
¥
¥
etc. Genuge Leisten, Crelles J., 36 (1848), 97-112.
etc., ¥
Ohyama, Y., Differential Relations of Theta Functions, Osaka J. Math., 32 (1995), 431450.
Ohyama, Y., Nonlinear systems related to second order linear equations, Osaka J. Math.,
33 (1996), 927-949.
$2 sqrt[4]{q}+2 sqrt[4]{q^{9}}+2 sqrt[4]{q^{25}}+$
[8]
[9]
nuna adreso:
Graduate School of Mathematics
Osaka University
Machikaneyama, Toyonaka 560-0043
Japan
E-mail: ohyamacmath.sci.osaka-u.ac.jp
(Ricevita la 16-an de septembro, 1997)
(Reviziita la 9-an de augusto, 2001)