数理解析研究所講究録 1212 巻 2001 年 43-49

43
Summary note on the moduli of punctured tori and related
Fuchsian differential equations
H.Shiga(Chiba Univ), T.Tsutsui(Chiba Univ) and J. Wolfart(Frankfurt Univ.)(千葉大・自然科学 志賀弘典、
千葉大・理
筒井亨、
フランクフルト大

1J. ヴォルファルト)

Setting of the problem, due to Keen, Rauch and Vasquez[KRV]

Let us consider an elliptic curve with Legend$E(t)$

module$t$: $y^{2}=x(x-1)(x-t)$ ,and suppose $E(t)=\mathrm{C}/L$ for a certain lattice
one point punctured torus. We have

$L=\mathrm{Z}+\mathrm{Z}\tau$$E’(t)=\mathrm{H}/G_{0}$on C. Let $E’(t)=E(t)-\{\infty\}$ be the,for some discrete group Go acting on the upper half plane H.
is generated by two elements
andthose correspond to a homology basis of $E(t)$ . We set $p=trace(B\mathit{2})$ , $q=trace(B_{2})$ ,$trace(B_{1}B_{2})$ . The triple $(p, q, r)$ is called the Fricke parameter
for $E’(t)$ . We are requested toget an explicit description of the Fricke parameter in terms of the complex moduli .
This problem is studied mainly by [KRV]. We are interested in it because it is the simplest
case of the study of the moduli space of punctured Riemann surfaces and at the same time
is a first step for the study of modukar functions for a quadrangle group. We do’ show any
essentially new result on this original problem than those of [KRV]. But we show some relations
between this problem and the theory of deformations for Fuchsian difffferential equations with anapparent singularity.
$G_{0}$$B_{1}$$B_{2}$

$r$

$=$

$t$

$\mathrm{t}$2Lam\’e equation

Let us consider the universal covering space H of C-L, and let pr be the natural projection.
By taking the quotient we have the isomorphism $\mathrm{H}/G_{0}\sim(\mathrm{C}-L)/L$ .

Theorem 2. 1 (Keen, Rauch and Vasquez $[KRV]$)Let be a local inverse of $pr$ :-L. Thensolutions of$\nu$$\mathrm{H}arrow \mathrm{C}$$\nu(\zeta)$

is given by a mtio of two independent(2.1)$\frac{d^{2}w}{d\zeta^{2}}+\frac{1}{4}(\wp(\zeta;L)+C(L))w=0$

for some nice constant $C(L)$ . We have$\lambda^{2}C(\lambda L)=C(L)$$\lambda$$\in \mathrm{C}^{*},C(\overline{L})=\overline{C}(L)$

.

(2.1)

44
with unique regular singuNote that this is a difffferential equation on the elliptic curve
equal
group
is
to Go .
monodromy
)
its
(the
and
orbit of 0 underlarity at$\mathrm{C}/L$$L$$\overline{0}$$-\mathrm{R}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$Setto the Fuchsian difffferential equation on$z_{1}=\wp(\zeta)$$\mathrm{P}^{1}-$. Let us change the difffferential equation (2.1) into the form$\eta’+p(z_{1})\eta’+q(z_{1})\eta=0$

in terms of the variable

$Z1$

,

(2.3)

$\eta(z_{1})=w(\zeta))$. So (2.1) is transformed to$\frac{d^{2}\eta}{dz_{1^{2}}}+\frac{1}{2}(\frac{1}{z_{1}-e_{1}}+\frac{1}{z_{1}-e_{2}}+\frac{1}{z_{1}-e_{3}})\frac{d\eta}{dz_{1}}+\frac{C+z_{1}}{16(z_{1}-e_{1})(z-e_{2})(z-e_{3})}\eta=0$. (2.4)

Here we used the convention $e_{1}=\wp(1/2)$ , $e_{2}=\wp(\tau/2)$ , $e_{3}=\wp((1+\tau)/2)$ . By the transformation$z_{1}=(e_{2}-e_{1})x+e_{1}$ we obtain$\frac{d^{2}\eta}{dx^{2}}+P(x)\frac{d\eta}{dx}+Q(x, H)\eta=0$$P(x)= \frac{1}{2}(\frac{1}{x}+\frac{1}{x-1}+\frac{1}{x-t})$,,$t(t-1)H$$Q(x, H)= \frac{1}{16x(x-1)}-\overline{x(x-1)(x-t)}$

.

(2.5)

where $t=(e_{3}-e_{1})/(e_{2}-e_{1})$ and.

$C=-e_{3}+ \frac{(e_{3}-e_{1})(e_{2}-e_{3})}{e_{2}-e_{1}}H$(2.5) is the general form of the Fuchsian difffferential equation with the Riemann scheme$(\begin{array}{llll}0 1 t \infty 0 0 0 1/41/2 1/2 \mathrm{l}/2 1/4\end{array})$

.

(2.6)

-Reduction of the
Now our problem can be devided in two problems[Problem 1]:
Describe the Fricke parameter in terms of the nice constant$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{m}-$$C$in (2.1).

[Problem 2]:
Describe the nice constant (that is equivalent to describe the nice accessary parameter $H$ )in terms of the moduli variable in (2.5).
$C$

$t$

3

Hermitian condition

We investigate the [Problem 2] in a little bit difffferent aspect.
be a reference point in H. Let
and $t<0<1<\infty$ . Let
For the moment we supposebe the circuit
around starting from $x_{0}(i=t,0,1, \infty)$ . Let
be a circuit on
at $x=x_{0}$ .
independent
solutionsofsystemfixedmatrix of the equation (2.5) with respect to a$B_{2}=A_{1}A_{0}$, $B_{3}=$Let be the projective monodromy group of (2.5), and set $B_{1}=AoAt$ }.
. Let Go denote the subgroup of generated by the system$t\in \mathrm{R}$$\mathrm{C}-\{t, 0,1\}$$\gamma_{\dot{l}}$$x_{0}$$A_{:}$$i$$G$$B_{1}B_{2}=A_{\mathrm{t}}A_{1}$$G$

Definition 3. 1 The tr.ple
parameter for the Fuchsian differential equation (2.5)$(p, q, r)=(\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{e}(B_{1}),\mathrm{h}\mathrm{a}\mathrm{c}\mathrm{e}(B_{2}),$$\{B_{1}, B_{2}\}$$\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{e}(B_{3}))$is called the Fricke

45
Proposition 3. 1 In the above situation we have the following normal form of the system} in the conjugacy class.
{At,$A_{0\mathrm{t}}$$\mathrm{A}_{\mathrm{t}}$,$A_{0}=(\begin{array}{ll}1 00 -\mathrm{l}\end{array})$$A_{1}=(\begin{array}{ll}a_{1} a_{2}a_{2} -a_{1}\end{array})$, $A_{t}=$(3.1)$(\begin{array}{ll}b_{1} b_{2}b_{3} -b_{1}\end{array})$

with the condition$a^{2}1+a^{2}2=1,b^{2}+1b2b3=1$$a_{2}(b_{2}-b_{3})--\pm 2i$

.

,
(3.2)

We have the exception of the system$A_{0}=(\begin{array}{ll}1 00 -1\end{array})$,$A_{1}=(\begin{array}{ll}1 i0 -1\end{array})$,$A_{t}=(\begin{array}{ll}b_{1} b_{2}b_{3} -b_{1}\end{array})$

.

(3.3)

Remark 3. 1 1) We note that the accessary parameter $H$ detemines the monodromy groupand the subgroup. The conjugacy class of detemines the Fricke parameter up to signature,
conversely the Fricke parameter detemines(but ituniquely up to the change of
and).
determines unique2) If $H(t)$ is a nice accessary parameter for the moduli mentioned in Section 2, the groupcoincides with the descrete groupfor $E’(t)$ .
$H$3) The parameteris nice for if and only if the group Go is a Fuchsian group. Especial$G$$G$$G_{0}$$G$$b_{2}$$b_{3}$$G_{0}$$t$$G_{0}$$G\circ$

$t$

$ly$

, if the system {At,

$A_{0}$,$A_{1}$} has an$invar\cdot ant$

Hemitian form

parameter.
4) If is real and $H$ is nice, the Schwarzian image ofwith index $(2, 2, 2, \infty)$ .
$t$$\mathrm{H}$$(\begin{array}{ll}1 00 -1\end{array})$, we obtain the niceby (2.5) makes a Poincar\’e quadrangle

Proposition 3. 2 The system (3.1) with (3.2) preserves the Hemitian $fom$$(\begin{array}{ll}1 00 -1\end{array})$

if and only if we have$a_{1},ia_{2}$4,$b_{1}\in \mathrm{R}$,$b_{3}=-\overline{b_{2}}$

.

(3.1)

Fricke parameter

Proposition 4. 1 ([KRVJ)For a nice parameter $H(t)$ we have(0) $p^{2}+q^{2}+r^{2}-pqr=0$ ,(i), $r( \tau)=p(-\frac{1}{1+\tau})$ ,(ii) $\varphi(\rho)=(3,3,3)$ , $\varphi(i)=(2\sqrt{2},2\sqrt{2},4)$ ,(iii) $\varphi(\Re(\tau)=0)=\{pq=2r\}$ , $\varphi(\Re(\tau)=-1/2)=\{q=r\}$ , $\varphi(|\tau|=1)=\{p=q\}$ .
$q( \tau)=p(-\frac{1}{\tau})$

Remark 4. 1 The property (0) is valid also for the equation with any accessary parameter $H(t)$ .
$-\mathrm{A}\mathrm{p}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$of Hill’s$\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{d}-$46
We suppose the lattice L takes the form L
The equation (2.1) can be regarded
as an equation for real functions
0 with real variable x and the periodicity)u. By the Fourier expansion we have$\ovalbox{\tt\small REJECT}$$\mathrm{Z}_{\mathrm{v}\mathrm{r}}+\mathrm{Z}rrik$$d^{\mathit{2}}u$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{i}+Q\{x$$Q(x +7\mathrm{i})\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$Q(\mathrm{z})$

.

(4.1)

$\frac{d^{2}u}{dx^{2}}+(\theta_{0}+2\sum_{n=1}^{\infty}\theta_{n}\cos(2nx))u=0$

We note that the solution may not be periodic.
-Floquet transformationLet , be a system of independent solutions at $x=0$ . Then we have$u_{1}$$u_{2}$$(\begin{array}{l}u_{1}(x+\pi)u_{2}(x+\pi)\end{array})=F$

with$(\begin{array}{l}u_{1}(x)u_{2}(x)\end{array})$(4.2)$F\in \mathrm{G}\mathrm{L}(2, \mathrm{C})$The above transformation $F$ is called the Floquet transformation. This is nothing but the circuit.
transformation $C=A At$ , and we note$F$
Let us consider the eigen function for in the form
$F\in \mathrm{S}\mathrm{L}(2, \mathrm{C})$$u(z)=e^{\mu z} \sum_{n=-\infty}^{\infty}b_{n}e^{2n\dot{|}z}$

So $F\circ u(z)=e^{\mu\pi}u(z)$ , and we have the eigen value
, and
value

$e^{\mu\pi}$

.

(4.3)

. Consequently we have another eigen$e^{-\mu\pi}$Trace(F)=Trace(C)=e$+e^{-\mu\pi}$

.

(4.4)

Let (4.3) substitute in (4.1), then we have the following system of infinite number of linear
equations for:$\{b_{n}\}$$( \mu+2ni)^{2}b_{n}+\sum_{m=-\infty}^{\infty}\theta_{m}b_{n-m}=0$,$n=0,$ $\pm 1,$$\pm 2$,(4.5)$\cdots$

We define a formal determinant$\cdot$.
$\frac{(\dot{|}\mu+2)^{2}-\theta_{0}}{\mathrm{B}^{\Gamma_{-}}\mathrm{f}_{0}2^{2}-\theta_{0}-\theta}\ldots$$\Delta(i\mu)=\mathrm{D}\mathrm{e}\mathrm{t}(.\cdot.\cdot..\cdot...\cdot\cdot.\cdot.\cdot$$\frac{(\dot{|}\mu+-\theta_{0}\frac{-\theta_{1}}{2^{2}-\theta_{0}0)^{2}}}{\overline{\pi}^{\theta}0^{2}-\theta_{0},\mp_{0}}$Theorem 4. 1 (Hill, see $a/so$ Whitacker- Watson$fom$
is convergent. Thenand that$4m^{2}$$\sum_{-\infty}^{\infty}|\theta_{n}|$$\cdot..\cdot)$$\frac{(\dot{l}\mu 2)^{2}-\theta_{0}\neq_{\theta}\underline{2}\varpi^{-\mp_{0}}-\theta_{0}^{-}-\theta}{2^{\mathrm{B}}-\theta_{0}}$,$\mathrm{E}^{\mathrm{I}}7^{\overline{\mathrm{r}}_{\theta}^{\theta}}\mp_{0}\mp_{0}$$l\mathit{9}.\mathit{4}$) We suppose that$\Delta(i\mu)$$e^{\mu\pi}$(4.6)$\theta_{0}$does not take theconverges, and it holds$\Delta(i\mu)=\Delta(0)-\sin^{2}(\frac{1}{2}\pi i\mu)/\sin^{2}(\frac{1}{2}\pi\sqrt{\theta_{0}})$Consequently.

$\cdot.\cdot..\cdot$

.

(4.7)

is an eigen value if and only if we have$\Delta(0)-\sin^{2}(\frac{1}{2}\pi i\mu)/\sin^{2}(\frac{1}{2}\pi\sqrt{\theta_{0}})=0$

As an easy corollary of this theorem we have

(4.8)

47
Theorem 4. 2 (see also Keen, Ranch and Vasquez $[KRV]$)

Trace(F)=2-A(0)(4.9)$\sin^{2}(\frac{1}{2}\pi\sqrt{\theta_{0}})$Remark 4. 2 Suppose is pure imaginary. Then and $C(L)$ take real values. The Lam\’eequation (2.1) can be considered as a $Hill’s$ equation by restricting it on the real axis (butrewriting in terms of functions. In that case the Trace(F) becomes to be the first trace of
the Fricke parameter. We can proceed this method to get the traces and . Even for a generalaccessary parameter we get real values for and , but in general we obtain with complex
values. We can’t see from that formula the $criter\cdot on$ that to be a real number.
$t$$\tau$$\sigma$

$p$

$q$

$p$

$r$

$r$

$q$

$r$

5

R-H problem of moving singularity type

We consider the following problem. Set a Riemann scheme(5.1)$(\begin{array}{llll}0 1 t \infty 0 0 0 1/41/2 1/2 1/2 1/4\end{array})$

for a Fuchsian differential equation$y’+p\mathrm{o}(x, t)y’+q0(x, t, H_{0})y=0$

1 –1/2

1 –1/2

1 –1/2$p_{0}(x, t)=\overline{x}\overline{x-1}\overline{x-t}++$$q0(x, t, H_{0})= \frac{1/16}{x(x-1)}-\frac{t(t-1)H_{0}}{x(x-1)(x-t)}$(5.2)

without apparent singularity. Let us regard $(t, H_{0})$ to be the deformation parameter of (5.2).
Any representation of $\pi 1(\mathrm{P}-\{0,1, t, \infty\}, *)$ on
admitting the Riemann scheme (5.1)
appears as a monodromy group of (5.2) ?
$GL(2, \mathrm{C})$5.1

Relation with the equation with an apparent singularity

Set a Riemann scheme with an apparent singularity$x=\lambda$(5.3)$(\begin{array}{lllll}0 1 t \lambda \infty 0 0 0 0 1/4-1/2 1/2 1/2 2 1/4\end{array})$

for a Fuchsian differential equation$y”+p(x, t, \lambda)y’+q(x, t, \lambda, \mu, H)y=0$

.

(5.1)

For more general Riemann scheme$(\begin{array}{lllll}0 \mathrm{l} t \lambda \infty 0 0 0 0 \rho_{\infty}k_{0} k_{1} \theta 2 \rho_{\infty}+k_{\infty}\end{array})$

we have the corresponding differential equation with$p(x, t)= \frac{1-\kappa_{0}}{x}+\frac{1-\kappa_{1}}{x-1}+\frac{1-\theta}{x-t}-\frac{1}{x-\lambda}$

(5.5)

48
$q(x, t, H)= \frac{\kappa}{x(x-1)}-\frac{t(t-1)H}{x(x-1)(x-t)}+\frac{\lambda(\lambda-1)\mu}{x(x-1)(x-\lambda)}$$H=[\lambda(\lambda-1)(\lambda-t)\mu^{2}-\{\kappa_{0}(\lambda-1)(\lambda-t)+\underline{1}$$t(t-1)$$\kappa_{1}\lambda(\lambda-t)+(\theta-1)\lambda(\lambda-1)\}\mu+\kappa(\lambda-t)]$

,
(5.6)

$\kappa=\frac{1}{4}(\kappa_{0}+\kappa_{1}+\theta-1)^{2}-\frac{1}{4}\kappa_{\infty}^{2}$for the coefficient function. Here and $H$ are accessary parameters, and $H$ is described in termsby the fact that is non-logarithmic singularity. At this moment our Riemann scheme
of ,
,
.
$\mu$

$t$

$\lambda$$\lambda,\mu$$\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{s}-\kappa 0=\kappa_{1}=\theta=1/2$$\kappa_{\infty}=0$. The solution
Proposition 5. 1 Let us fix , and supposeof (5.4) at $x=0$.
$where\sim$,isonmeans the universal covehnp. So $u(x, 0)$ is
solution of (5.4).
$u(x, \lambda)$$|\lambda|<\epsilon$$t$$U-\{0\}\mathrm{x}\sim|\lambda|<\epsilon$$holomo\gamma ph\dot{\mathrm{t}}\mathrm{C}$$a$Remark 5. 1 We can not perfom this procedure in case x $=0$ is a logarithmic singularity.

5.2

Isomonodromy deformation

Suppose we have $t\neq 0,1$ , andby the Hamiltonian system$\infty$$\lambda\neq 0,1,t$,$\infty$. The isomonodromy defermation of (5.4) is given$\frac{d\lambda}{dt}=\frac{\partial H}{\partial\mu}$,$\frac{d\mu}{dt}=-\frac{\partial H}{\partial\lambda}$

.

(5.7)

. Let us take an initial
Now we consider (5.7) in the tublar neiborhood$(t,\lambda,\mu)=(t0,0,\mu
0)$, then we can find the integral curve by following (5.7).
data$\{|\lambda|<\epsilon\}\mathrm{x}(t, \mu)-\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}$Proposition 5. 2 For any initial data $(t, \lambda,\mu)=(t0,0,\mu 0)$ $(to\neq 0)$ the integral curve intertransversally, so any monodromy group of (5.2) is obtained as the one ofsects the axis
an isomonodromic defomation of (5.4)$\lambda=0$[Proof]. We have$\frac{\partial H}{\partial\mu}(t,0,\mu)=t/2,-\frac{\partial H}{\partial\lambda}(t,0,\mu)=-1/16+(\mu-\mu^{2}t)$

So it holds $*^{d}(t,0,\mu)\neq 0$ for t$\neq 0$

.

(5.8)

.
q.e.d.

Remark 5. 2 This Proposition offers a mapping from the monodromy space of (5.2) to that of(5.4)’ According to the Riemann-Hilbert correspondence this map induces the natural injective
map to the complex variety of the set of representations of. It seems to be
an open mapping. So our problem in the beginning of this section will have the positive answer.
$\pi_{1}(\mathrm{P}-0,1, t, \infty, *)$6

Some comments(1) In the Riemann scheme (5.3) we can let

the deformation
we have$\lambda’arrow 0$$\lambda$

move to the infinity. This process is visualized byusing the transformed variable and singularity: $x’=t/x$ ,$(\begin{array}{lllll}0 \mathrm{l} t \lambda’ \infty 0 0 0 0 -1/40 1/2 1/2 2 1/4\end{array})$.

$\lambda’=t/\lambda$

. So

(6.1)

49
In that limit procedure we don’t have the Proposition corresponding to Prop. 5.1. The Hamil-

tonian equation reduses a system corresponding the logarithmic derivative of a certain hyperge. So the integral orbit for the system does not go out fromometric differential equation on, and its orbit does not mean the isomonodromy. We don’t know the meaning of
the wall$\lambda’=0$$\lambda’=0$

this degenerate eqation.
(2) Let us consider the isomonidromy orbit of (5.4) for the nice accessary parameter. We
always have a fixed monodromy group. Suppose is real, then the nice parameter $H$ is also real.
Its image of the upper half plane by the Schwarz map diffffers depending on the value of $H$ . We
don’t know what happens during this deformation for the figures of Schwarz map.
$t$(3) It is quite difficult to proceed the observation of (2). We prefer to consider more basic
case. Namely the hypergeometric difffferential equation with an apparent singularity. There arearticles by Klein [K], Schilling [S] and Ritter [R] those study this problem. Also we regard it
simplest case study of the difffferential equation which induces a quadrangle group.

$\mathrm{a}$(4) Let us fix two Riemann schemes they are mutually contiguous, and let $P_{1}(x)$ and $P_{2}(x)$ be
the solutions. So these two functions have multibaluedness with the same monodromy behavior.
If we take the linear combination $aP_{1}(x)+bP_{2}(x)$ , it satisfies a difffferential equation with the same
monodromy. Klein [K] appointed that this difffferential eqation will have the apparent singularityand vice versa. So we are interested where we have the apparent singularity depending on , .
$b$

$a$

References[1] Dubrovin, B. and Mazzocco, M., Monodromy of certain Painlev\’ereflection groups, preprint (1998) arxivrmath.

-

VI transcendents and$\mathrm{A}\mathrm{G}/9806056$[2] Hitchin, N.J., The self-duality equations on a Riemann surface, Proc. London Math.
$\mathrm{S}\mathrm{o}\mathrm{c}.,55(1987)$, 59-126.

[3] Iwasaki, K., Kimura, H., Shimomura, S. and Yoshida, M., From Gauss to Painlev\’e, Vieweg(19911).
[4] Kawai, S., Isomonodromic deformation of Fuchsian type projective connections on elliptic
curves, RIMS Kokyuroku 1022(1997), 53 57.
-[5] Klein, Vorlesung \"uber die Hypergeometrische Function, Springer.

[6] Keen, L., Rauch, H.E. and Vasquez, A.T., Moduli of punctured tori and the accessory
parameter of Lam\’e’s equation, Trans. Act. A.M.S., 255(1979), 201 -230.
[7] K. Okamoto, On Fuchs’ problem on a torus I, Funkcial Ekvac 14(1971), 137-152.

[8] K. Okamoto, Sur le probl\‘eme de Fuchs sur un tore II, J. Fac. Sci. Univ. Tokyo, Sect.IA
24(1977), 357-?.

[9] Ritter, E., Ueber die hypergeometrische Function mit einem Nebenpunkt, Math. An, 1-36.
$\mathrm{n}.,48(1897)$[10] Schilling, F., Ueber die Theorie der symmetriscen -Functionen mit einem einfachen Neben, 481-522.
punkte, Math.
$\mathrm{S}$$\mathrm{A}\mathrm{n}\mathrm{n}.,52(1899)$[11] Whitacker,E.T. and Watson, G.N., A course in modern analysis, Cambridge[12] Yoshida, M., Fuchsian Difffferential Equations, Vieweg (1987)$\mathrm{U}.\mathrm{P}.(1950)$.