ガロア・タイヒミュラー塔の普遍定義体について
—極大退化曲線の変形と織田予想
京大数理研伊原康隆
(YASUTAKA IHARA)
東大数理中村博昭 (HIROAKI NAKAMURA)
\S
概要。
本稿では、
$n$個のマークされた点をもつ種数
$g$の代数曲線のモジュライ空
間 $M_{g,n}$に関する、互いに関連した
2
つの結果について報告します。
その1つは、$M_{g,n}$ のコンパクト化 $\overline{M}_{g,n}$における “ $0$次元カスプ
”,
即ち極
大退化曲線をパラメトライズする
$\overline{M}_{g,n}$の点の形式近傍に
(有限通りの付加 構造に対応して)標準的な座標系が導入できること。
もう1
つは、 曲線の副 $\mathbb{C}$基本群の外部自己同型群における
$\pi_{1}$$(M_{g,n})$ の普遍モノドロミー表現の「定義体」
として現れる
$\mathbb{Q}$の代数拡大体
$\mathbb{Q}_{g,n}^{(\mathrm{c})}$ が、2-$-2g-n<0$
である限り、
$(g, n)$によらないであろう、 という織
孝幸氏の予
測にたいする –つめ結果です。
(
ここで $\mathrm{C}$ はfull
即ち商、部分、拡大で閉じ
ている有限群のクラスとする。
)
新しく得られたのは
定理
.
$\mathbb{Q}_{g,n}^{(\mathrm{c})}\subset \mathbb{Q}_{0}^{(\mathrm{C})},3$.
逆方向の包含関係については、
$n>0$,
(!:profinite またはpro-l の仮定の
もとで松本真氏
$([\mathrm{M}\mathrm{a}])_{\text{、}}$中村高尾尚武氏上野亮
–
氏
$([\mathrm{N}\mathrm{T}\mathrm{U}]\sim[\mathrm{N}])$等に
よって、すでにその成立が知られています (
参:
数理研講究録884)。 証明は、極大退化曲線の標準変形空間内での形式
(管状)近傍の基本群を、
ガロアの作用も込めて精密に記述する事によって得られます。また、
この記述
はさらにいくつかの数論的副産物をもちます。詳しくは RIMS-preprint:
On
deformation of maximally
degenerate stable marked curves
and
Oda’s
problem
\S 1
INTRODUCTION1.1. In this article, we shall study deformation of a maximally degenerate stable
marked curve, from the point of view of Galois representation, i.e., with the aim of
comparing Galois actions on the fundamental groups of the original and deformed
curves.
We shall start with a construction involving an $expli_{C}\dot{f}t$ parametrization of a
universal deformation ofsuch a degenerate curve $X^{0}$ (see \S 1.2,
\S9)\sim.
Then we shallstudy a certain 1-parameter subfamily of deformation from the Galois theoretic
viewpoint. Our first step for this is to construct a “tangential base point” on
the total space of deformation, outside $X^{0}$ but near each of its singular points.
Our explicit parametrization of deformation is crucial in this construction. Paths
connecting these base points will then be compared with paths around and paths
inside $X^{0}$. (This, of course, involves comparison of Galois actions.) They are all
“within” the formal neighborhood of$X^{0}$. Since the family is 1-dimensional so that
$X^{0}$ is a divisor, Grothendieck-Murre theory [GM] can be fully used. This study is
presented in
\S 3
(the main result is quoted in\S 1.3
below).This will then be applied to the followingprediction by Oda (“$\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e}\mathrm{n}$ lines” in
\S 2
of [O]$)$. Themoduli stack$l\sqrt I_{g,n}$ over$\mathrm{Q}$of$n$-pointmarkedsmooth curvesofgenus$g$has a canonical $l$-adictowerofcoveringsarising from the monodromy representation
of$\pi_{1}(\mathrm{J}/I_{\mathit{9}^{n}},)$ on the pro-l fundamentalgroup ofan $n$-point punctured smooth curve
ofgenus $g$. He predicted that the constant field
$\mathrm{Q}_{g,n}^{(l}\mathrm{P}^{\Gamma\circ-}$)
of this tower is independent of $(g, n)$, as long as $(g, n)$ is “hyperbolic”, i.e.,
$2-2g-n<0$
. (The special caseof this prediction, the independence of $\mathrm{Q}_{0,n}^{(_{\mathrm{P}^{\mathrm{r}\mathrm{o}-l}})}$ on $n(\geq 3)$, had previously been
communicated to Ihara by Deligne [De].) We shall prove that $\mathrm{Q}_{g,n}^{(l}\mathrm{p}\mathrm{r}\mathrm{o}-$) $\subset \mathrm{Q}_{0,s^{\mathrm{r}}}^{(\mathrm{p}-}\circ l$).
Combinedwiththealready established inclusion $[\mathrm{N}\mathrm{T}\mathrm{U}]\sim[\mathrm{N}]$, [Ma], this will confirm
his prediction $\mathrm{Q}_{g,n}^{(\mathrm{p})}\mathrm{r}\circ-1=\mathrm{Q}_{0,3}^{(l}\mathrm{P}^{\mathrm{r}\mathrm{o}-}$) under the additional assumption $n\geq 1$ (see
\S 1.4
and
\S 4
for details).1.2. Now let us be more precise on each of the main points. We shall study, in \S 2,
deformation of any “$\mathrm{P}_{01\infty}^{1}$-diagram” over $\mathrm{Q}$, which is a synonym for “maximally $\mathrm{d}\mathrm{e}_{\mathrm{o}}\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}$ stable marked curve over $\mathrm{Q}$ with which each irreducible component is
smooth”. By definition, a $\mathrm{P}_{01\infty}^{1}$-diagram over $\mathrm{Q}$ is a pair $(X^{0}, \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(X0))$ of
(i) ageometrically connected reducedpropercurve$X^{0}$ over$\mathrm{Q}$, with only ordinary
double.
singularities Sing$(X^{0})$, and(ii) a finite (possibly empty) set Mark$(X^{0})$ of smooth $\mathrm{Q}$-rational points of $X^{0}$
specified $(” \mathrm{m}\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{d}")$,
satisfyingthe condition that each irreducible component $X_{\lambda}^{0}$ of$X^{0}$ can be identified
with the projective line $\mathrm{P}^{1}$ over
$\mathrm{Q}$ in such a way that
(Sing$(x0)\cup \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(x^{0})$) $\cap X_{\lambda}^{0}=\{0,1, \infty\}$
.
It is well-known that each $\mathrm{P}_{01\infty}^{1}$-diagram over $\mathrm{Q}$ has a universal deformation
over the spectrum $S$ of the algebra of formal power series $\mathrm{Q}[q_{1},$
$\cdots,$ $q_{m}\mathbb{I}$, where
$m=|\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(X^{0})|$. But the general belief seems to have been that there is no
canon-ical choice, over $S=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{Q}[q_{1},$
words, the algebra $\mathrm{Q}\mathbb{I}q_{1},$ $\cdots$ ,$q_{m}$
I
is canonical but the generators $q_{1},$$\cdots,$$q_{m}$ arenot. We shall show that just by adding a pair of combinatorial structures $(J, i)$
on $(X^{0}, \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(X0))$ one can define a “canonical” universal family $(X’/S, \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(x))$
(Mark$(X)$ consists of sections $Sarrow X$ extending Mark$(x^{0})$). Thus, once $(J, i)$
is given, each of $q_{1},$$\cdots,$$q_{m}$ can be regarded as a distinguished formal
holomor-phic function on the formal neighborhood of the point of the moduli stack $M_{g,n}^{stb\iota\epsilon}a$
that corresponds to $(X^{0}, \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(X0))$
.
Here,$g$ is the arithmetic genus of $X^{0}$ and
$n=|\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(X^{0})|$.
The additional structure consists of
(a) a tangential structure $J$ on $X^{0}$; giving $J$ is equivalent to choosing, for each
$\lambda$ and $P\in \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(X0)\cap X_{\lambda}^{0}$, one point from the two-point set $((\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(x0)-P)\mathrm{u}\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(X^{0}))\cap X_{\lambda}^{0}$ ,
and
(b) an ordering of Sing$(x^{0})$; i.e., a bijection $i:\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(x^{0\sim})arrow\{1,2, \cdots, m\}$.
Our construction of $(X/S, \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(x))$ is via its formal completion along $X^{0}$,
which is obtained by pasting “standard” affine formal deformations in a natural
way (Theorems $1(\S 2.3),$ $1’(\S 2.4)$).
This is ageneralization ofTateelliptic curves, and corresponds to a special case
of Mumford curves (see
\S 2.4.2).
From the point of view of Mumford uniformizationtheory [Mu], we are only choosingaspecial Schottkygroup. But ourconstruction is
at least what makes it directly applicable toour main purpose, and also, hopefully,
what makes it clear why this special choice of a deformation is canonical.
We shall actually include curves and deformations over Z.
1.3. Now restrict $(X/S,.\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(x))$ to the diagonal $\mathrm{s}_{\mathrm{P}^{\mathrm{e}\mathrm{c}}}\mathrm{Q}[q\mathrm{I}$ of $S$ defined by
$q_{1}=\cdots=q_{m}=q$, and let $(C, \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(c))$ be the generic member of the restricted
family. Then $C$ is a proper smooth curve over the quotient field $\mathrm{Q}((q))$ of $\mathrm{Q}[q\mathrm{I}$,
and Mark$(C)$ consists of finitely many $\mathrm{Q}((q))$-rational points. For any field $I1’$
(al-ways of characteristic $0$ in this article), denote by$\overline{I\prime_{\searrow}’}$
its algebraic closure, and by
$G_{K}=\mathrm{G}\mathrm{a}1(\overline{I\prime_{\backslash }’}/I\mathrm{c}’)$ its absolute Galois group. Let $C$ be any class of finite groups
which is almost full, i.e., closed under the formation of $\mathrm{t}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}_{\mathrm{O}}\sigma$ subgroups, factor
groups and finite direct products. Then the main result of
\S 3
reads as follows. Theorem $2’$(\S 3.5). If$\sigma\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$ acts trivially on the maximal pro-C quotien$t$of th$\mathrm{e}$ fundamen$t\mathrm{a}l$ groupoid of $\mathrm{P}^{1}\otimes\overline{\mathrm{Q}}-\{0,1, \infty\}w.\mathrm{r}.t$. the set of Deligne’s
tangential base points, then $\sigma$ has an extension $\tilde{\sigma}\in \mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}((q))}/\mathrm{Q}((q)))$ that acts
triviallyon the $\max’\mathrm{i}m\mathrm{a}i$pro-C quotien$t$ofthefundamen$t\mathrm{a}l_{\mathrm{o}}\sigma ro$upoidof$C\otimes\overline{\mathrm{Q}((q))}-$ $\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(C)\mathrm{w}.\mathrm{r}.t$. th$\mathrm{e}$ set of
“
$t\mathrm{a}\iota \mathrm{I}_{\circ}\sigma e\mathrm{n}t\mathrm{i}al$ base $p_{o\mathrm{j}n}t_{S}$” $\tilde{\mu}(\mu\in \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}(x^{0}))$ defin$ed$ in
\S 3.3.
In $p$articular, th$e$ outer action $\mathit{0}\mathit{4}\tilde{\sigma}$ on the maximal pro-C quotient of thefundamental group of$C\otimes\overline{\mathrm{Q}((q))}-\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(c)$ is trivial.
Remark. $(\tilde{\mu})$ and the choice of $\tilde{\sigma}$ are related to eachother (see
\S 3
for details).Actually what we obtain is not just the comparison of the kernels of Galois
actions, but the comparison of the actions themselves. An explicit description
of the Galois action on the fundamental groupoid of $C\otimes\overline{\mathrm{Q}((q))}-\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(C)$, in
$\mathrm{P}^{1}\otimes\overline{\mathrm{Q}}-\{0,1, \infty\}$, together with applications, will be given in a subsequent article (in preparation).
1.4. Now we come back to Oda’s prediction. Let $(g, n)$ be a pair ofnon-negative
integers satisfying
$2-2g-n<0$
, and $\Lambda/f_{g,n}$ be the moduli stack over $\mathrm{Q}$ of propersmooth curves of genus $g$ with $n$ (ordered) marked points. Then the fibering
$M_{g,n+1}arrow\lambda/I_{g,n}$ defined by “forgetting the $(n+1)$-th marked point” gives a
uni-versal family of $n$-point punctured smooth curves of genus $g$ over $M_{g,n}$
.
If $\xi$ isany geometric point of$M_{g,n}$ and $C_{\xi}$ is the fiber above $\xi$, there is a canonical exact
sequence of profinite groups (algebraic fundamental groups)
(1) $1arrow\pi_{1}(c_{\xi},\tilde{\xi})arrow\pi_{1}(M_{g,n}+1,\tilde{\xi})arrow\pi_{1}(M_{g,n}, \xi)arrow 1$,
where $\tilde{\xi}$ is any geometric point of
$C_{\xi}$. This defines an outer action of $\pi_{1}(M_{g,n}, \xi)$
on $\pi_{1}(c_{\xi},\tilde{\xi})$ (by conjugation), and hence also that on the maximal pro-C quotient $\pi_{1}^{(C)}(c_{\epsilon},\tilde{\xi})$, for any almost full class $C$ of finite groups:
(2) $\varphi$ : $\pi_{1}(\Lambda/I\xi)g,n’arrow$ Out
$\pi_{1}^{(C)}(c\epsilon,\tilde{\xi})$.
Now the projection on $\mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$ of $\mathrm{I}<\mathrm{e}\mathrm{r}\varphi$ is of the form $\mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q}_{g,n}^{(C)})$, with a
Galois extension $\mathrm{Q}_{\mathit{9}_{1}}^{(c)}n$
over $\mathrm{Q}$ which is independent of the choice of $\xi$. The basic
predictionby Oda is that $\mathrm{Q}_{g,n}^{(c_{)}}$ would not depend on$(g, n)$. But since $\mathrm{Q}_{gn}^{()}c$
)
$\subset \mathrm{Q}_{g,n+}^{(c)}1$
and since, for each $g\geq 0$, there exists a $\mathrm{P}_{01\infty}^{1}$-diagram of type $(g, n.)$ over $\mathrm{Q}$ for
sufficiently large $n$ (\S 2.1), Theorem $\underline{9}’$ cited above (applied to $\xi$ corresponding to
$(C\otimes\overline{\mathrm{Q}((q))}, \mathrm{M}\mathrm{a}\mathrm{r}\mathrm{k}(C)))$ gives immediately the following
Theorem $3\mathrm{A}(\S 4.2)$
.
$\mathrm{Q}_{g,n}^{(c)}\subset \mathrm{Q}_{0,3}^{(c)}(+)$.Here, $\mathrm{Q}_{0,3}^{(c)}(+)$ is the kernel of the $\mathrm{G}\mathrm{a}1(\overline{\mathrm{Q}}/\mathrm{Q})$ action on the maximal pro-C
quo-tient of the fundamental groupoid of $\mathrm{P}^{1}\otimes\overline{\mathrm{Q}}-\{0,1, \infty\}$w.r.t. the set of Deligne’s
tangential base points.
When $C$ consists of all finite $l_{-^{\sigma}}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{o}\mathrm{P}^{\mathrm{S}}$, where $l$ is a
$\mathrm{f}\mathrm{i}_{d}\backslash r\mathrm{e}\mathrm{d}$ prime, then $\mathrm{Q}_{g,n}^{()}c=$ $\mathrm{Q}_{g})n(\mathrm{p}\mathrm{r}\circ-\iota)$ and $\mathrm{Q}_{0^{\mathrm{p}-}}^{(\mathrm{r}\mathrm{o}\mathrm{t}},3$
) $=\mathrm{Q}_{0,3}^{(_{\mathrm{P}^{\mathrm{r}}}\circ-l)}(+)$ (even if$l=2$). We thus obtain
Theorem $3\mathrm{B}$
.
$\mathrm{Q}_{g,n}^{(\mathrm{p}\circ- l)}\Gamma\subset \mathrm{Q}_{0,3}^{(\mathrm{p})}\mathrm{r}\circ-\iota$.This inclusionhas previously been proved in the special case where $2g\equiv 0$ (mod
$l-1)$, using appropriate $l$-covers of $\mathrm{P}^{1}-\mathrm{f}0,1,$
$\infty$
}
([Ma]).A comparison “at each level” of the weight filtrations of $\mathrm{Q}_{g,n}^{(_{\mathrm{P}^{\mathrm{r}}})}\mathrm{O}-l$ and $\mathrm{Q}_{0^{\mathrm{p}_{3}-}}^{(\mathrm{o}l},\mathrm{r}$) will
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