数理解析研究所講究録
第 819 巻 1993 年 106-119

106

ON MORDELL-WEIL GROUPS OF ABELIAN SCHEMES
BY

MASA-HIKO SAITO
Kyoto University\S 1 Mordell-Weil group.
Let
be a connected smooth quasi-projective variety defined over the field ofcomplex numbers C. An abelian scheme
$S$

$f$

: $Aarrow S$

is a smooth projective group scheme over with connected fiber. For every closed
point $s\in S$ , the fiber $A_{s}=f^{-1}(s)$ is an abelian variety defined over C.
$S$(1.1) Definition. For an abelian scheme : $Aarrow S$ , we define the Mordell-Weil,of K-rational poin of the generic fiber$gro$ up $MW(A/S)$ by the groupwhere $K=C(S)$ is the function field of and denote the generic poin of .
$f$

$ts$

$A_{\eta}(K)$$S$$A_{\eta}$$t$$\eta$$S$

By definition, the Mordell-Weil group $MW(A/S)$ is isomorphic to the group{ $s:S\cdotsarrow A$ rational section of },$f$and Hartogs’ theorem and GAGA imply that this group is isomorphic to the group of
regular sections of .
$f$an abelian variety defin $ed$(1.2) Definition. Let $K$ be the functi on field of , andover K. $K/C$ -tra ce ofis a pair $(B, \tau)$ consisting of an abelian variety $B$ defined$C$over and a $hom$ omorphism
$S$

$A$

$A_{K}$$A_{K}$$\tau$: $Barrow A_{K}$over $K$ which as the followin universal prop erty. Given an abelian varie $ty$
defined over
and a om omorphism : $Carrow A_{K}$ , then there exists a uniq $ue$
; $Carrow B$ defin $ed$ oversuch th athomomorphism.
$defi_{li}ed$

$C$

$h$

$C$

$\phi_{*}$

$g$

$h$

$\phi$$C$$\phi=\tau\phi_{*}$The existence of $(K/C)$ -trace is proved by Chow. Moreover we have the following
fundamental result due to Lang and N\’eron.
Typeset by$\mathcal{A}_{ki^{S-}}$

TEX

107
(1.3) Theorem-Definition. (cf. Lang-N\’eron [La, pl39, $Tb.2J$) Let $K$ be th function field of $S,$
an abelian variety defined over $K$ , and $(B,\tau)$ a $(K/C)$ -trace of. Then $A(K)/\tau(B(C))$ is fin itely generat ed. If we write as$e$$A_{K}$$A_{K}$$A_{K}/\tau(B(C))\simeq Z‘\oplus$

we call th rank
$e$

$r$

(Torsion),of the free part of $A_{K}/\tau(B(C))$ the Mordell-Weil $mnk$.

This is a function field analogue of Mordell-Weil theorem for abelian variety definedover a number field. In the number field case, there is a beautiful conjecture due toBirch&Swinnerton-Dyer about the relation between the order of zero of L-function
and the Mordell-Weil rank. On the other hand, Mordell-Weil groups admit heightpairings, which have been deeply studied by Shioda [Sd 2] and Cox-Zucker [C-Z] forelliptic surfaces and recently for families of Jacobian of curves (see [Sd3]). In thisnote, we will give a Hodge theoretic interpretation of Mordell-Weil groups by using
Zucker’s relative Hodge theory. We will also give a few applications in [Sa.MH] andrecent result on the group of the component of the N\’eron model.

\S 2 Relative Hodge theory after Deligne and Zucker.
We shall give a Hodge theoretic interpretation of a Mordell-Weil group $MW(A/S)$ .
denote the local system of the first
Given an abelian scheme : $Aarrow S$ , let
homology of fibers of . Then from the relative exponential sequence we have theexact sequence of sheaves on$R_{1}f_{*}Z_{X}$

$f$

$f$

$S^{an}$(2.1)$0arrow R_{1}f_{*}Zarrow \mathcal{L}ie_{A/S}arrow \mathcal{O}_{S}^{an}(A)arrow 0$.

Setting $V_{Z}=R_{1}f_{*}Z_{X}$ and using the isomorphisn$\mathcal{L}ie_{A/S}\simeq R^{1}f_{*}\mathcal{O}_{A}^{an}$,we have the following exact sequence:$0arrow H^{0}(S, V_{Z})arrow^{p_{0}}H^{0}(S, R^{1}f_{*}\mathcal{O}_{A}^{an})arrow H^{0}(S, \mathcal{O}_{S}^{an}(A))$(2.2)$arrow^{\delta}H^{1}(S, V_{Z})arrow^{p_{1}}H^{1}(S,R^{1}f_{*}\mathcal{O}_{A}^{an})$

It is well-known that the exact sequence (2.1) is equivalent to giving data of variation of Hodge structure (VHS) of weight $(- 1)$ , moreover it is polarized by a relative
ample line bundle on .
$A$(2.3) Definition. A polarized variation of Hodge structure (VHS) of weight-l andconsisting of:of types $(- 1,0),$ $(0,- 1)$ over is data
$S$$(V_{Z}, A,\mathcal{F}^{0})$t08(i) a local system of free Z-modules on ,(ii) a flat Z-valued non-degenerate symplectic form A on $W_{Z}$ ,such that(iii) and a locally free subsheaf$S$$\mathcal{F}^{0}\subset \mathcal{F}^{-1}$$:=V_{Z}\otimes z\mathcal{O}_{S}$,$\mathcal{F}^{0}\oplus\overline{\mathcal{F}^{0}}\cong V_{Z}\otimes_{Z}\mathcal{O}_{S}$satisfying that
(HRBR): for any non-zero local section$u\in \mathcal{F}^{0}$, we have$A(u, u)=0$ ,$-(\sqrt{-1})A$( , Of) $>0$ ,$u$(GT): and the Griffiths’ transversality$\nabla(\mathcal{F}^{p})\subset \mathcal{F}^{p-1}\otimes\Omega_{S}$Note that the subsheaf$\mathcal{F}^{0}\subset V_{Z}\otimes \mathcal{O}_{S}$.

is given by$ker(V_{Z}\otimes \mathcal{O}_{S}arrow \mathcal{L}ie_{A/S}\simeq R^{1}f_{*}\mathcal{O}_{A}^{an})$,hence one has$R^{1}f_{*}\mathcal{O}_{A}^{an}\simeq Gr_{F}^{-1}$.

For a general polarized VHS, we have the following(2.4) Theorem. Letisbe a polarized $VHS$ over of weight $m$ . Assume thatcompact. Then the cohomology group $H^{q}(S, V_{Z})$ admits a Hodge structure of weight$q+m$ an a primitive decomposition.
$S^{an}$$S$$V_{Z}$$d$(2.5) Remark. This theorem is a starting point of studies of Hodge structures on
the cohomology groups with coefficient in VHS. When $\dim S=1$ but may be noncompact, then Zucker extended Deligne’s result to the cohomology groups,
which is isomorphic to an intersection cohomology group $IH^{q}(S, V_{Z})$ . Now these kindof results have been extended to more general cases. (see [K-K], [Sa.Mol, 2]).
$S$

$H^{q}(\overline{S},j_{*}V_{Z})$Let : $Aarrow S$ be an abelian scheme andthe corresponding polarized
VHS. We define the filtration on the holomorphic de Rham complexby$f$$(V_{Z}, A, \mathcal{F}^{0})$$\Omega_{S}(V_{C})$$F^{r}\Omega_{S}^{p}(V_{C})=\Omega_{S}^{p}\otimes \mathcal{F}^{r-p}$.

(Griffiths’ transversality assures that they actually form subcomplexes.) Assume thatis compact. Then we have an isomorphism

$S$

(2.6)

$H^{n}(S, V_{C})\cong H_{(2)}^{n}(S, V_{C})$for all

$n$

.

109
By using(2.7)$L^{2}$-harmonic theory, one can show that there exists a Hodge decomposition$H^{n}(S, V_{C})\cong H_{(2)}^{n}(S, V_{C})=\oplus_{p+q=n-1}H^{p,q}$.

,
Since one has a quasi-isomorphism
induces a filtration on $H^{n}(S, V_{C})$ and the Hodge compothe filtration
nents are given by$V_{C}\simeq\Omega_{S}(V_{C})$$F^{r}\Omega_{S}(V_{C})$$H^{p,q}\simeq H^{n}(S, Gr_{F}^{p}\Omega_{S}(V_{C}))$.

For example, $H^{0}(S, V_{C})$ has a 2-step filtration $0=F^{1}\subset F^{0}\subset F^{-1}$ whose successive quotients are:$H^{0,-1}=Gr_{F}^{0}=F^{0}=H^{0}(\mathcal{F}^{\triangleleft}arrow\Omega_{S}\otimes Gr_{F}^{-1})$$H^{-1,0}=Gr_{F}^{-1}=F^{-1}/F^{0}=H^{0}(Gr_{\mathcal{F}}^{-1})$

.

,

.

has a 3-step filtration $0=F^{2}\subset F^{1}\subset F^{0}\subset$$F^{-1}=H^{1}$ whose successive quotients are:

where
(2.8)
(2.9)

(2.10)$Gr_{F}^{-1}=\mathcal{F}^{-1}/\mathcal{F}^{0}$$H^{1}(S, V_{C})$$H^{1,-1}=Gr_{F}^{1}=F^{1}=H^{1}(0arrow\Omega_{S}^{1}\otimes F^{1}arrow\Omega_{S}^{2}\otimes Gr_{F}^{-1})$$H^{0,0}=Gr_{F}^{0}=F^{0}/F^{1}=H^{1}(\mathcal{F}^{0}arrow\Omega_{S}^{1}\otimes Gr_{F}^{-1})$$H^{-1,1}=Gr_{F}^{-1}=F^{-1}/F^{0}=H^{1}(Gr_{F}^{-1})$

,

,

.

Considering $H^{1}(S, V_{Q})$ as a lattice of $H^{1}(S, V_{C})$ , we set(2.23)$H^{1}(S, V_{Q})^{0,0}=H^{1}(S, V_{Q})\cap H^{0,0}$.

Let: $H^{n}(S, V_{C})arrow H^{-1,n}=H^{n}(S, Gr_{\mathcal{F}}^{-1})$ be the natural projection map inducedby the Hodge spectral sequence. Set also
$p_{n}$(2.11)(2.12)$A_{const}=coker\{p_{0} :$H^{1}(S, V_{Z})^{0,0}=ker\{p_{1} :H^{0}(S, V_{Z})arrow H^{0}(Gr_{F}^{-1})\}$,

H^{1}(S, V_{Z})arrow H^{1}(S, Gr_{F}^{-1})\}$.

Then by Hodge theory one has(2.13)$H^{1}(S, V_{Q})^{0,0}=H^{1}(S, V_{Z})^{0,0}\otimes z$

Q.

Under these notations, we can state the following theorem which gives a very naturaldescription of $MW(A/S)$ . (Cf. [Zl, Cor. 10.2].)(2.14) Theorem. $Assume$ th at is compact. Then(i) $A_{const}$ in (2.11) is an abelian varie $ty$ over $C$ , and(ii) we have a natu $ral$ exact sequence of abeli an group$S$(2.15)$0arrow A_{const}arrow MW(A/S)arrow H^{1}(S, V_{Z})^{0,0}arrow 0$.

110(2.16) Corollary. If$S$

is compact, th followings are equivalent.
$e$(i) $H^{0}(S, V_{C})=0$ .
.
(ii)(iii) $K/C$ -tra ce $A_{const}$ is zero.
(iv) $MW(A/S)\simeq H^{1}(S, V_{Z})^{0,0}$ , so it is a fnitely generated abelian group.
$H^{0}(S, R^{1}f_{*}\mathcal{O}_{S}^{an})=0$

If moreover $H^{1}(S, V_{Q})^{0,0}=0$ , then $MW(A/S)$ is a ffite $gro$ up.

\S 3 Mordell-Weil groups of Kuga flber spaces.
Letbe a semisimple Q-algebraic group such that $D:=G_{R}/K$ becomes a hermitis, roughly speaking, aian symmetric domain. A Q-symmplectic representation ofhomomorphism : $G_{Q}arrow Sp(2g, Q)$ which induces an equivariant holomorphic map$G_{Q}$$G_{Q}$$\rho$$h:\mathcal{D}arrow Tt_{g}=Sp(2g, R)/K’$.

via , we obtain a family of abelian
Pulling back the universal family. Taking a torsion free discrete group $\Gamma\subset\rho^{-1}(Sp(2g, Z))$ , one canvarieties, which we call a Kuga fiber space ofobtain an abelian scheme :abelian varieties associated a symplectic replesentation .
Shioda [Sdl] proved that Mordell-Weil groups of elliptic modular surfaces are finite.
Silverberg [Sil, 2&3] showed the finiteness of Mordell-Weil groups of Kuga fiber spaces
which are characterized by endomorphism algebras and polarizations, introduced by
Shimura [Shl], [Sh2].
cohomology
By using the result in \S 2, Borel-Wallach vanishing theorem [B-W] forgroups and also (Mixed) Hodge theory, the author proved the following$\tilde{A}_{g}arrow?t_{g}$$h$$Aarrow \mathcal{D}$$f$$A_{\Gamma}arrow S_{\Gamma}=\Gamma\backslash \mathcal{D}$$\rho$$L^{2}$(3.2) Theorem. (cf. [Sa. $MH$, 1991]). For a $Kuga$ fiber space associated to a standard
Q-symplectic representa tion, the Mordell-Weil $gro$ up is finite except possibly for on
$e$

case.
On the other hand, Mok and To obtained the following theorem independently.
(3.3) Theorem. ($[Mo,$ $1990J$, [Mo-T, 1991]). For any Kuga fiber space with a trivial$K/C$ -trace, the Mordell-Weil group is finite.

Mok announced the above result in [Mo], but in the first version of full paper[Mo-T], there was a misunderstanding about Kuga fiber spaces, that is, they tacitly
assumed that the R-valued points
has no compact factor, which is not true inmany important cases.
$G_{R}$

111\S 4 Mordell-Weil groups of Elliptic surfaces.
: $Aarrow S$ be an abelian scheme of relative dimension . In this section weassume that $\dim S=1$ . If is not compact, in order to have a similar description
as in Theorem (2.14), we have to introduce some compactification of both abelianschemes and VHSs. The canonical “compactification” (or extension in precise) of anabelian scheme is given by “N\’eron model” due to N\’eron (cf. [A], [B-L-R]), and thecanonical extension of local system was given by Deligne [D1], and Zucker [Z1] extend
the Hodge theory for this extension.
In order to illustrate this, we will explain about elliptic surfaces. Hence we also
assume that ( the relative dimension of f) is equal to 1. Denote by the compactification of , and set. Then we have the following diagram:
Let

$f$

$g$

$S$

$g$

$\overline{S}$$=$$\Sigma=\overline{S}-S$

$S$

$Y$

–

(4.1)

$\overline{A}$$rightarrow$$\frac{\downarrow\overline{f}}{S}$$A$$\downarrow S^{f}$$\Sigma^{\downarrow}$$\sim j$$arrow$

Here is a smooth projective surface which has no exceptional curve of the first kindin fibers and we set $Y=\overline{A}-A$ . The fiber space :is called an elliptic
surface. If we denote by
the smooth part of , and by
the connectedcomponent in which the zero section is passing. Then
is a smooth commutative
N\’erongroup scheme over which has the Neron’s universal property, so we callmodel. In this case, we have the following isomorphism:$\overline{A}$$\overline{A}arrow\overline{S}$$\overline{f}$$\overline{A}^{\#}\subset\overline{A}$$\overline{A}_{0}^{\#}\subset\overline{A}^{\#}$$\overline{f}$$\overline{A}^{\#}$$\overline{A}^{\#}$

$S$

(4.2)

$MW(A/S)\simeq${$s:\overline{S}arrow\overline{A}^{\#}$, a holomorphic section$of\overline{f}$}.

Moreover we define the narrow Mordell-Weil group by(4.3)$MW_{0}(A/S)\simeq${$s:\overline{S}arrow\overline{A}_{0}^{\#}$, a holomorphic section$of\overline{f}$}.

Setting $V_{Z}=R_{1}f_{*}Z_{A}$ , we have the following exact sequence due to Kodaira(4.4)$0arrow j_{*}V_{Z}arrow R^{1}\overline{f}_{*}\mathcal{O}_{A}arrow \mathcal{O}_{\overline{S}}(\overline{A}_{0}^{\#})arrow 0$.

Zucker [Z1] showed that
underlies a cohomological Hodge complex or a Hodge
module in the sense of Mo. Saito [Sa.Mol, 2]. In particular, the cohomology group
has a pure Hodge structure of weight $q-1$ . Let
denote a Deligne’s
quasi-canonical extension of . Then the Gauss-Manin connection :extends to$j_{*}V_{Z}$$H^{q}(\overline{S},j_{*}V_{Z})$$\overline{V_{\mathcal{O}}}$$\nabla$$V_{\mathcal{O}}$(4.5)$\nabla$:.

$\overline{V_{\mathcal{O}}}arrow\overline{V_{\mathcal{O}}}\otimes\Omega\frac{1}{s}(log\Sigma)$$V_{\mathcal{O}}arrow V_{\mathcal{O}}\otimes\Omega_{S}^{1}$

112
Let us set. Thenhave the following isomorphism:$\overline{\mathcal{F}^{p}}=j_{*}\mathcal{F}^{p}\cap\overline{V_{\mathcal{O}}}$(4.6)$\overline{\mathcal{F}^{p}}$is a locally free extension of 1‘ to$\overline{Gr_{\mathcal{F}}^{-1}}:=\overline{\mathcal{F}^{-1}}/\overline{\mathcal{F}^{0}}\simeq R^{1}\overline{f}_{*}\mathcal{O}_{A}$$\overline{S}$. Then we.

We set, as in (2.11) and (2.12),(4.7)$A_{const}$(4.8)$H^{1}(\overline{S},j_{*}V_{Z})^{0,0}$$:=H^{0}(\overline{S}, \overline{Gr_{F}^{-1}})/H^{0}(\overline{S},j_{*}V_{Z})$$:=ker\{H^{1}(\overline{S},j_{*}V_{Z})arrow H^{1}(\overline{S},\overline{Gr_{\mathcal{F}}^{-1}})\}$.

From Zucker’s results, one can see that $A_{const}$ is an abelian variety defined over $C$
and
(4.9)
where

$H^{1}(\overline{S},j_{*}V_{Z})^{0,0}\otimes Q\simeq H^{1}(\overline{S}, j_{*}V_{Q})\cap H^{0,0}$$H^{0,0}$is the Hodge component of type $(0,0)$ of(4.10)$H^{0,0} \simeq H^{1}(\nabla :,$H^{1}(\overline{S},j_{*}V_{C})$\overline{\mathcal{F}^{0}}arrow\overline{Gr_{\mathcal{F}}^{-1}}\otimes\Omega\frac{1}{s}(log\Sigma))$and.

As in Theorem (2.14), we have the following proposition from the exact sequence(4.4).
Proposition (4.11). Under the above notation, we have the following exact sequ ence:$0arrow A_{const}arrow MW_{0}(A/S)arrow H^{1}(j_{*}V_{Z})^{0,0}arrow 0$

Corollary (4.12). Under the above notation, the followings are equivalent.
(i) $H^{0}(\overline{S},j_{*}V_{C})=0$ .
(ii).
$A_{const}$(iii) K/C-traceis zero.
(iv) $MW_{0}(A/S)\simeq H^{1}(\overline{S},j_{*}V_{Z})^{0,0}$ , so it$H^{0}(\overline{S}, R^{1}\overline{f}_{*}\mathcal{O}_{\overline{S}^{an}})=0$

Ifmoreover
Remarkhave$H^{1}(\overline{S},j_{*}V_{Q})^{0,0}=0$$(4\cdot 13)$

. Since$\overline{f}$

:

$is$

a finitely generated abelian $gro$ up.

, then $MW_{0}(A/S)$ and $MW(A/S)$ are finite groups.

$\overline{A}arrow\overline{S}$is an elliptic surface, if$H^{0}(\overline{S}, R^{1}\overline{f}_{*}\mathcal{O}_{\overline{S}^{an}})=0$.

$\overline{f}$is not trivial, then we113Hence if is not trivial, we have an isomorphism $MW_{0}(A/S)\simeq H^{1}(\overline{S},j_{*}V_{Z})^{0,0}$ .
In this case, Shioda [Sdl] showed that the full Mordell-Weil group $MW(A/S)$ is
isomorphic to$\overline{f}$$NS(\overline{A})/T$

where is a subgroup of the N\’eron-Severi group
generated by the zero section
and all components of fibers. Note that we have an isomorphism
and the Leray spectral sequence for respects Hodge structure (cf. [Z1]). In oursituation, we have a homomorphism ([Sd2], [C-Z])$T$$NS(\overline{A})$$H^{2}(\overline{A}, Z)^{0,0}\simeq NS(\overline{A})$$\overline{f}$$\delta$: $MW(A/S)arrow H^{1}(\overline{S},j_{*}V_{Z})^{0,0}\otimes Q$

and the narrow Mordell-Weil group is given by. On$Q$ , one has a bilinear form induced by that on, and hence we can definea paring on the Mordell-Weil group $MW(A/S)$ which makes $MW(A/S)$ a lattice.
Shioda called this the Mordell-Weil lattice of
which has been deeply studied in[Sd 2].
$\delta^{-1}(H^{1}(\overline{S},j_{*}V_{Z})^{0,0})$$H^{1}(\overline{S},j_{*}V_{Z})^{0,0}\otimes$$H^{2}(\overline{A}, Z)^{0,0}$$\overline{A}/\overline{S}$\S 5 A sketch of a proof of Theorem (3.2).
We will give a sketch of a proof of Theorem (3.2). We may assume that a Qalgebraic groupis simple and a Q-symplectic representation is primary i.e. sumof irreducible representations which are mutually isomorphic.
Denote by :a corresponding Kuga fiber space associated to a$:=R_{1}f_{*}Z_{A}$ . Assume thattorsion free discrete subgroup
. Set
.
case,
In thisif we can show that
$G_{Q}$$f$$A_{\Gamma}arrow S_{\Gamma}=\Gamma\backslash \mathcal{D}$$\dim S_{\Gamma}\geq 2$$V_{Z}$$\Gamma\subset G_{Q}$$H^{q}(S_{\Gamma}, V_{C})=0$

for $q=0,1$ ,we obtain the finiteness of the Mordell-Weil group $MW(A/S)$ .
be the Baily-Borel-Satake compactification of . Since codimLet
we have isomorphisms$\overline{S_{\Gamma}}$$S_{\Gamma}$$H^{q}(S_{\Gamma}, V_{C})\simeq IH^{q}(\overline{S_{\Gamma}}, V_{C})$$(\overline{S_{\Gamma}}/S_{\Gamma})\geq 2$,

for $q=0,1$ ,wheredenote intersection cohomology groups. On the other hand, byZucker conjecture [L], [Sa-St], one has isomorphisms between intersection cohomology
groups and -cohomology groups, which are calculated by some representation theory[B-W]. In fact, by using that, we can show that
$IH(\overline{S_{\Gamma}}, V_{C})$$L_{2}$$H_{(2)}^{q}(S_{\Gamma}, V_{C})=0$

for $q<R$-rank of$G_{R}$.

114
Therefore, we have done if R-rank is greater than one. And if R-rank is one, we havethe isomorphism$G_{R}\simeq SU(n, 1)\cross K$, hence forwhere $K$ is compact. In this case we can not expect the vanishing of
example in case thatis compact, we have to use sharper criterion like$H^{1}$$S_{\Gamma}$$H^{1}(S_{\Gamma}, V_{Q})^{0,0}=0$.

This is done by a careful study of the $(0,0)$ -component and for detail see [Sa.MH].
is compact.)(The same argument goes through even if $\dim S_{\Gamma}=1$ but$S_{\Gamma}$

isis not compact and R-rank of
Next, we have to deal with the case whenone, that is, the case when $G_{Q}=SU(n, 1, Q)$ .
We will only give a sketch of the proof when $G_{Q}=SL_{2}(Q)$ i.e. when :is an elliptic modular surface. The proofs of other cases are a little bit tricky, but the
key idea is the same as in the following.
is finite.
We only have to show that the narrow Mordell-Weil group. Thanks to (4.10) and (4.12), this will be proved if
Here we put $A:=A_{\Gamma}$ and$G_{R}$$S_{\Gamma}$$f$$A_{\Gamma}arrow S_{\Gamma}$$MW_{0}(A_{\Gamma}/S_{\Gamma})$$S=S_{\Gamma}$$H^{0.0} \simeq H^{1}(\overline{\nabla} :Note that the sheaves
complex over$\overline{\mathcal{F}^{0}}$and.

\overline{\mathcal{F}^{0}}arrow\overline{Gr_{F}^{-1}}\otimes\Omega\frac{1}{s}(log\Sigma))=0$$\overline{Gr_{F}^{-1}}\otimes\Omega\frac{1}{s}(log\Sigma)$

are invertible. Since the Gauss-Manin

$S$

V:

$\mathcal{F}^{0}arrow Gr_{F}^{-1}\otimes\Omega_{S}^{1}$is induced by a non-trivial homogeneous variation,must induce an isomorphism.
On the other hand, by the uniqueness of the canonical extension, has to extend to
an isomorphism V, therefore we have$\nabla$$\nabla$$H^{1}(\overline{\nabla})=0$

as desired.
Remark (5.1). The above argument for elliptic modular surfaces implies that the
Hodge decomposition on
are given by$H^{1}(\overline{S},j_{*}V_{C})$$H^{1}(\overline{S},j_{*}V_{C})=H^{1,-1}\oplus H^{-1,1}$.

The space $H^{1,-1}$ is isomorphic to the space of holomorphic cusp forms. This is theeasiest case of Eichler-Shimura isomorphism which was reformulated by Zucker [Z1]in this form.

115\S 6 N\’eron models of Jacobians of curves over function flelds.
In this section, we will discuss N\’eron models of Jacobians of curves over function
fields. A typical examples are given by elliptic surfaces as in \S 4, but we will deal withcurves with arbitrary genus.
As before, let be a connected smooth curve, its smooth compactification, andset. We denote by $K$ the function field $C(S)$ of (or S).
Consider a projective smooth morphism : $Xarrow S$ whose geometric fibers aresmooth connected curves with genus $g\geq 1$ . Moreover we always consider the following
diagram;$S$$\overline{S}$$\Sigma=\overline{S}-S$

$S$

$f$

$Y$

$\overline{X}$$rightarrow$

(6.1)

$X$

$rightarrow$$\frac{\downarrow\overline{f}}{S}$$\downarrow S^{f}$$\Sigma^{\downarrow}$$arrow$$rightarrow j$

Hereis a smooth projective surface without exceptional curves of the first kind infibers ofis a compactification of , which is a projective flat morphism. The
generic fiberof : $Xarrow S$ (or :) is a proper smooth curve definedover the field $K$ . The Jacobian of curveis defined to be$\overline{X}$$\overline{f},$$f$$\overline{f}$$X_{K}$$f$$\overline{X}arrow\overline{S}$$\overline{f}$$X_{K}$$J_{X_{K}}$$:=Pic_{X_{K}/K}^{0}$) denote
over $K$ . Let $Pic_{X/S}$ (resp.
the relative Picard functor for (resp. ) (cf. [B-L-R, 8.1]). Then since : $Xarrow S$ isa projetive smooth morphism, the functor $Pic_{X/S}$ is represented by a smooth separated
S-scheme which is also denoted by $Pic_{X/S}$ (cf. [B-L-R, 9-3]).
Moreover one has a decomposition

which is an abelian variety of dimension
$f$

$g$

$Pic_{\overline{X}/\overline{S}}$$f$$\overline{f}$$Pi_{C_{X/s=\square Pic_{X/s}^{n}}}n\in Z$

wheredenote the open and closed subscheme of $Pic_{X/S}$ consisting of all line
bundles of degree . The subscheme
becomes an abelian scheme over , which
is denoted by $J_{X/S}$ ([B-L-R, 9-4]), and moreover has a canonical S-ample rigidified
line bundle on . The abelian scheme : $J_{X/S}arrow S$ is the N\’eron model of $J_{X_{K}}/K$
over ([B-L-R, 9.5, Th.1]). In particular, we have a canonical isomorhpism$Pic_{X/S}^{n}$

$n$

$S$

$Pic_{X/S}^{0}$

$S$

$L$

$J$

$\pi$

$S$

M $W(J_{X_{K}}/K)=J_{X_{K}}(K)arrow\sim J_{X/S}(S)$where $J_{X/S}(S)$ denote the grcup of regular sections of .
In order to obtain the N\’eron model over , we have to extend the abelian scheme: $J_{X/S}arrow S$ to some group scheme over .
$f$$\overline{S}$$\overline{S}$

$\pi$

116
denote the scheme theoretic fiber of with the, let
For each$g.c.d(m_{i})=$decomposition into irreducible components. We assume that for all$XarrowS$
has a section. In this case by using a
1. This condition is satisfied if e.g. :is an algebraic spaceresult due to Raynaud theorem (cf. [B-L-R, 9.4, Th. 2])ofover andis a separated -scheme. Consder a subfunctor$s\in\Sigma$$X_{s}=\Sigma_{i=1}^{l}m_{i}X_{i}$$s$$s\in\Sigma,$$f$$Pi_{\overline{X}/\overline{S}}$$\overline{P}$$\overline{S}$$\overline{S}$$Pic_{\overline{X}/\overline{S}}$$Pic \frac{0}{X}/\overline{s}$. Then canwhich is defined as the kernel of the degree morphism. Moreover let
in
be locally considered as a scheme theoretic closure ofdenote the scheme theoretic closure of zero section $\epsilon:Sarrow Pic_{X/S}^{0}$ in .
Now we can state the fundamental result.
$\overline{P}$$deg:Pic_{\overline{X}/\overline{S}}arrow Z$$Pic_{X/S}^{0}$$Pic_{\overline{X}/\overline{S}}$$\overline{P}$$\overline{E}$

Theorem6.2. ([B-L-R, 9.5, Th.4]) Under the above nota tion and $ass$ umptions, $we$
have the followin :
$s$

$g$

exists as a separated S-group scheme and is the N\’eron(i) The quotientmodel ofover .

$\overline{J}=\overline{P}/\overline{E}$$\overline{S}$$J_{X_{K}}$(ii)$Pic \frac{0}{X}/\overline{s}$is a separated S-scheme and coincides with the identi $ty$ compon enof the N\’eron model$\overline{J}$of$J_{X_{K}}$$t\overline{J}^{0}$.

The group of connected components of N\’eron model.
In [B-L-R, 9.6], they calculated the group of connected component of the singularfiber of N\’eron modelusing the intersection number of the singular fiber
We will show that there exists another approach by using the monodoromy on a
nearby fiber. This approach seems to be very hopeful for general abelian scheme which
is not neccesarily a Jacobian.
Under the same notations and assumptions as in Theorem 6.2, we have the following
exact sequence of the sheaf on.
$\overline{J}/\overline{J}^{0}by$$X_{s}$$S^{an}$(6.3).

$0arrow R^{1}f_{*}Zarrow R^{1}f_{*}\mathcal{O}_{X}^{an}arrow \mathcal{O}_{S}^{an}(J_{X/S})arrow 0$

It is easy to show that the identity component:
following exact sequence on$\overline{J}^{0}$of the N\’eron model$\overline{S}^{an}$

(6.4)

.

$0arrow j_{*}R^{1}f_{*}Zarrow R^{1}\overline{f}_{*}\mathcal{O}_{X}^{an}arrow \mathcal{O}_{S}^{an}(\overline{J}_{X/S}^{0})arrow 0$

Moreover one can define $Tor=\oplus_{s\in\Sigma}Tor_{s}$ by(6.5)$0arrow\overline{J}_{X/S}^{0}arrow\overline{J}_{X/S}arrow Torarrow 0$.

$\overline{J}$fits into the117denote the small nbd of a critical value $s\in\Sigma$ with coordinate . Fix aLetpoint $t\in\Delta-\{0\}$ and consider the monodromy transformation $T=T_{s}$ : $H^{1}(X_{t}, Z)arrow$$H^{1}(X_{t}, Z)$ . Note that $H^{1}(X_{\ell}, Z)$ is a free Z-module of rank $2g$ . Consider the following
homomorphism$t$$\Delta$$N$$:=T-I_{2g}$ : $H^{1}(X_{t}, Z)arrow H^{1}(X_{t}, Z)$.

Then one has isomorphisms(6.6)$KerN\simeq(j_{*}R^{1}f_{*}Z_{X})_{s}$,$CokerN\simeq(R^{1}j_{*}(R^{1}f_{*}Z_{X}))_{s}$.

From the above sequence, one has the following theorem.

Theorem 6.7. Under the above notation and aes umptions, we have an isomorph$ism$
$s$

(6.8)

$Tor_{s}\simeq Torsion$

part of $CokerN$

Remark 6.9. A degenerate elliptic curve of type

$T=(\begin{array}{ll}-1 -b0 -1\end{array})$

while the$g_{\vee}\overline{|}roup$

has a local monodromy$I_{b}^{*}$,of connected compnents is one of$Z/2\oplus Z/2$

, or $Z/4$

depending on the parity $of-b$ . Ueno and Namikawa classified all degenerate curvesof genus 2 with explicit equations and local monodromy. One can try to calculate the$Tor$ for the stable curve of genus 2with the local monodromy$T–(\begin{array}{llll}1 0 2 -l0 1 -1 20 0 1 00 0 0 1\end{array})$$\Rightarrow$$T\circ r_{s}$$\cong$$Z/3Z$ .

.

118
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–,–,$L_{2}$

DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KYOTO UNIVERSITY, KYOTO, 606,
JAPAN
E-mail address: mhsaito@kusm.kyoto-u.ac.jp