数理解析研究所講究録 1319 巻 2003 年 14-23

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HEIGHT FUNCTIONS OVER FUNCTION FIELDSDEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY
ATSUSHI MORIWAKIFor details of this talk, see [1], [2], [3] and [4].

1. FUNCTION FIELDSFirst of all, we fix two kinds of functions fields, namely, an arithmetic
function field and ageometric function field.
An arithmetic function field is afinitely generated extension field of Q.
Ageometric function field is afinitely generated extension field of analgebraically closed field.
$\bullet$$\bullet$

2. HEIGHT FUNCTION ON$\mathrm{P}^{1}(\mathbb{Q})$

First, let us review aheight of arational number. Roughly speaking, itmeasures the complexity of rational numbers, and you may agree with thefollowing:

The complexity of rational
Hence, forshould be$a/b\in \mathbb{Q}$

( ,
$a$

.

$\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}.=$$b\in \mathbb{Z}$and$a\mathbb{Z}+b\mathbb{Z}=\mathbb{Z}$$h= \log\max\{|a|, |b|\}$This gives rise to aheight function$h^{\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}}$), the complexity.

on$\mathrm{P}^{1}(\mathbb{Q})=\{(a:b)|a, b \in \mathbb{Q}, (a, b) \neq(0,0)\}$namely, for $x=(a:b)$ with ,
$a$

$b$

$\in \mathbb{Z}$and$a\mathbb{Z}+b\mathbb{Z}=\mathbb{Z}$$h^{\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}}(x)= \log\max\{|a|, |b|\}$

3. HEIGHT FUNCTION ON
For this purpose, we need to ask again what,.

$\mathrm{P}^{1}(\mathbb{Q}(t))$,$h$of $a/b$

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DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY ATSUSHI MORIWAKI

3.1. Geometric case. (Complexity
For $x=$ $(f(t) : g(?))$ with $f(t)$ , $g(t)$$h^{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$is NOT an extension of$h^{\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}}$$=\mathrm{D}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$$\in \mathbb{Z}[t]$

)

and

$f$

when we view

3.2. Arithmetic case. (ComplexityFor, we set(?), $g(t)$ relatively prime,$\mathbb{Q}$as asubfield of$=\mathrm{D}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}+\mathrm{L}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}$$\mathbb{Q}(t)$.

of coefficients)$f= \sum_{i}a_{i}t^{i}\in \mathbb{Q}[t]$$|f|_{\infty}= \max_{\mathrm{i}}\{|a_{i}|\}$.

Then, as before, we may consider$\max\{\deg(f(t)), \deg(g(t))\}$$+ \log\max\{|f|_{\infty}, |g|_{\infty}\}$,which is NOT good from the geometric view point. Thus, we need amore
sophisticated invariant to measure the largeness of coefficients. For this.
purpose, let us fix apositive $(1, 1)$ -form
withonThen, we set$\Omega$$\mathrm{P}^{1}(\mathbb{C})$$v(f)= \exp(\int_{\mathrm{P}^{1}(\mathbb{C})}\log|f|\Omega)$$\int_{\mathrm{P}^{1}(\mathbb{C})}\Omega=1$.

We can see $||f||_{\infty}.=.v(f)$ . Hence, we may define$h^{\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}}(x)= \max\{\deg(f(t)),\deg(g(t))\}$$+ \int_{\mathrm{P}^{1}(\mathbb{C})}\log\max\{|f(t)|, |g(t)|\}\Omega$.

4. AQUICK REVIEW OF ARAKELOV GEOMETRY
4.1. Arithmetic curve. Let $K$ be anumber field andthe ring of$K$
integers in . Let. Let
be the set of all embeddings
be aflat and finitely generated-module of rank 1. For an embedding, the tensor productin terms of the embedding isdenoted by. Let. The collection
be ahermitian metric of. For
is called ahermitian line bundle on
simplicity, it is denoted by .
Let be anon-zero element of . Then, let us consider:$O_{K}$$L$$Karrow \mathbb{C}$$K(\mathbb{C})$$O_{K}$$\sigma\in K(\mathbb{C})$$L\otimes_{K}\mathbb{C}$$L\otimes_{\sigma}\mathbb{C}$$\sigma$$L\otimes_{\sigma}\mathbb{C}$$||\cdot||_{\sigma}$$(L, \{||\cdot||_{\sigma}\}_{\sigma\in K(\mathrm{C})})$$C=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(O_{K})$$\overline{L}$

$L$

$s$

$\log\#(L/sL)-\mathrm{y}^{\backslash }.\log(||s\otimes_{\sigma}1||_{\sigma})$.

$\sigma$Then, by the product formula, it does not depend on the choice of , sothat it is denoted by.
$s$$\overline{\deg}(\overline{L})$

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HEIGHT FUNCTIONS OVER FUNCTION FIELDS

4.2. General case.
$X$ :aprojective and flat integral scheme over such thatis smooth over .
$(Z, T)$ : for anon-negative integer , apair $(Z, T)$ is called an arithmetic
is a
is acycle of codimension andcycle codimension if.
current of type $(p-1,p-1)$ on$\mathbb{Z}$$Xarrow \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathbb{Z})$$\mathbb{Q}$

$p$

$Z$

$p$

$T$

$p$

$X(\mathbb{C})$: the set of all arithmetic cycles of codimension .

$\hat{Z}^{p}(X)$$p$$\hat{R}^{p}(X)$generated by the following elements:: the subgroup of$((/),-[\log|f|^{2}])$, where is anon-zero rational function on an(1)
integral closed subscheme of codimension $p-1$ and $[\log|f|^{2}]$is the current defined by$\hat{Z}^{p}(X)$$f$$\mathrm{Y}$.

$[ \log|f|^{2}](\gamma)=\int_{Y(\mathbb{C})}(\log|f|^{2})\gamma$(2), where ce and are currents of type $(p-2,p-1)$
and $(p-1,p-2)$ respectively.
$(0, \partial(\alpha)+\overline{\partial}(\beta))$

Note that
Here we define$\hat{Z}^{0}(X)=\mathbb{Z}(X, 0)$$\beta$and $\hat{R}^{0}(X)=0$ .

$\overline{\mathrm{C}\mathrm{H}}^{p}(X):=\hat{Z}^{p}(X)/\hat{R}^{p}(X)$.

-hermitian line bundle on $X$ , that is, is aline
Let
be a
is a
-hermitian metric of. We definebundle on $X$ andahomomorphism.:in the following way: Let $(Z, T)$ be an element of. For simplicity,$Z$
we assume that is integral. Then, taking a non-zero rational section of, we consider an arithmetic cycle of codimension $p+1$ :$\overline{L}=(L, ||\cdot||)$$L$$C^{\infty}$$C^{\infty}$$||\cdot||$$\hat{c}_{1}(\overline{L})$$L_{\mathbb{C}}.\mathrm{o}\mathrm{n}X(\mathbb{C})$$\overline{\mathrm{C}\mathrm{H}}^{p}(X)arrow\overline{\mathrm{C}\mathrm{H}}^{\mathrm{p}+1}(X)$$\hat{Z}^{p}(X)$$s$$L|_{Z}$

(

where
Let

$[\log(||s||_{Z}^{2})]$$\overline{L}_{1}$,$\ldots$,

on $Z,$$\mathrm{d}\mathrm{i}\mathrm{v}(s)$$-[\log(||s||_{Z}^{2})]+c_{1}(\overline{L})\wedge T$),

is acurrent given by.
be-hermitian line bundles on $X$ . Then,$\phi$$\overline{L}_{\dim X}$$\mapsto\int_{Z(\mathbb{C})}\log(||s||_{Z}^{2})\phi$$C^{\infty}$$-\dim X$$\hat{c}_{1}(\overline{L}_{1})\cdots\hat{c}_{1}(\overline{L}_{\dim X})\in \mathrm{C}\mathrm{H}$(X).

Moreover, we have ahomomorphism$-\dim X$$\overline{\deg}:\mathrm{C}\mathrm{H}$$(X)arrow \mathbb{R}$given by$\overline{\deg}$(, $T)= \sum_{P}n_{P}\log\neq(\kappa(P))+\frac{1}{2}\int_{X(\mathbb{C})}T$.

$\sum_{P}n_{P}P$DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY ATSUSHI MORIWAKI

Thus, we have the number$\overline{\deg}(\hat{c}_{1}(\overline{L}_{1})\cdots\hat{c}_{1}(\overline{L}_{\dim X}))$

which is called the intersection number ofintersection number$\overline{L}_{1}$,,$\ldots$,$\overline{L}_{\dim X}$. Note that the$\overline{\deg}(\hat{c}_{1}(\overline{L}_{1})\cdots\hat{c}_{1}(\overline{L}_{\dim X}))$

can be defined even if$Xarrow \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\mathbb{Z})$is not smooth over Q.

5. POLARJZATION AND HEIGHT FUNCTION
$K$

$d$

$B$

: an arithmetic function field, i.e., afield finitely generated over Q.
: the transcendental degree of $K$ over Q.

: aprojective and flat integral scheme over$K$.

$\mathbb{Z}$whose function field ison: a nef hermitian line bundle on $B$ , i.e. the Chern form
for every integral
is semi-positive and$B$ .
1-dimensional subscheme on: Apair
is called apolarization of $K$ , denoted by .
For
, we define$\overline{H}$$c_{1}(\overline{H})$$\overline{\deg}(\hat{c}_{1}(\overline{H})\cdot(Z, 0))\geq 0$$B(\mathbb{C})$$Z$$(B,\overline{H})$$\overline{B}$$(5, \overline{H})$$(\phi_{0}, \ldots, \phi_{n})\in K^{n+1}\backslash \{0\}$$h^{\overline{B}}(\phi_{0},$\ldots ,$\phi_{n}):=$$\{-\mathrm{o}\mathrm{r}\mathrm{d}_{\Gamma}(\phi_{i})\}\overline{\deg}(\hat{c}_{1}(\overline{H}|_{\Gamma})^{d})$$\sum_{\Gamma}\max_{\dot{1}}$$+ \int_{B(\mathbb{C})}\log(\max_{i}\{|\phi_{i}|\})c_{1}(\overline{H})^{\wedge d}$.

( ’s run over all prime divisors on $B$ )It is easy to see$\Gamma$$h^{\overline{B}}(x\phi_{0}, \ldots , x\phi_{n})=h^{\overline{B}}(\phi_{0}, \ldots, \phi_{n})$.

Thus we get$h^{\overline{B}}$:$\mathrm{P}^{n}(K)arrow \mathrm{R}$.

In the case where $K$ is anumber field,
is the arithmetic heightfunction.
In the case whereis an arithmetic surface andis aconstant multiple of the geometric height function as$.(0<-c<1)$ ,$h^{\overline{B}}$$\star$$B$$\star$$h^{\overline{B}}$$\overline{H}=(O_{B}, c|\cdot|_{\mathrm{c}\mathrm{a}\mathrm{n}})$

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HEIGHT FUNCTIONS OVER FUNCTION FIELDS

6. ANOTHER DESCR1PT10N
apolarization:$*\mathrm{F}\mathrm{i}\mathrm{x}$$*\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{t}\mathrm{y}$and line bundle over $K$$\{\begin{array}{l}X..\mathrm{a}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{t}\mathrm{y}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}KL..\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{b}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{l}\mathrm{e}\mathrm{o}\mathrm{n}X\end{array}$$*\mathrm{M}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}$Apair$(\mathcal{X}, \overline{\mathcal{L}})$$*\Delta_{P}$For

of $(X, L)$foris called amodel of $(X, L)$ .

$P\in X(\overline{K})$$P\in X(\overline{K})$, the Zariski closure of the image$\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(\overline{K})arrow X\mathrm{L}arrow \mathcal{X}P$

is denoted by$\Delta_{P}$Then we define.
$h_{(\mathcal{X},\overline{\mathcal{L}})}^{\overline{B}}$:$X(\overline{K})arrow \mathbb{R}$

to be
,

$:= \frac{\overline{\deg}(\hat{c}_{1}(\overline{\mathcal{L}}|_{\Delta_{P}})\cdot\hat{c}_{1}(f^{*}(\overline{H})|_{\Delta_{P}})^{d})}{[K(P).K]}.$$h_{(\mathcal{X},\overline{\mathcal{L}})}^{\overline{B}}(P)$. Note that if
where is the canonical morphism
model of $(X, L)$ , then there is aconstant with$f$$\mathcal{X}arrow B$$(\mathcal{X}’,\vec{L})$$C$$|h_{(\mathcal{X},\overline{\mathcal{L}})}^{\overline{B}}(P)-h_{(\mathcal{X},\overline{\mathcal{L}}’)}^{\overline{B}},(P)|\leq C$$(\forall P\in X(\overline{K}))$is anotherDEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY ATSUSHI MORIWAKIThis means thaton

$X(\overline{K})$$h_{(\mathcal{X},\overline{L})}^{\overline{B}}$19

is uniquely determined modulo bounded functions, so that we may write it as$h_{L}^{\overline{B}}$.

7. NORTHCOTT’S THEOREM,
Theorem 1(Northcott’s theorem). We assume that is big, $i.e.,$$O(m^{d})$ andfor a sufficient large , there is a non-zero $s\in H^{0}(B, H^{\otimes n})$ with$||s||_{\sup}<1$ . Then,for any $M$ and , the set$\overline{H}$$\mathrm{r}\mathrm{k}_{\mathbb{Z}}H^{0}(B$

$n$

$e$

$\{P\in X(\overline{K})|h_{L}^{\overline{B}}(P)\leq M, [K(P) : K]\leq e\}$is finite.
Theorem 2(Refinement). We assume thatis big. Then, for a fied ,$\overline{H}$$e$. $K$]$\leq e$}$\frac{\log\neq\{P\in X(\overline{K})|h_{L}^{\overline{B}}(P)\leq h,[K(P)}{h^{d+1}}\cdot$

is bounded above as$h$goes to the infinity.

8. THE NUMBER OF ALGEBRAIC CYCLESIn the similar techniques, we have the following:
Theorem 3(Geometric version). Let $X$ be a projective scheme over a finite
field and $H$ a very ample line bundle on X. For a non-negative integer
we denote by $n_{k}(X, H, l)$ the number of effective -dimensional cycles with$k_{f}$$\mathrm{F}_{q}$$l$$\deg(H^{\cdot}.

{}^{\mathrm{t}}V)=k$.

Then, there is a constant $C$ depending only on
that

$l$

and

$\dim_{\mathrm{F}_{q}}H^{0}(X, H)$

such$\log_{q}(n_{k}(X, H, l))\leq Ck^{l+1}$

for all$k$$\geq 1$.

Theorem 4(Arithmetic version). Let $X$ be a projective and flat integral
scheme over andan ample-hermitian line bundle X. For a real
number , we denote by
the number of effective l-dimensional
cycles with$\mathbb{Z}$$h$$\overline{H}$$C^{\infty}$$n\leq h(X,\overline{H}, l)$$\overline{\deg}(\hat{c}_{1}(H)^{l}..

V)\leq h$.

Then, there is a constant $C$ such that$\log(n_{\leq h}(X, \overline{H}, l))\leq Ch^{l+1}$

for all$h\geq 1$.

HEIGHT FUNCTIONS OVER FUNCTION FIELDS

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Remark 5. The above two theorems might give rise to new zeta functions.
For example, in Theorem 3if we set$Z(X, H, l)(T)= \sum_{k=0}^{\infty}n_{k}(X, H, l)T^{k^{l+1}}$,

then $Z(X, H, l)$ is aconvergent power series at 0. Moreover, in Theorem 4,
if we set$\zeta(X, \overline{H}, l)(s)=\sum_{V}\exp(-s\cdot\overline{\deg}(\hat{c}_{1}(H).l.

is aconvergent Dirichlet series on ${\rm Re}(s)>>0$ , where-dimensional cycles.

V)^{l+1})$

runs over all effective

$V$

$l$

9. HEIGHT FUNCTION ON AN ABELIAN VARIETY
We assume that $X$ is an abelian variety . Let be asymmetric ample
line bundle on . Then, as in the usual theory of height functions, we havethe canonical quadratic function
$A$

$L$

$A$

$\hat{h}_{L}^{\overline{B}}$:$A(\overline{K})arrow \mathrm{R}$.

Actually, it is defined by$\hat{h}_{L}^{\overline{B}}(P):=\lim_{narrow\infty}\frac{h_{L}^{\overline{B}}(nP)}{n^{2}}$By Northcott’s theorem, if$\overline{H}$is big, then$\hat{h}_{L}^{\overline{B}}(P)=0$

From now on, we assume that.

$\overline{H}$.

$P\in A(\overline{K})_{\mathrm{t}\mathrm{o}\mathrm{r}}$$\Leftrightarrow$is big. Here we set$\langle x, y\rangle_{L}^{\overline{B}}=\frac{1}{2}(\hat{h}_{L}^{\overline{B}}(x+y)-\hat{h}_{L}^{\overline{B}}(x)-\hat{h}_{L}^{\overline{B}}(y))$

Then,we set$\langle$,$\rangle_{L}^{\overline{B}}$gives rise to an inner product$A(\overline{K})$(&R. For$\delta_{L}^{\overline{B}}(x_{1}, \ldots, x_{l}):=\det(\langle x:, x_{j}\rangle_{L}^{\overline{B}})$

10. BOGOMOLOVTheorem 6. Let be a subgroupof . Let us fix a basis$\Gamma$,$\ldots$,$x_{l}\in A(\overline{K})$,.

$+\mathrm{M}\mathrm{o}\mathrm{R}\mathrm{D}\mathrm{E}\mathrm{L}\mathrm{L}$, and a subvariety
finite rank in. If the setof$\{\gamma_{1}, \ldots,\gamma_{n}\}$$A_{\overline{K}}$$x_{1}$$A(\overline{K})$$\mathrm{Y}$$\Gamma\otimes \mathbb{Q}$$\{x\in \mathrm{Y}(\overline{K})|\delta_{L}^{\overline{B}}(\gamma_{1}, \ldots, \gamma_{n}, x)\leq\epsilon\}$is Zariski dense in for every positive number , thenis a translationof an abelian subvariety of A-g by an element of , where$\mathrm{Y}$$\mathrm{Y}$$\epsilon$$\Gamma_{\mathrm{d}\mathrm{i}\mathrm{v}}$$\Gamma_{\mathrm{d}\mathrm{i}\mathrm{v}}=\{x\in A(\overline{K})|\exists n\in \mathbb{Z}_{>0}nx \in\Gamma\}$.

DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY ATSUSHI MORIWAKI

Corollary 7(Bogomolov’s conjecture). Let
the set$\mathrm{Y}$be a subvar iety of$A_{\overline{K}}$. If$\{x\in \mathrm{Y}(\overline{K})|\hat{h}_{L}^{\overline{B}}(x)\leq\epsilon\}$is Zariski dense infor every positive number , thenby a torsion point.
of an abelian subvariety of$\mathrm{Y}$$\mathrm{Y}$$\epsilon$

is a translation$A_{\overline{K}}$Corollary 8(Mordell-Lang conjecture). Let be a complex abelian variety,a subvar iety of A. Then, thereanda subgroup of finite rank insuch thatare abelian subvarieties , ,of and , ,$A$$\Gamma$$\mathrm{Y}$$A(\mathbb{C})_{f}$$C_{1}$$\ldots$$C_{n}$$A_{f}$$\gamma_{1}$$\ldots$$\gamma_{n}\in\Gamma$$\mathrm{Y}(\mathbb{C})\cap\Gamma=\cup(C_{i}+\gamma_{i})i=1n$and$\mathrm{Y}(\mathbb{C})\cap\Gamma=\cup(C_{\dot{l}}(\mathbb{C})+\gamma_{i})\cap\Gamma i=1n$.

11. OUTLINE OF THE proof
Step 1: Prove Bogomolov’s conjecture, i.e. the case where $\Gamma=0$ .
Step 2: Verify the special case of Mordell-Lang conjecture:If
is dense in , then is atranslation of an abeliansubvariety.
Step 3: Poonen’s$\mathrm{Y}(K)$$\mathrm{Y}$$\mathrm{Y}$$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{a}+\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p}1+\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p}2$

12. POONEN’S IDEA
K : afield finitely generated over Q.
$\overline{B}=(B,\overline{H})$: abig polarization of K ($\overline{H}$: big).

A : an abelian variety over K.
L : asymmetric ample line bundle on A.

: asubgroup of finite rank insuch that there is afinitely gen.
erated subgroupof $A(K)$ with
Let be an infinite subset of. We say is small with respect toif there is adecomposition
for each $s\in S$ with the following$A(\overline{K})$$\Gamma$$\Gamma_{0}\otimes \mathbb{Q}=\mathrm{I}\otimes \mathbb{Q}$$\Gamma_{0}$

$S$

$S$

$A(\overline{K})$$s=\mathrm{z}(\mathrm{s})$$\Gamma$$\mathrm{z}(\mathrm{s})$properties:(1)
for all $s\in S$ ;(2) for any $\epsilon>0$ , there is afinite proper subset $S’$ of$\gamma(s)\in\Gamma$$\hat{h}_{L}^{\overline{B}}(z(s))\leq\epsilon$

for all $s\in S$$\backslash S’$.

$S$such that

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HEIGHT FUNCTIONS OVER FUNCTION FIELDS

Let

$F$

be afinite extension of $K$ . ForOF[x)$x\in A(\overline{K})$22, we set$:=\{\sigma(x)|\sigma\in \mathrm{G}\mathrm{a}\mathrm{l}(\mathrm{i}\mathrm{f}/F)\}$.

For an integer $n\geq 2$ , let $\sqrt n:A^{n}arrow A^{n-1}$ be ahomomorphism given by$\beta_{n}(x_{1}, \ldots, x_{n})=(x_{2}-x_{1}, x_{3}-x_{1}, \ldots, x_{n}-x_{1})$For asubset

$T$

of

$S$

and afinite extension$F$.

of $K$ , we set$D_{n}(T, F)=\cup\sqrt n(O_{F}(s)^{n})s\in T^{\cdot}$

the Zariski closure of $D_{n}(T, F)$ .
Moreover, we denote by
Apair $(S, K)$ is said to be minimized ifofand any finite extension $F$ of $K$ ,(1) for any infinite subset;
for all integers $N\geq 1$ .
(2)is small with respect to , thenNote that if an infinite subset of
there are an infinite subset of , afinite extension $F$ of $K$ , and apositive
integer $N$ such that $([N](T), F)$ is minimized.
$\overline{D}_{n}(T, F)$

$S$

$T$

$\overline{D}_{2}(T, F)=\overline{D}_{2}(S, K)$$\overline{D}_{2}([N](S), K)=\overline{D}_{2}(S, K)$

$S$

$T$

$A(\overline{K})$$\Gamma$$S$

such
Theorem 9(Poonen-Moriwaki). Let be an infinite subset ofthat is small with respect to F. If $(S, K)$ is minimized, then there is ansuch that $\overline{D}_{n}(S, K)=C^{n-1}$ for all $n\geq 2$ .
abelian subvariety of$S$$A(\overline{K})$

$S$

$C$

$A_{\overline{K}}$The above theorem is aconsequence of Bogomolov’s conjecture.

Three ingredients:
1the above theorem
2the special case of Mordell-Lang conjecture
3ageometric trick to remove ameasure-theoretic argument in PoOnen’s paper
imply the main theorem.
More precisely, we can prove it in the following way:
Replacing $K$ by afinite extension of $K$ , we may assume that there is a
. We set
finitely generated subgroupof $\Gamma\cap A(K)$ with$\Gamma_{0}\otimes \mathbb{Q}=\Gamma\otimes \mathbb{Q}$$\Gamma_{0}$Stab(Y) $=\{a\in A|\mathrm{Y}+a=\mathrm{Y}\}$ .
, it is sufficient to show the following claim.
Considering$=\{0\}$, then is apoint.
Claim: If Stab(Y)$\dim\mathrm{Y}>0$. Then, replacing $K$ by afinite extension of
We assume that
$K$ , we can find an infinite subsetofwith the following properties:.
(1) is small with respect to(2) is Zariski dense in Y.
$A/\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{b}(\mathrm{Y})$$\mathrm{Y}$

$S$

$S$

$S$

$\Gamma_{\mathrm{d}\mathrm{i}\mathrm{v}}$$\mathrm{Y}(\overline{K})$DEPARTMENT OF MATHEMATICS, KYOTO UNIVERSITY ATSUSHI MORIWAKI

4(3) $(S, K)$ is minimized.
ofwith $\overline{D}_{n}(S, K)=C^{n-1}$ for
Then, there is an abelian subvariety
all $n\geq 2$ . If $\dim C=0$ , then $S\subseteq A(K)$ . Thus, by the special case of
Mordell-Lang conjecture,
is atranslation of an abelian subvariety $B$ of. Then, Stab(Y) $=B$ . Thus, $\dim B$ $=0$ , which implies $\dim \mathrm{Y}=0$ , sothat we have acontradiction.
Next we assume that $\dim C>0$ . Let us fix apositive integer with
: $Aarrow A/C$ be the natural homomorphism and$n>2\dim(A)$ . Let$T=\pi(\mathrm{Y})$ .
. Then, we have aLet
be the fiber product over inmorphism:given by$\beta_{n}(x_{1}, \ldots, x_{n})=(x_{2}-x_{1}, \ldots, x_{n}-x_{1})$ .
. Then,, let
be the Zariski closure ofSince. Thus, we get$C$$A_{\overline{K}}$$\mathrm{Y}$$A_{\overline{K}}$

$n$

$\pi$

$T$

$\mathrm{Y}_{T}^{n}$$\beta_{n}$$\mathrm{Y}^{n}$$\mathrm{Y}_{T}^{n}arrow A^{n-1}$$O_{K}(s)^{n}\subseteq X_{T}^{n}$$\bigcup_{s\in S}O_{K}(s)^{n}$$\mathrm{Y}$$\sqrt n(\mathrm{Y})\supseteq C^{n-1}$$\dim(X_{T}^{n})\geq\dim(C^{n-1})$.

On the other hand, since Stab(Y) $=\{0\}$ ,$\dim(X/T)\leq\dim(C)-1$

.

Thus,
$\dim(X_{T}^{n})-\dim(C^{n-1})$$=(n \dim(X/T)+\dim(T))$$-(n-1)\dim(C)$$\leq\dim(C)+\dim(T)-n$$\leq 2\dim(A)-n<0$ .

This is acontradiction.

REFERENCES[1 A. Moriwaki, Arithmetic height functions over finitely generated fields, Invent, math.
140 (2000), 101-142.
[2 A. Moriwaki, Ageneralization of conjectures of Bogomolov and Lang over finitely
generated fields, Duke Math. J. 107, (2001), 85-102.
[3 A. Moriwaki, The canonical arithmetic height of subvarieties of an abelian variety
over afinitely generated field, J. reine angew. Math. 530 (2001), 33-54.
[4 A. Moriwaki, Diophantine geometry viewed from Arakelov geometry, (Japanese)S\={u}gaku 54 (2002), no. 2, 113-129

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