数理解析研究所講究録
第 819 巻 1993 年 80-95

80

From Subvarieties Of Abelian Varieties
To Kobayashi Hyperbolic Manifolds:
After P. Vojta and G. Faltings

Lin Weng

The aim of this note is two folds: the first is to understand Faltings’ theorem about rational
points of subvarieties of abelian varieties, which is conjectured by Lang; the other is to see what
we can use from Faltings’ proof if we deal with some conjectures of Lang about rational points ofKobayashi hyperbolic manifolds.

For doing this, we first recall the following

Faltings’ Theorem. $([F91b])$ Let $X$ be a subvariety of an abelian variety over a field
finitely generated over Q. Then $X$ contains a finite number of translations of abeliansubvarieties which contain all but a finite number of points of $X(F)$ .
$F$

On the other hand, for Kobayashi hyperbolic manifolds, we have the following
Lang’s Conjecture. $([L74])$ Let $X$ be a projective variety defined over a number field . If
$X$ is a Kobayashi hyperbolic manifold, then $X$ has only finitely many k-rational points.
$k$

An obvious connection between the above Faltings theorem and Lang conjecture is that if $X$ isa subvariety of an abelian vaierty which does not contain any translation of abelian subvarieties of, then $X$ is a Kobayashi hyperbolic manifold, hence by the above conjecture, $X$ has only finitely
many k-rational points, while that $X$ has only finitely many k-rational points is an immediate
consequence of Faltings’ theorem. So as one may imagine, the meanning of the title of this paper
is not in this sense.
$A$

$A$

What is the meaning of the title? Roughly speaking, for k-rational points of abelian varieties,
there is a natural N\’eron-Tate pairing among them; while for k-rational points of Kobayashi hyperbolic manifolds, there is a natural Kobayashi distance among them, provided that we can givea good definition for p-adic Kobayashi hyperbolic semi distances. By some classical results, weknow that the N\’eron-Tate pairing is quite rigid, while the Kobayashi distance involves very strong
global properties of the space. So philosophically, once we can find some methods to pass concepts
from the N\’eron-Tate pairing to the Kobayashi hyperbolic semi distance, then one should also canverify the Lang conjecture above.

Basically this note comes from several discussions with Lang in the past two years, I would
like to thank him warmly.

8
I. Faltings’ Theorem

As we stated above, the following result may be thought of as a special situation of Faltings’
theorem.
be an abelian variety defined over a number field . If $X$ is an
Theorem. $([F91a])$ Let
subvariety of which does not contain any translation of abelian subvarieties of , then $X$
has only finitely many k-rational points.
$A$

$k$

$A$

$A$

On the other hand, we can also deduce Faltings’ theorem from this result by applying one
of Kawamata’s structure theorem about subvarieties of abelian varieties. So, note that only thisresult has a closed relation with Lang’s conjecture above, in this section, we only recall the proofof this theorem.
We begin with some facts and notation from arithmetic geometry [We 92].

. Ratiopnal Points. From Grothendieck’s viewpoint, each point of $X$ defined over a field $K$
is just a morphism from $Spec(K)$ to $X$ . Thus, arithmetically, if $X$ has its arithmetic model
$a$

$\pi$

:

$\mathcal{X}arrow Spec(\mathcal{O}_{k})$

,

where
denotes the ring of integers of a number field , then a k-rational point corresponds toa section of ,$s(x)$ :,$k$$\mathcal{O}_{k}$$\pi$$Spec(\mathcal{O}_{k})arrow \mathcal{X}$so that $\pi os(x)=Id_{Spec(\mathcal{O}_{k})}$ . Suppose
is a hermitian line sheaf on , we also call thewith the arithmetic sectionofcorresponding to the rational pointintersection ofat , which is in fact the degree of the pullback hermitian line sheaf on
as the degree ofthe arithmetic curve
via $s(x)$ .
$\mathcal{X}$$(\mathcal{L}, \rho)$$c_{1}^{Ar}(\mathcal{L},\rho)$$(\mathcal{L}, \rho)$$x$$\mathcal{X}$$E_{x}$$x$$Spec(\mathcal{O}_{k})$$b$. Arithmetic Setup. Supposeon the product$X^{m}$

is a positive integer. Later we will consider the arithmetic. So the arithmetic setup now becomes that:$m$is a normal irreducible
its rings of integers, that
Assume that is a number field,
projective -scheme whose generic fiber
is an abelian veriety, and that is a closed irreducible
does not contain any translate of any abelian subvariety. For a fixedsubscheme such thatvery ample line sheaf on
which is symmetric on the generic fibre, choose a proper normal,
trivial over , such that the Poincar\’e line sheavesonmodffication
extend to line
sheaves on , whereis defined by$k$$\mathcal{A}$$\mathcal{O}_{k}$$A_{k}$$\mathcal{O}_{k}$$\mathcal{X}$$\mathcal{X}_{k}$$\mathcal{L}$$\mathcal{A}$$k$$\mathcal{B}arrow \mathcal{A}^{m}$$\mathcal{P}_{ij}$$B_{k}$$\mathcal{B}$$\mathcal{P}_{ij}$$\mathcal{P}:j:=(x_{i}+x_{j})^{*}(\mathcal{L})-pr_{i}(\mathcal{L})-pr_{j}(\mathcal{L})$.

, there is the associated N\’eron model A. Thus, in addition, we may also assume
Further, foras an open subset. Let be the closure ofin , and $y\circ be$ the openthat contains
subset by taking the intersection with. Obviously, contains of the closure of all k-rational
points in.
$\mathcal{A}_{k}$$A^{m}$$\mathcal{B}$$\mathcal{X}^{m}$$\mathcal{Y}$$A^{m}$$\mathcal{Y}$$\mathcal{B}$$y\circ$$X_{k}^{m}$$i=1,$,. The Index. If is a homogenous polynomial with variables$m,$ $j=1,$, thesuch that for a fixed , the degree of onis , then for any point$i(x, f;d_{1}, \ldots, d_{m})$ or $i(x, f)$ , is defined as the maximal
index of at with respect towithrational number such that for any set of integers
$f$

$c$

$x_{ij},$

$f$

$i$

$f$

$d_{1},$$x$$\sigma$$\ldots,$$x_{ij}$$\ldots,$$\ldots,$$x\in P:=P^{n_{1}}\cross\ldots\cross P^{n_{m}}$$d_{i}$$d_{m},$$j_{1},$$\ldots,j_{m}$$\frac{j_{1}}{d_{1}}+\frac{j_{2}}{d_{2}}+\ldots+\frac{j_{m}}{d_{m}}<\sigma$,$n_{i}$$[$

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and any choice of differential operators$D_{i}$of degree$\leq j:$, on the i-th factor$P^{n}\cdot$,$D_{1}oD_{2}o\ldots oD_{m}(f)$vanishes in .
$x$

. Arithmetic Height. Let $X$ be an algebraic cycle of pure codimension on the projective,
space. Then the height of $X$ with respect to a hermitian line sheaf, is defined by
$d$

$p$

$P$

$:=P^{n}\cross Spec(\mathcal{O}_{k})$$(\mathcal{L}, \rho)$$h_{(\mathcal{L},\rho)}(X)$$h_{(\mathcal{L},\rho)}(X):=\deg(X_{Ar}c_{1}^{Ar}(\mathcal{L},\rho)^{\dim X+1})\in R$

where$X_{Ar}$,is the as sociated arithmetic cycle of $X$ , which may be defined as follows:

Suppose $X$ is an irreducible subvariety of codimension in $P$ . If $X$ lies in a fiber over a$:=\deg(X_{k})\cdot h^{p}$ denote the harmonic
finite place of , then $X_{Ar};=(X, 0)$ . Otherwise, let$X$Green) current$(p,p)$ -form on
uniuqe.
Then there exists a
representing
so that
$p$

$k$

$v$

$h_{X}$$P_{\infty}$$dd^{c}gx-\delta_{X}=-h_{X}$Thus, (X,$g_{X}$$g_{X}\in\tilde{\mathcal{D}}(X_{R})$$s$.

) $\in CH_{Ar}^{p}(P)$ . We then let $X_{Ar}:=(X, gx)$ .

As an immediately consequence, we have the following
Proposition. For any effective algebraic cycle $X$ ,$h_{(\mathcal{O}(1),\rho)}(X)\geq 0$

where$\rho$,

denotes the canonical metric induced from the Fubini-Study metric on$P$.

be a hermitian vector sheaf on an arithmetic variety . Then the metricnaturally induces a metric on the vector space of global sections of the pullback vector sheaf
at infinity. Usually, we denote this norm as. Note that this norm depends also on the metric$e$. Norm. Let$\mathcal{X}$$(\mathcal{E}, \rho)$$s$$\rho$$||s||$of the manifold at infinity.
Next we explain the basic strategy for proving Faltings’ theorem. The method Faltings used is
Diophantine Approximations, which was recently re-emphasized by Vojta in his proof of Mordell’s
conjecture. To explain it, we recall the following

Roth’s Theorem. Let$\alpha$be a fixed algebraic number. Given$\epsilon>0$, one has the inequality$| a-\frac{p}{q}|\geq\frac{1}{q^{2+\epsilon}}$for all but a finite number of fractions $p/q$ in lowest form with $q>0$ .
In order to prove Roth’s theorem, we first find a polynomial $P(X)$ , such that

1. The coefficients are not too large;
2. $P(X)$ vanishes with higher order at$(\alpha, \ldots, \alpha)$;

Thus if there are too many approximation rational points
such that$a_{i}/b_{i}$, we may choose$\beta_{i}$$:=a_{i}/b_{i}$

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$a$
$b$

. $h(\beta_{i})>>0$ ;. $h(\beta_{i+1})/h(\beta_{i})>>0$ .

Hence, by Dyson’s lemma, we have

3. $P(X)$ cannot vanish with higher order atthat$\zeta:=D^{j}$’.. .

$(\beta_{1}, \ldots,\beta_{m})$. That is, there exist$D^{j_{n}}P(\beta_{1}, \ldots, \beta_{m})\neq 0$$j_{1},$$\ldots,j_{m}$so.

, we get a contradiction: On one hand, because, hence the height should have an upper bound; while on
many derivatives vanish at
the other hand, by 3, the height should be bounded below.

Therefore, by consider the height of$\zeta\in Q$$(\alpha, \ldots, \alpha)$

Faltings’ proof actually has the same pattern. First, note that polynomials are just the global
sections of very ample line sheaves on products of projective spaces, we may use sections of ample
line sheaves. At this point, Vojta’s makes his first essential contribution: There are more ample
line sheaves on products of varieties than on products of projective spaces. In order to get 1 and
2, Vojta uses an arithmetic Riemann-Roch theorem, while Faltings uses a generalization of Siegel
lemma about the control of the size of solutions for a system of linear equations by the size of theinteger coefficients. Finally, to give 3, Vojta uses a generalization of Dyson’s lemma, while Faltings
uses his Product Theorem.
$s$I.l. Several Intermediate Results$\backslash$I.1.1. Find Ample Line Sheaves
At this point, we have Vojta’s remarkable discovery. That is, on the product of varieties,
there may exist more ample line sheaves than on the product of projective spaces. The importancehere is that, classically, we only use homogeneous polynomials to do diophantine approximations,but homogeneous polynomials are just global sections of (very ample) line sheaves on productsof projective spaces. So if, instead of working only with homogeneous polynomials, we consider
sections of ample line sheaves on products of varieties in question, we may have more choices.

With this advantage of using global sections of ample line sheaves, which do not just comefrom the pull-back of ample line sheaves on products of projective spaces, we actually need to pay
a little bit: Note that the index is defined for homogeneous polynomials, if we choose a global
section of any ample line sheaf, we first need to embed this ample sheaf to a pull-back line sheaffrom a product of projective spaces; second, when we use the embedding to study the index forat any point, we need to control the norm of $I(s)$ well in order to use the classical approach. On
the other hand, as we need to pass from the complex situation to the arithmetic situation, we must
consider the denominators of the coeffients. Basicall, in Faltings’ proof, Faltings uses his Theorem
4.4 and Prop. 5.2 to deal with the problems above.
$I$

$s$

Theorem 1 $(=Theorem4.4[F91a])$ Suppose $m$ is big enough. For any very ample symmetric, there exists a positive number , such that, for any, there exists
line sheaf on, defined bya real number which makes the Faltings line sheaf,$A_{k}$$\mathcal{L}$$\epsilon<\epsilon_{0}$$\epsilon_{0}$$\mathcal{L}(-\epsilon, s_{1}, s_{2}, \ldots, s_{m})$$s$$\mathcal{L}(-\epsilon, s_{1}, s_{2}, \ldots, s_{m}):=-\epsilon\sum_{l}s_{1}^{2}pr^{*}|(\mathcal{L})+\sum_{i}(s_{i}x;-s_{i+1}x_{i+1})^{*}(\mathcal{L})$ample on$\mathcal{X}_{k}^{m}$, whenever$\frac{s_{1}}{s_{2}}\geq s,$$\frac{s_{2}}{s_{3}}\geq s,$$\ldots,$$\frac{s_{m-1}}{s_{m}}\geq s$

.

,

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Remark 1. In fact, it suffices to choose $m$ large enough so that the map$\alpha_{m}$: $X^{m}arrow A^{m-1}$

defined by$\alpha_{m}(x_{1}, x_{2}, \ldots, x_{m});=(2x_{1}-x_{2},2x_{2}-x_{3}, \ldots, 2x_{m-1}-x_{m})$is finite. On the other hand, by the condition for $X$ , i.e. $X$ is a subvariety ofcontain any translate of abelian subvariety, we see that such an exists.

$A$

which does not$m$satisfing
Remark 2. For , we may determine it by the fact that there exists a positive$Y\subset
X^{m}$, the intersection
and any product subvariety
the following condition: For any
numb er$\epsilon_{0}$$\epsilon_{0}$$\epsilon\leq\epsilon_{0}$$\mathcal{L}(-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{dim(Y)}Y$

is positive. The existence of such an is guaranteed by Remark 1 and the fact thatis” the pullback byof an ample line sheaf.
$\epsilon_{0}$$\mathcal{L}(0, s_{1}, s_{2}, \ldots, s_{m})1$“$\alpha_{m}$Now by the fact that$(s:x_{i}-s_{i+1}x_{t+1})^{*}(\mathcal{L})+(s_{i+1}x)^{*}(\mathcal{L})=2s_{i}^{2}\mathcal{L}:+2_{S_{1+1}^{2}}\cdot \mathcal{L}_{i+1}$,, and note that the terms on the left hand side are generated by their global
wheredenotessections, we see that, for big enough, on , there are natural injections of; without a common zero, where
into. In particular, we get some
. Thus by a twisted
sub-line sheaves of the pullback of an ample line sheaf
Koszul complex a standard discussion, we may have the following$pr_{i}(\mathcal{L})$$\mathcal{L}_{i}$

$A$

$d$

$4d \sum_{i}s_{i}^{2}\mathcal{L}_{i}$$I_{d}$$\mathcal{L}(-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{d}$$d_{i}=4_{S_{1}^{2}}\cdot d$$:= \sum_{i}d;\mathcal{L}$$\pi^{*}\mathcal{O}(d_{1}, d_{2}, \ldots, d_{m})$Proposition 2 (=Proposition 5.2 $[F91a]$ ). There exist some effective bounded positive, a suitable constant , and an exact sequence
integers
$a,$

$b$

$c$

$0arrow\Gamma(\mathcal{X}_{k}^{m}, \mathcal{L}(-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{d})arrow\Gamma(\mathcal{X}_{k}^{m}, \otimes;\mathcal{L}_{i}^{d_{i}})^{a}arrow\Gamma(\mathcal{X}_{k}^{m}, \otimes_{i}\mathcal{L}_{i}^{3d}:)^{b}$,such that the follows hold:
1. The norms of the maps are bounded by $\exp(c\sum_{i}d_{i})$ .
2. The difference of the natural norm on$\Gamma(\mathcal{Y}, \mathcal{L}(-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{d})$and that induced on it by the restriction fromis bounded by $\exp(c\sum_{i}d_{i})$ .
, then on the open
maps
over
to
3. If a section of$\exp(c\sum_{i}d_{i})$.
subsetof , the denominator of is bounded by$\Gamma(\mathcal{Y}, \otimes_{i}\mathcal{L}_{i}^{d}:)^{a}$$\mathcal{L}(-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{d}$$s$$\mathcal{Y}^{o}$$\mathcal{X}_{k^{m}}$$\Gamma(\mathcal{Y}, \otimes_{i}\mathcal{L}^{d_{i}}|)^{a}$$s$$\mathcal{Y}$I.1.2. An Upper Bound For The Index

Once we have Step 1, it is very easy for us to deduce an upper bound for the index of a global
section with a suitable restiction on the norm by certain standard methods with the help of Siegel’s
lemma.
Theorem $3.(=Theorem5.3[F91a])$ . For any point $x=(x_{1}, x_{2}, \ldots, x_{m})$ in the smooth locus, if all, and a positive number so thatare large enough, then
of
such
that
there exists for big a section$\sigma$$\mathcal{X}_{k^{m}}$$d$$0<\sigma<\epsilon<\epsilon_{0}$$s_{i}/s_{t+1}$$s\in\Gamma(\mathcal{Y}^{o}, \mathcal{L}(\sigma-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{d})$85
1. $i(x, s)<\sigma$ .
2. $||s|| \leq\exp(c\sum_{j}d_{i})$ , for a suitable constant$c$which depends only on$\sigma$and .
$\epsilon$

I.1.3. A Lower Bound For The Index

contains infinitely many k-rational points,
With above, in this step, we need to show that ifthen we may get a lower bound for the global sections at a suitable point. Then finally, the finite
statement comes by getting a contradiction from comparing the lower bound and the upper boundfor the index. So at first, we need to choose a suitable point.
$\mathcal{X}_{k}$By the Mordell-Weil theorem, $A(k)\otimes R$ is a finite dimensional vector space with an innerproduct given by the N\’eron-Tate pairing, so by a sphere packing, we can find k-rational points, such thatwith heights$x_{1},$$x_{2},$$h_{1},$$x_{m}$$\ldots,$$h_{2},$$\ldots,$$h_{m}$

1.
is big enough;2.
are all bounded below by ;. $<x_{i},$ $x_{i+1}>\geq(1-\epsilon/2)||x_{i}||\cdot||x_{i+1}||$ , which means that all$h_{1}$$s^{2}$$h_{i}/h_{i-1}$$3$$x_{i}$

are almost in the same direction.

But for the purpose here, these conditions are not enough. In fact, as we need to have an
induction on the dimension of $X$ to deduce the assertion, so we may put a certain condition onwith respect to projections, which comes from the idea of the following$x_{i}$

Product theorem $(=Theorem3.1[F91a])$ . Suppose $P,$
, is aproduct of projective spaces over a field of characteristic zero,a linesheaf on with positive integers , and a non-zero global section of over $P$ . Let
denote the subset consisting of the points with the index of at at least . Then for any
positive number, there exists an , depending on , such that the following holds:$P;=P^{n_{1}}\cross P^{n_{2}}\cross\ldots\cross P^{n_{m}}$

$k$

$P$

$d_{i}$$\mathcal{L}$$Z_{\sigma}\subset P$$\mathcal{L}$

$s$

$s$

$p$

$\epsilon>0$$:=\mathcal{O}(d_{1}, d_{2}, \ldots, d_{m})$

$r$

$p$

$\sigma$$\epsilon$

Suppose is an irreducible component of, which is also an irreducble component of$d_{m-1}/d_{m}\geq r$ , we have
Then if
1. $Z=Z_{1}\cross Z_{2}\cross\ldots\cross Z_{m}$ is a product of closed subvarieties;
2. The degree $\deg(Z_{i})$ are bounded by some constant only depending on .
$Z$

$Z_{\sigma+e}$$d_{1}/d_{2}\geq r,$ $d_{2}/d_{3}\geq r,$$Z_{\sigma}$.

$\ldots,$$Z_{i}\subset P^{n_{i}}$$\epsilon$

With this, we define an essential projection in the following sense:
Suppose $X,$
, is a projective variety. There exist a projection : $Xarrow P^{\dim(X)}$ and a
hypersurface
not containing $X$ , with $\deg(Y)\leq(n-d)\deg(X)$ , such that the ideal of $Y$
annihilates. In particular, is \’etale outside $X\cap Y$ . Furthermore $Z=\pi(X\cap Y)\subset P$ is a
hypersurface of degree $\leq(n-d)\deg(X)^{2}$ whose ideal annihilates.
$X\subset P^{n}$$\pi$$Y\subset P^{n}$$\Omega_{X/P}$$\pi$$\Omega_{X/P}$

Proposition 4 $(=Proposition2.2[F91a])$ . The essential projection exists for a setup (X,$P^{n}$).

With this proposition, for each choose an essential projection: $Xarrow P_{i}=P^{\dim(X)}$ , and, defined by a homogeneous polynomial ; with integral coefficients, whosea hypersurface. We also let : $X^{m}arrow P=P_{1}\cross P_{2}\cross\ldots\cross P_{m}$ denote the product of .
ideal annihilates
Thus by induction on $\dim(X)$ , we may further assume that there are infinitely many k-rational
points not in any. In this way, we put an additional condition for the rational points
above:$i$$\pi_{i}$$G$$Z_{i}\subset P_{i}$$\Omega_{X/P}$$\pi$$\pi_{i}$$\pi_{\dot{\iota}^{-1}}(Z_{i})$$x_{i}$

4.

$\pi_{i}(x_{i})$is not in$Z_{i}$.

86
On the other hand, when we do arithmetic, the above essential projection may not work well.
We need a process by removing the denominators for the local equations. Thus, for example, when
we discuss the essential projection, we find that a good projection in arithmetic in the following
sense is useful:
Assume$\mathcal{X}_{k}\subset P_{k}^{n}$is irreducible. Choose a k-rational point

$x$

in

$P_{k}^{n}$so that

1. is not in .
.
2. The homogeneous coordinates of are all integers of absolute value at mostis bounded below by a positive constant only
3. For each finite place , the distance.
depending on$\mathcal{X}$$x$$\deg(\mathcal{X}_{k})[k :

$x$

Q]$

$d_{v}(x, \mathcal{X})$$v$$\deg(\mathcal{X}_{k})$Hence, the projection
$a$

$b$

$c$

$\pi$

with

$x$

as the center satisfies the following conditions:

.
:
makes
. The projection of tohas degree at most $\deg(\mathcal{X}_{k})[k:Q]$ .
. There exists a nontrivial homogeneous polynomial of degree with coefficients inthat $F$ vanishes on the projection.
.

$\pi$

$\pi(\mathcal{X}_{k})\subset P^{n-1}$$P_{k}^{n}arrow P_{k}^{n-1}$$\mathcal{X}_{k}$$l$$P_{k}^{n-1}$

$F$

$l$

$Q$

such

Therefore, for a good projection, we may give a necessary estimate at infinity. As a corollary,combining with the facts about essential projections, we have the following

andProposition 5. There exists a composition of good projections :a homogeneous polynomial $F$ of degree at most $(n-d)\deg(X_{k})[k : Q]$ , whose coefficients are, such that does not vanish identicallyrational integers bounded in size by,in. There also exists a hypersurface $Z\subset P$ , of degree less thanbut annihilates$(n-d)\deg(\mathcal{X}_{k})^{2}[k : Q]^{2}$ , defined by a polynomialwith coefficients in and bounded insize by $\exp(c_{1}h(\mathcal{X})+c_{2})$ , such that annihilates
.
$\pi$

$F$

$\exp(c_{1}h(\mathcal{X})+c_{2})$$\mathcal{X}_{k}$$\mathcal{X}_{k}arrow P:=P_{k}^{\dim(\mathcal{X}_{k})}$$\Omega_{\mathcal{X}/P}$

$G$

$G$

$Z$

$\Omega_{\mathcal{X}/P}$

With these conditions for rational points, by a standard discussion in the sense of arithmetic
intersection theory, and note that there is a difference between the arithmetic height and the
N\’eron-Tate height, we may easily have the following
Proposition 6. With the same notation as above, the arithmetic degree of the hermitian line
sheaf
at $x=(x_{1}, x_{2}, \ldots, x_{m})$ is bounded above by$\mathcal{L}(\sigma-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{d}$$d( \sigma-\frac{\epsilon}{2})m+c\sum_{:}d_{i}$.

Next, we give a lower bound for the index $i(x, s)$ . We hope that $i(x, s)\geq\sigma$ . In practice, wewill give a local discussion from the above conditions, and hence show the following

Proposition 7. With the same notation as above, choose
If are positive integers so that

$G_{i}$

for each$e_{i}$$\sum_{:}\frac{e_{i}}{d_{1}}\leq i(x, s)$,then the arithmetic degree of the hermitian line sheaf$\mathcal{L}(\sigma-\epsilon, s_{1}, s_{2}, \ldots, s_{m})^{d}\otimes\otimes_{i}\mathcal{L}_{i}^{e;\deg(G_{i})}$$\pi_{i}$. Suppose $i(x, s)<\sigma$ .

87
at $x=(x_{1}, x_{2}, \ldots, x_{m})$ is bounded below by$-c \sum_{:}d_{i}$.

is bigis small enough andSo, by comparing Proposition 6 and Proposition 7, oncefrom Proposition
enough, we may get a contradiction by the upper bound norm condition for
5 and the fact that the second term of$h_{1}$$\sigma$$G_{i}$$d( \sigma-\frac{\epsilon}{2})m+c\sum_{:}d$:so that the whole expression becomes negative if
if of size
completes the proof.
$d/h_{1}$$h_{1}$

is sufficiently large. This

I.2. The Proof Of Intermediate Results: A Sketch

Due to the lack of space, we omit this section.

II. From Subvarieties of Abelian Varieties
To Kobayashi Hyperbolic Manifolds: A Speculation

In this section, we look at the present possibility of using diophantine approximations toprove the Lang conjecture stated in the introduction about rational points of Kobayashi hyperbolic
manifolds.
II.1. Kobayashi Hyperbolic Manifolds
Let $D$ denote the (open) unit disc in the complex plane C. We introduce the Poincar\’ehyperbolic norm on $D$ as follows:

If $z\in D$ and $v\in T_{z}(D)$ is a tangent vector at , which in this case can be identified with acomplex number, then$z$$|v|_{hyp,z}$$;= \frac{|v|_{ecu}}{(1-|z|^{2})}$

wheredenotes the euclidean norm on C. Similarly, for any positive number , we let $D(O, r)$
be the open disc of radius with center . The Poincar\’e hyperbolic metric on $D(O, r)$ is definedby$|v|_{euc}$

$r$

$0$

$r$

$|v|_{hyp,r,z};= \frac{r|v|_{ecu}}{(r^{2}-|z|^{2})}$Thus multiplication by$r$$m_{f}$

gives an analytic isometry between$D$: $Darrow D(0,r)$

and $D(O, r)$ .

On the other hand, for the unit disc, we have the following

88
Schwarz Lemma. Every analytic map : $Darrow D$ with $f(O)=0$ satisfies $|f(z)|\leq|z|$ for all
and $|f’(0)|\leq 1$ . Furthermore, if there is at least one point $c\in D^{*}$ , the punctured disc,with $|f(c)|=|c|$ , or 1 $f’(O)|=1$ , then is a rotation around .
$f$$z\in D$$f$

$0$

The proof of this lemma can be obtained by the maximal modules principle and the fact thatin $D$ such that
there exists a sequence of point
for $karrow\infty$ .
$|z_{k}|arrow 1$$z_{k}$As direct consequences of this result, we have the following
Theorem. The analytic automorphic group of $D$ is given byAut$D=\{\frac{az}{e^{\overline{b}_{1}z_{\theta}}}b\in C,|a|^{2}-=\{\frac{+_{z}\overline{a}_{-}^{:_{w^{a}’}}+b}{\overline{w}z-1}:w\in D,0\leq|_{\theta^{b}}|_{<}^{2}=_{2\pi^{1}\}^{\}}}$.
Schwarz-Pick Lemma. Let $f$ : $Darrow D$ be an analytic morphism of the disc onto itself.
Then$\frac{|f’(z)|}{1-|f(z)|^{2}}\leq\frac{1}{1-|z|^{2}}$Therefore, we know that analytic automorphisms of $D$ are also isometry with respect to thehyperbolic metric. So, by the double transitive of Aut(D), we can cauculate the hyperbolic distancefor any two point $b\in D$ as follows:$a,$First, if$s$is a positive real number in$D$, then$d_{hyp}(0, s)= \int_{0}^{s}\frac{1}{1-t^{2}}dt=\frac{1}{2}\log\frac{1+s}{1-s}$.

So, in general,$d_{hyp}(a, b)=d_{hyp}(0, | \frac{b-a}{1-\overline{a}b}|)$$= \frac{1}{2}\log\frac{1+\frac{|b-a|}{|1-\overline{a}b|}}{1-\frac{|b-a|}{|1-\overline{a}b|}}$.

Also we know that, for $r>1$ ,$d_{hyp,D}(0, \frac{1}{r})=d_{hyp,D(0,r)}(0,1)$.

Starting from this, for any connected complex space $X$ , we may introduce the Kobayashi
hyperbolic semi distance, $d_{hyp}(x, y)$ , as follows:
Let $x,$ $y\in X$ . We consider a sequence of holomorphic maps
$f_{i}$and points$p_{i},$$q_{i}\in D$: $Darrow X$ , $i=1,2,$$\ldots,$$m$such that$f_{1}(p_{1})=x$,$f_{i}(q_{i})=f_{i+1}(p_{i+1})$,$f_{m}(q_{m})=y$.

89
In other words, we join and by a Kobayashi chain of discs. Add the hyperbolic distances
to define the Kobayashi
between ; and , and take the inf over all such choices of
$x$

$p$

$y$

$f_{1},$$q_{i}$$p_{i},$$q_{i}$

hyperbolic semi distance$d_{Kob,X}(x, y)=d_{X}(x, y)= \inf\sum_{i=1}^{m}d_{hyp}(p_{i}, q;)$Obviously,$d_{X}$.

satisfies the properties of a distance, except that $d_{X}(x, y)$ may be$0$

if $x\neq y$ .

There are several important properties for the Kobayashi hyperbolic semi distance.

1. Every analytic morphism is distance decreasing for the Kobayashi hyperbolic semi distances.
is the largest semi distance on $X$ such that every analytic morphism : $Darrow X$ is distance2.
decreasing.
$f$$d_{X}$

3. $d_{D}=d_{hyp}$ .

Actually, the above properties may character the Kobayashi hyperbolic semi distance.

With above, we define Kobayashi hyperbolic manifolds as these complex manifolds onwhich the Kobayashi hyperbolic semi distance is a distance, that is, $d_{X}(x, y)=0$ if and only if$y\in X$ .
$x=y$ for all points$x,$

II.2. A Possible Definition of p-adic Kobayashi Semi DistanceFor many reasons, p-adic Kobayashi semi distances are quite important. In particular, inorder to do arithmetic for Kobayashi hyperbolic manifolds, we need to give the corresponding
definition for the Kobayashi hyperbolic semi distance in p-adic situations. In this subsection, wegive a possible definition for p-adic Kobayashi hyperbolic semi distance.
In more details, this definition has its foundation on the theory of rigid analytic spaces. We
know that for p-adic situation, since usually the corresonding objects are discrete, it is very hard
to do calculus: There are too many strang phenomenons if we use a very simple translationfrom -adic cauculus to a p-adic one. In this direction, Tate, in his remarkable paper “Rigid
Analytic Spaces”, motivated by the question how to characterize elliptic curves with bad reduction,
discovered a new category of analytic-algebraic objects with a structure rich enough to make doalgebraic geometry in the sense of Grothendieck possible. More precisely, we may use the affinoids
to build rigid analytic spaces, and hence analytic morphisms in this category should come fromthe associated morphisms for certain affinoid algebras. We also have Tits-Bruhat buildings andtheir geometric realization. For these, see [BGR 84] and [Be 90].
$\infty$

On the other hand, calculus for this theory is still not well studied. The point is that now we
do not have a solid foundation for calculus, say, the concept for distances and so on.
The first breaking point in this direction, according to my knowledge, is Drinfel’d’s works
about the upper half plane, which may be thought as the moduli space of elliptic modules (Drinfel’dmodules of rank 2). Parallel to the complex situation about the moduli space of elliptic curves,
we may introduce calculus to study the problems at hands. For more details, see the survey paper[DH 84].
Recently, p-adic calculus becomes more and more important: We have arithmetic geometry
in the sense of Arakelov, in which the arithmetic can be realized as a global version of all $v-$

90
adic calculus, where are infinity places and finite places. Nevertheless, we also have the p-adic
geometry.
superrigidity theorem, which has its roots from the classical rigidity theorem for$v$$\infty$

As a more precise example, Rumely also studies p-adic calculus in order to deal with a certain
arithmetic problem, which has its root from some results of Fekete, Szeg\"o, and Cantor. In [Ru 89],
we may find a definition of capaticity for certain subsets over all places. He actually goes quitefar, even through with the restriction on dimension one objects. Among others, we may find thatp-adic Green’s functions, the canonical distances very useful, even through a canonical distance isin fact not a distance at all.
Now what we can do for p-adic Kobayashi hyperbolic semi distance. First, for p-adic geometry,
we have the corresponding concepts about analytic spaces, analytic morphisms and so on, whichhave their names rigid analytic spaces, analytic morphisms (affinoid morphisms), etc.. Instead ofrecall them all, next we only mention the following for the purpose here:
In the complex situation, a complex space can be built up by patching complex discs. Similarly,
we build up a rigid analytic space $X$ by the following affionid domain:$D^{n}(k):=\{(x_{1}, \ldots, x_{n})\in k^{n} :\max_{1\leq i\leq n}|x;|\leq 1\}$,

where is a p-adic field which comes from the completion of a certain algebraic closed field. Letof formal power series in n-indeterminates overbe the subalgebra of the k-algebra, defined by$k$$T_{n}$$k[[x_{1}, \ldots, x_{n}]]$$k$$T_{n}(k)$$:=k<x_{1},$ $\ldots,$ $x_{n}>$$:=${$\sum_{i_{1},\ldots,i_{n}\geq 0}a_{i_{1},\ldots i_{n1^{1}}^{X^{1}\ldots X^{\int_{\hslash}}}}n$:$a;_{1}\ldots;_{n}\in k,$ $|a_{i_{1}},\ldots,;_{n}|arrow 0$, if $i_{1}+..,$ $+i_{n}arrow+\infty$ }.

Usually, we callthe free Tate algebra in indeterminates over . Obviously, there exists anatural bijection between $D^{n}(k)$ and the maximal spectrum of . (If we use the language in [Be90], we may do even better.)$T_{n}$$k$$n$$T_{n}$

So, in general, we can use the Grothendieck language to define the rigid analytic space in thesense that, locally, it is a spectrum of
for a certain , and these local patches may be glued
by affinoid morphisms, which come from algebraic morphisms among. Hence we also get thedefinition for analytic morphisms in the rigid analytic space category.
$T_{n}$$n$$T_{n}’ s$

As direct consequences of the definition, we have the following
Maximal Modules Principle. The maximum of the values taken a strictly convergent
power series $f$ is assumed on the subset$\{(x_{1}, \ldots , x_{n})\in D^{n}(k) :|x_{1}|=\ldots=|x_{n}|=1\}$of the unit ball $D^{n}(k)$ .

Identity Theorem. If $f\in T_{n}$ vanishes for all $x\in D^{n}(k)$ , then $f=0$ . In particular, the map
associating to a series $f\in T_{n}$ its corresponding function from $D^{n}(k)$ to is an injection.
$k$

Furthermore, for automorphisms of the unit disc, we have the following

91
Proposition. The series
itself if and only if$f= \sum_{1}^{\infty_{=0}}a_{i}x^{i}\in T_{1}$$|a_{0}|\leq 1,$$|a_{1}|=1$

defines a bi-affinoid map of the unit disc into

, and $|a_{i}|<1$ for all $i>1$ .

With the above preparation, we define the padic Poincar\’e hyperbolic distance on theuint disc $D^{1}(k)$ by$d_{hyp,v}(a, b):= \log_{v}\frac{1+|b-a|_{v}}{1-|b-a|_{v}}$where if$q_{v}$

denotes the number of elements of the primitive residus field of$\log_{v}$$:=\log_{q_{\nu}}$$k,$$\log_{v}$is defined by.

Next let us “justify” this definition. First, for simplicity, we define$\log_{v}=\{\begin{array}{l}log_{q_{w}}log_{e}log_{e^{2}}\end{array}$$i^{fvisfinite;}ifvisarea1va1ue_{v^{;}alue}i_{fvisacomp1ex}$.
Thus by the definition for the complex situation, we know that$d_{hyp}(a, b)==d_{hyp}(|) \frac{1}{2}\log\frac{1+0,|\frac{b-a}{1_{\frac{|b-a|-\overline{a}b}{|1-\overline{a}b|}}}}{1-\frac{|b-a|}{|1-\overline{a}b|}}$$= \log_{v}\frac{1+\frac{|b-a|_{v}}{|1-\overline{a}b|_{v}}}{1-\frac{|b-a|_{v}}{|1-\overline{a}b|_{v}}}$,

where denotes the complex place. Note that in this expression,
is so in $D$ , therefore when
we take a finite place, by the ultrametric inequality, we see that $|1-x|_{v}=1$ for any point in D.
$\overline{a}b$

$v$

$al$

Now suppose that this definition makes sense, then in a similar manner, we may introduce thep-adic Kobayashi hyperbolic semi distancefor a general space $X$ . Thus we need to show that
Kobayashi hyperbolic semi distance has similar properties as these for the Kobayashi
thehyperbolic semi distance over C. In particular, we need to have a corresponding result in thep-adic category for the following:$d_{X}$$1\succ adic$

Every analytic morphism is distance decreasing for the Kobayashi hyperbolic semi distances.
For this, we first need to show that an automorphism of the unit disc in rigid analytic space
theory actually keeps the distance above. This is rather obvious by the Proposition above: We
may first assume that $f(O)=0$ , i.e. $f\in T_{1}$ and $a_{0}=0$ . Thus, we find that, in this case, should
be an element ofand so theterm dominates others.
$f$

$f$

$T_{1}$$a_{1}x$

Thus, we need prove that the corresponding p-adic Schwarz Lemma holds.

p-adic Schwarz Lemma. Let : $Darrow D$ be an analytic morphism in the sense of rigid
analytic spaces. If $f(O)=0$ , then $|f(x)|_{v}\leq|x|_{v}$ for all $x\in D$ .
$f$

92
Proof. Here is a completion of an algebraic closed field is very important. In fact, in thisof such thatcase, we know that for each point $a\in|k|$ , there exists a sequence of points$k$$x_{j}$$k$1.
.
$\lim_{j}|x_{j}|_{v}=a$2.
.
$|x_{k}|_{v}\neq a$

Form here, it is not difficult to show that the closure ofas sertion by the fact that the maximal modules principle holds.

$|k|_{v}$is$R_{\geq 0}$. Then we may have theII.3. Classical Relations of Arithmetic Heights And Distances
Due of the lack of space, we omit this section.

II.4. The Situation For Kobayashi Hyperbolic Manifolds: A Strategy

In this section we propose certain steps towards Lang’s conjecture for projective Kobayashi
hyperbolic manifolds.
By the discussion in the last subsection, we know that we may first need the following steps,
which are quite possible:, find a direct relation between Kobayashi hyperbolic distance
1. For any two k-rational points$d(x, y)$ and the arithmetic intersection. And this relation should come from the local
contributions.
2. To give a relation between Kobayashi hyperbolic distance and $ar\dot{Y}thmetic$ heights.
$x,$

$y$

$E_{x}E_{y}$Now let us discuss the above items in more details.

1. We need to define Kobayashi hyperbolic semi distance for all places. Once we have such a
definition, with the discussion from the previous subsection, we need it to be good in the following

sense:
Let $X$ be a projective Kobayashi hyperbolic manifold defined over a number field , then adefinition for p-adic Kobayashi hyperbolic semi distances is good if the following is satisfied:$k$Suppose $X$ has its arithmetic modeloverthe arithmetic intersection of the corresponding sections
relation with$Spec(\mathcal{O}_{k})$$\mathcal{X}$. For any two rational point$E_{x},$$E_{y}$$d_{HYP}(x, y):= \sum_{v}d_{hyp,v}(x_{v}, y_{v})$of X over$Spec(\mathcal{O}_{k})$of $X$ ,has a natural
$x,$

$y$

,

where
and
denotes the reduction of and at the place , and only finite terms on the righthand side is non-zero. Also this global relation should come from the local contributions.
$x_{v}$$y_{v}$

$x$

$y$

$v$

Even through in II.2, we proposed a defintion for p-adic Kobayashi hyperbolic semi distances,
but it is very hard to show that this definition is good in the sense above. Since we may expectthat any relation should come from the local contributions, so we can go a little bit further: We
give a more precise statement about the word “good“.

We start with one dimensional case and with a fixed place, e.g. with Riemann surface .
By the well-known uniformization theorem, we know that the upper half planeis the universal$C$$\mathcal{H}$93covering of any Riemann surface with genus at least 2, outside of the cusps. Thus, we can choosea uniformization: $?i/\Gammaarrow C-S$ ,$\pi$

is a discrete group of $PSL_{2}(R)$ and is a finite set of cusps on $X$ . So we may use astandard process to offer the Green functions of by certain twisting process. (For more details,see [La 75] or [Gr 86].) We will not go further in this direction. Instead, we look at the situation
over $?i$ , which may give us a basic idea to general problems, as corresponding objects for Riemannsurfaces may be obtained from an average process (modulo divergence).
where

$S$

$\Gamma$$C$

First, we know that on $\prime H:=\{z=x+iy;y>0\}$ , the natural metric$d \mu=\frac{dzd\overline{z}}{y^{2}}$gives a hyperbolic mesaure on$w\in H$ is given by$’\kappa$. By an easy calculation, we know that the hyperbolic distance of$z,$$d_{hyp}(z, w)= \cosh^{-1}(\frac{|z-w|^{2}}{2{\rm Im} z{\rm Im} w}+1)$.

So note that the corresponding “ Green function” may be given by$g(z, w)=-2 \log|\frac{z-w}{\overline{z}-w}|$,

we see that$e^{\pm d_{hyp}(z,w)}= \frac{1+e^{-g(z,w)/2}}{1-e^{-g(z,w)/2}}$.

Thus by passing a similar relation to Riemann surfaces, say, by averaging overmodulo thedivergence, note that the Green function has its contribution to the arithmetic intersection, so
over $C$ , we may “get” the following relation$\Gamma$$(e^{c_{1}d_{hyp}(x,y)}-1)(e^{c_{2}[E_{*},E_{Y}]_{\infty}}-1)=O(1)$,. In particular, we see thatfor certain constants“iff“ $d_{hyp}arrow 0$ . Then we may
see that if there are infinitely many k-rational points with the arithmetic heights decay rapidly,
then the Kobayashi hyperbolic distances among them should be small enough. Hence we may havea kind of special sphere packing:$c_{1},$$c_{2}$$[E_{x}, E_{y}]arrow\infty$

is compact, so we always can choose an equal-dimension sphere in $X$ , which contains
infinitely many k-rational points. But on the other hand, for a unit sphere, the Kobayashi hyperbolic distance of the center and the point nearby the boundary should be arbitary large, so by the
discussion above, k-rational points cannot scatter in this way.
$X$

Suppose we now have a good definiton so that the above assertion holds, then to apply
Diophantine Approximations to show the finiteness theorem, we may go a step further:
By the Product Theorem, we may choose the k-rational points
$a$
$b$

$c$

.

$x_{1},$$\ldots,$$x_{m}$

so thatin the same unit ball of dimension $\dim X$ with mutually small hyperbolic distance.
. $h(x:)$ go to infinite rapidly;. are not too twisted. That is, they close to a certain fixed proper subvariety.
$x_{i}$$x_{i}$

94$d$. Once we take the good projection in the sense of Faltings, they are not in the corresponding
hypersurface .
$Z_{1}$

2. Now let us look at how one could have a natural connection beteween Kobayashi hyperbolic
distance and the arithmetic height function. We make this connection by introducing a definition
proposed by Philippon [Ph 90]:
Suppose : $Xarrow P^{n}$ is a closed embedding of an m-dimansional smooth variety $X$ in
defined over a number field . Then we may first define the elimination form of $X$ , denoted: A point ofbe the dual projective space ofas $E(X)$ by the following process: Let
defined by the equation $\zeta z=0$ for $z\in P^{n}$ . Let ,ofcorresponds a hyperplaneso thatbe the subvariety consisting of the points$P^{n}$$\phi$$k$$P^{n}$$\check{P}^{n}$$\check{P}^{n}$$H_{(}$$\zeta$$Y$$P^{n}$$Y\subset(\check{P}^{n})^{m+1}$$(\zeta_{0}, \ldots, \zeta_{m})$$(\cap;H_{\zeta_{i}})\cap X\neq\emptyset$.

Then
is a hypersurface defined over , which is defined by a multihomogeneous polynomial $E$
with degree $d=\deg(X)$ . With $E$ , we may define the Philippon height of $X$ as
$Y$

$k$

$h_{PH}(X):= \sum_{vffiite}\log|E|_{v}+\sum_{iv,nflnite}\int_{(S^{n})^{m+1}}\log|E(v, X)|d\mu$,

where as usual
denotes the maximum v-adic norm of the coefficients of $E,$ $E(v, X)$ denotes
the v-conjugation of $E(X)$ , anddenotes the natural $(U(n+1))^{m+1}$ metric on copys of the unit
sphere .
$|E|_{v}$$d\mu$$S$

Theorem. ([So 91]) The natural relation between the Philippon height and the arithmetic
height is given by$h(X)=h_{PH}(X)+ \frac{1}{2}(n+1)\sum_{j=1}^{n}\frac{1}{j}[k :Q]\deg X$.

With above, note that the essential fact behind the proof of the following theorem is that, on
one hand, the arithmetic height is naturally associated with a certain distance, which is essentiallya quadratic form, while on the other hand the height should decay rapidly, say, exponentially, we
may conclude that now the situation for Kobayashi hyperbolic manifold is somehow as the one forthe following
Theorem. (Faltings [Fa $91a]$ ) Let be an abelian variety over a number field , and $E\subset A$
a closed subvariety. Suppose that for any place of and any positive , the number of
k-rational points $x\in A-E$ , for which the v-local distance $d_{v}(x, E)$ from to $E$ is less than
, is finite.
$A$

$k$

$v$

$k$

$\kappa$$x$$H(x)^{-\kappa}$

Thus, with enough choices of hermitian ample vector sheaves on products of varieties, it ishopeful to prove the Lang conjecture stated in the introduction. We hope that later this idea
would be applied to reprove the special case of Faltings’ theorem, the Mordell conjecture.

95
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translation, translated by N.I. Koblitz, 1990
[BGR 84] BOSCH, S., G\"UNTZER, U., REMMERT, R. Non-archimedean analysis, Grund. Math. Wiss.
261, Springer-Verlag, 1984

[DH 87] DELIGNE, P., HUSEM\"OLLER, D. Survey of Drinfel’d modules, Contemporary Math. vol.
67, 1987[Fa $91a$] FALTINGS, G. Diophantine approximation on abelian varieties, Ann. Math., 133 (1991),

549-576[Fa $91b$ ] FALTINGS, G. The general case of Lang’s conjecture, preprint, Princeton, 1991

[Ge 89] van der GEER, G. A letter to A. Parshin, 31011989[Gr 86] GROSS, B.H. Local heights on curves, in Arithmetic Geometry, edited by G. Cornell and J.
H. Silverman, Springer-Verlag, 1986

[La 74] LANG, S. Higher dimensional diophantine problems, Bull. Amer. Math. Soc. 80 No. 5, 1974
[La 75] LANG, S. $SL_{2}(R)$ , Addison-Wesley, 1975

[La 83] LANG, S. Fundamentals of diophantine geometry, Springer-Verlag, 1983

[La 88] LANG, S. Introduction to Arakelov theory, Springer-Verlag, 1988
[La 87] LANG, S. Introduction to complex hyperbolic spaces, Springer-Verlag, 1987

[Ma 91] MANIN, Y.I. Three-dimensional hyperbolic geometry as
Math., 104, 223-244 (1991)$\infty$-adic Arakelov geometry, Invent.

[Ph 90] PHILIPPON, P. Sur des hauteurs alternatives, preprint 1990
[Ru 89] RUMELY, R.S. Capacity theory on algebraic curves, Springer-Verlag, LNM$n^{o}$

1378, 1989

[Si 86] SILVERMAN, J.H. The theory of height functions, in Arithmetic Geometry, edited by G.
Cornell and J. H. Silverman, Springer-Verlag, 1986
[So 91] SOUL\’E, Ch. Geometrie d’Arakelov et theorie des nombers transcendants, preprint IHES,

1991[Vo 91] VOJTA, P. Siegel’s theorem in the compact case, Ann. Math., 133 (1991)[We 92] WENG, L. Bott-Chern Secondary Characteristic Objects and Arithmetic Riemann-Roch Theorem, manuscript 1992