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\centerline{\bf Distance functions on varieties over non-archimedian local
fields}
\medskip
\centerline{\bf Jos\'e Felipe Voloch}
\bs

Let $K$ be a field, complete with respect to an absolute value $|.|$.
Let $X$ be an algebraic variety defined over $K$. Then $X(K)$ has a 
natural topology coming from the topology of $K$ induced by $|.|$ and it
is possible, in many different ways, to describe this topology by a metric.
This has been studied, for instance, in [Si], where many functorial properties
are obtained. However, the main focus of [Si] is to obtain global height
functions, so the metrics considered are defined in completions of global
fields and care was used to study how things varied with respect to the place.
Also, archimedian valuations are considered.  All this results in
the results of [Si] stated as holding only modulo bounded functions.
Some of the results of [Si] were extended in [B].
Another approach, for projective
space, was suggested in [R] and studied further in [CV]. Again, the focus was
on global fields. The purpose of this note is to concentrate on the 
non-archimedian local case and obtain more refined results. We will use a
different development of the theory, but we will show that we recover the
metrics defined by the aforementioned authors, in particular showing that
they coincide, which is not obvious from their definitions. Our results will
be sharp and will not involve any unspecified bounded function. Most, if
not all, of our results will not be a surprise to the experts and, in fact,
we have implicitly used these results, e.g. in [V]. Nevertheless, it seems
appropriate to record these results with proofs, since no other source is
currently available in the literature. 
At the end we will discuss some
global results and give a sharpening of a result of Carlitz which suggests
an interesting conjecture.

In the case $K=\Qp$, the definition of our metric will be, informally,
that $d(P,Q) = p^{-m}$, if $P,Q$ are equal modulo $p^m$ but not
modulo $p^{m+1}$. To make sure that our definitions are independent of
any choice of coordinates and to deal with more general situations, we 
will use the language of schemes. Perhaps with a bit more effort one could
dispense with that.

Let $K$ be a field, complete with respect to an absolute value $|.|$.
That is, for any $x \in K, |x|$ is a real number and the 
following holds:
{
\item{(i)} For all $x \in K, |x| \ge 0$ and $|x| = 0$ if and only if $x=0$. 
\item{(ii)} For all $x,y \in K, |xy| = |x||y|.$
\item{(iii)} For all $x,y \in K, |x + y| \le \max \{|x|,|y|\}$, with equality
holding if $|x|,|y|$ are distinct.
\par}

Let $\O = \{x \in K \mid |x| \le 1\}$, be the ring of integers of $K$ and,
for any $\e, 0 < \e \le 1$, define $\M_{\e} = \{x \in K \mid |x| < \e \}$.
So, $\M_1 = \M$ is the maximal ideal of $\O$. Let $\O_{\e} = \O/\M_{\e}$
for $0 < \e \le 1$. If $X$ is a scheme over $\O$, we will denote by $X_{\e}$
the base change of $X$ to $\O_{\e}, X_{\e} = X \otimes_{\O} \O_{\e}$.

Let $X$ be a reduced scheme of finite type over $\Spec \O$. We will define
a metric on $X(\O)$. More generally, we will define, for a closed subscheme $Y$
of $X$ a distance function of elements of $X(\O)$ to $Y$ as follows.
Let $d(P,Y) = \inf \{ \e > 0 \mid P_{\e} \in Y_{\e}\}$, provided this set
is non-empty, otherwise set $d(P,Y) = 1$.

If we start with a variety over $K$ we can always take a model over $\O$
and work there. If the variety is projective, then any $K$-rational point
extends to an integral point of the integral model. In general, this will
not be the case, and thus our metric is only defined for integral points.
This is similar to a difficulty encountered in [Si] where in the 
quasi-projective
case, he obtained his distance functions modulo a ``distance to the boundary''.

\proclaim Theorem 1. Let $X$ be a reduced scheme of finite type over 
$\Spec \O$. The distance defined above has the following properties:
\item{(a)} If $Y$ is an effective Cartier divisor on $X$ and 
$U$ is an open subset of $X$ on which $Y$ is given by $(f), f \in \O_X(U)$, 
then $d(P,Y) = |f(P)|$ for $P \in U(\O)$.
\item{(b)} If $Y,Z$ are closed subschemes of $X$ then $d(P,Y \cap Z) = 
\max \{d(P,Y),d(P,Z)\}$.
\item{(c)} If $Y$ is another reduced scheme of finite type over 
$\Spec \O$ and $f: X \to Y$ is a morphism of $\Spec \O$-schemes, 
then for any closed subscheme $Z$ of $Y$,
$d(P,f^*(Z)) = d(f(P),Z)$, for all $P \in X(\O)$.
\item{(d)} As a function on $X(\O) \times X(\O), d(P,Q)$ defines a metric
which induces the topology coming from the topology of $\O$. Moreover, it
satisfies the ultrametric inequality $d(P,R) \le \max \{ d(P,Q), d(Q,R)\}$,
for all $P,Q,R \in X(\O)$ with equality holding 
if $d(P,Q), d(Q,R)$ are distinct.
\item{(e)} If $L/K$ is a Galois extension with ring of integers $\O'$,
$\s \in \Gal (L/K)$ and $Y$ is a closed subscheme of $X \otimes_{\O} \O'$, 
we have $d(P^{\s},Y^{\s}) = d(P,Y)$, for all $P \in X(\O')$.

{\it Proof:} To prove (a), note that the reduction of $f$ modulo $\M_{\e}$
defines $Y_{\e}$ in $U_{\e}$ and that $|f(P)| < \e$ if and only if the
reduction of $f(P)$ modulo $\M_{\e}$ is zero. Likewise, (b) is 
straightforward. For (c), it suffices to notice that $f(P_{\e}) = f(P)_{\e}$
and that $f^*(Z)_{\e} = f^*(Z_{\e})$, as follows from the functorial 
properties of base change. For $P,Q \in X(\O), P_{\e} \in Q_{\e}$ if and only 
if $P_{\e} = Q_{\e}$, so the ultrametric inequality follows from transitivity
of equality and the case of equality in the ultrametric inequality is a formal
consequence of it. Finally, it easy to show that the induced topology is that
coming from the topology of $\O$, by taking local coordinates and using (a).
To prove (e) it is enough to notice that the ideals $\M'_{\e}$ of $\O'$ are
Galois invariant since $K$ is complete and the rest follows from the 
functorial properties of base change.

{\it Remark:} It follows from items (a) and (b) of the theorem and
[Si], theorems 1.1 and 2.1 that our distance coincides with that of [Si].
Note however that [Si] always works with $-\log d$.

The distance of [R] is defined as follows. If $P,Q$ are points in $\P^n(K)$
represented by vectors $x,y \in K^{n+1}$, then $d'(P,Q) = |x \wedge y|/|x||y|$,
where vectors are given the sup-norm. When regarding a point in $\P^n(K)$
as a point in $\P^n(\O)$, one chooses a representative $x \in \O^{n+1}$
one of whose coordinates is a unit, thus $|x| = 1$. If such representatives
$x,y$ are chosen for $P$ and $Q$, then $d'(P,Q) = \max_{i \ne j} 
\{ |x_iy_j -x_jy_i| \}$. To show that this coincides with our definition,
note that (compare [Si], \S 3) $d(P,Q) = d((P,Q),\Delta)$, where $\Delta$ is
the diagonal in $\P^n \times \P^n$. Now, it follows that $d=d'$ since
$x_iy_j -x_jy_i=0, i \ne j$, gives a system of equations defining $\Delta$.
Theorem 1 (d) also gives that, if $A$ is a linear map on $\P^n$ over $\O$,
then $A$ induces an isometry on $\P^n(\O)$, which is a special case of
Theorem 3 of [CV], since $A$ is defined over $\O$ if and only if $\eta(A)=1$
in the notation of [CV]. More generally, automorphims of $X/\Spec \O$ are
isometries.

Another equivalent way of defining the distance is through intersection
theory. This is standard when the valuation on $K$ is discrete, but it does
work in general. Namely, if $P \in X(\O)$ is viewed as a section
$s_P : \Spec \O \to X$ and $Y$ is as above, then either the image of $s_P$
and $Y$ are disjoint (in which case $d(P,Y) = 1$) or the image of $s_P$ is
contained in $Y$ (in which case $d(P,Y) = 0$) or $s_P^*(Y)$ is a closed 
subscheme
of $\Spec \O$ supported at the closed point, thus defined by an ideal $I$
of $\O$, in which case $d(P,Y) = \inf \{ \e \mid I \subset \M_{\e} \}$,
as is readily checked. This shows that $d(P,Y) = 0$ if and only if 
$P \in Y(\O)$.

As an application we prove the following higher dimensional generalization
of Krasner's lemma:

\proclaim Corollary 1. Let $X$ be a reduced scheme of finite type over
$\Spec \O$. If $P$ and $Q$ are integral points of $X$ defined over a separable
algebraic extension of $K$ and $d(P,Q) < d(P,P')$ for every conjugate $P'$ of
$P$ over $K$, then $P$ is defined over $K(Q)$.

{\it Proof:} If $P$ is not defined over $K(Q)$, then there exists $\s$ in the
absolute Galois group of $K(Q)$ such that $P^{\s} \ne P$. It follows from
Theorem 1 that
$$d(P,P^{\s}) \le \max \{ d(P,Q), d(Q,P^{\s}) \} = d(P,Q)$$
and this contadicts the hypothesis.

\proclaim Corollary 2. Let $G$ be a reduced group-scheme of finite type over
$\Spec \O$. Then the distance is translation invariant.

{\it Proof:} This follows immediately from Theorem 1 (c), applied to the
morphism given by translation by an element of $G(\O)$.

{\it Remark:} It follows that, in the situation of the corollary,
$d(P,Q) = d(PQ^{-1},1)$ and, from the ultrametric inequality, the
sets $G^{(\e)} = \{ P \in G(\O) \mid d(P,1) < \e \}$ form a filtration of
$G(\O)$ by subgroups, which recovers the canonical filtration of the
formal group $G^{(1)}$ of $G$.

We should also remark that, given a distance function on points,
there is an obvious alternative way of defining distances
to subschemes as follows: $d'(P,Y) =  \inf_{Q \in Y(\O)} d(P,Q)$. 
It follows from theorem 1 that $d(P,Y) \le d(P,Q)$ for $Q \in Y(\O)$ 
by applying item (b) with $Z = Q$, so $d(P,Y) \le d'(P,Y)$ for all
$P \in X(\O)$. 
If the valuation on $K$ is discrete then, by [G1], corollary 1, there exists
$c,\d > 0$, such that $d'(P,Y) \le cd(P,Y)^{\d}$, for all $P \in X(\O)$ and
this was generalized to arbitrary $K$ as above in [Sc].

As a final application we discuss a global problem in ``non-linear
diophantine approximation''.

Let $X$ be an algebraic variety defined over a field $K$, either a number
field or a function field in one variable, with field of constants $k$.
Let $v$ be a place of $K$ and $Y$ a subvariety of $X$ defined over the
local field $K_v$. We shall be interested in points in $X(K)$ which are
$v$-adically close to $Y$ in terms of their heights. If $Y$ is defined over
$K$, it is easy to get an estimate that states that a point cannot be too
close to $Y$ unless it is in $Y$, which can be viewed as a generalization
of Liouville's theorem in diophantine approximation. When $K$ is a number
field, one can view Vojta's conjectures [Vo] as an analogue of Roth's theorem
in this context, but these remain largely unproved, except when $X$ is an
abelian variety. Our goal is to state and prove, under some hypotheses
in the function field case,
a result which can be viewed as an analogue of Dirichlet's theorem. Such
results are known if $X$ is projective space and $Y$ is a linear subspace,
whereas we try to obtain results in a more general context, which explains
the term ``non-linear". Such questions seem to have first been
raised by Schmidt [S1] and he obtained some results for hypersurfaces of
projective space of sufficiently high dimension in the number field case in
[S2]. The special case of projective quadrics has been extensively studied
under the guise of Oppenheim's conjecture and there the results are sharper
in the number field case, due to the work of Margulis and others
(see [M]).

Let $K = k(t)$, where $k$ is an algebraically closed field and $t$ is an
indeterminate and consider its completion $K_v = k((t))$ with respect to the
valuation $v$ centered at 0. We consider the distance function, as defined
above, for varieties over $K_v$. We define the global height of 
$P = (x_0: \ldots :x_n) \in {\bf P}^n(K)$
as $h(P) = \max \{ \deg x_i \}$ if $x_0, \ldots ,x_n \in k[t]$ have no
common factor.

\proclaim Theorem 2. Let $f_1, \ldots , f_s \in K[x_0, \ldots, x_n],
g \in K_v[x_0, \ldots, x_n]$ be homogeneous polynomials.
Let $X = \{ P \in {\bf P}^n \mid f_1(P) = \cdots = f_s(P) = 0 \},
Y = \{ P \in X \mid g(P) = 0 \}$. Let
$d_i = \deg f_i, e = \deg g$. Assume that
$ \sum d_i + e \leq n$ and that  $Y(K) = \emptyset$.
Then, for infinitely many
$P \in X(K), -\log(d(P,Y)) \geq \mu h(P) + O(1)$, where
$\mu = n+1 - \sum d_i$.

{\it Proof:} The proof is an extension of the classical proof of Tsen's
theorem (compare, e.g. [G2]). A similar argument occurs in [C], where
finite, instead of algebraically closed fields are considered. However, in [C],
it is only proved that there is a solution to the given inequality, as opposed
to infinitely many.

Let $H$ be a large integer and consider
$x_0, \ldots ,x_n \in k[t]$ of degree at most $H$, polynomials to be
determined. Their coefficients then give $(n+1)(H+1)$ unknowns in $k$.
The conditions $f_i(x_0, \ldots, x_n) = 0, i = 1, \ldots, s$ and
$v(g(x_0, \ldots, x_n)) \geq M$ impose
$H\sum_1^s d_i + M + O(1)$ conditions given by homogeneous
equations in the unkowns. A solution can be guaranteed to exist if we let
$M = H(n+1 - \sum_1^s d_i) + O(1)$.
Let $x_0, \ldots, x_n$ be a solution
and $d$ their greatest common divisor. Then putting $P = (x_0: \ldots :x_n)$
we have $h(P) = \max \{ \deg x_i \} - \deg d \leq H - \deg d$ and
$-\log d(P,Y) \geq M -v(d)e \geq
M -\deg d e \geq \mu h(P) + O(1)$.
What remains to be shown is that this process leads to infinitely many
distinct points $P$. Suppose not. Assume that the same point $P$ gives rise
to infinitely many $x_0, \ldots, x_n$ and therefore to infinitely many $d$.
From $v(g(x_0, \ldots, x_n)) \geq M$ we get that, since $g(x_0, \ldots, x_n)
\neq 0$,
$ev(d) \geq M + O(1)$ hence $e\deg d \geq M + O(1)$,
 and since $\deg d \leq H$ we get $eH \geq M + O(1)$,
dividing by $H$ and letting $H$ go
to infinity then gives $e \geq n+1 - \sum_1^s d_i$, contradicting the
hypothesis. This completes the proof.\par
This result is best possible in its generality. For example, let $n = 1,s = 0$
and $g(x_0,x_1) = x_0^2 + tx_1^2$, then $X = {\bf P}^1$ and $-\log d(P,Y)
\le 1, \forall P \in X$. There are other examples in all dimensions. It is
conceivable that under additional hypotheses there will be a similar result
for higher degree. The same polynomial $g$ can be used for $n=2$, say, to show
that the constant $\mu$ cannot be improved in general.
However, one may conjecture that if $Y(K_v)$ is Zariski dense in $Y$, one may
weaken the restriction $\sum d_i + e \leq n$ to $\sum d_i \leq n$. 
One may also conjecture that similar statements also hold in the number field
case.
 

\bigskip
\centerline{\bf References.}
\bigskip
\noindent
[B] M. L. Brown, {\it Construction de hauteurs locales et globales sur les
vari\'et\'es alg\'ebriques}, C. R. Acad. Sci. Paris {\bf t. 307} (1988) 87-90.
\medskip
\noindent
[C] L. Carlitz, {\it Some applications of a theorem of Chevalley},
Duke J. Math. {\bf 18} (1951), 811 - 819.
\medskip
\noindent
[CV] K. K. Choi and J. D. Vaaler, {\it Diophantine approximation in
projective space}, to appear.
\medskip
\noindent
[G1] M. J. Greenberg, {\it Rational points in Henselian discrete valuation
rings}, Publ. Math. IHES, {\bf 31} (1967) 59-64.
\medskip
\noindent
[G2] M. J. Greenberg, {it Lectures on forms of higher degree}, Benjamin,
New York, 1969.
\medskip
\noindent
[M] G. Margulis, {\it Dynamical and ergodic properties of subgroup actions}
in {\it Proc. Int. Congress of Mathematicians, Kyoto, 1990}, Springer, Tokio,
1991.
\medskip
\noindent
[R] R. S. Rumely, {\it Capacity theory on algebraic curves}, LNM 1378,
Springer, Berlin, 1989.
\medskip
\noindent
[S1] W. M. Schmidt, {\it Approximation to algebraic numbers}, L'Enseignement
Mathematique {\bf 17} (1971) 187-253.
\medskip
\noindent
[S2] W. M. Schmidt, {\it Diophantine inequalities for forms of odd degree},
Advances in Math. {\bf 38} (1980) 128-151.
\medskip
\noindent
[Sc] N. Schappacher, {\it Some remarks on a theorem of M. J. Greenberg},
Queen's Papers in Pure and Applied Mathematics, {\bf 54} (1980) 101-114.
\medskip
\noindent
[Si] J. H. Silverman,
 {\it Arithmetic distance functions and height functions in
Diophantine geometry}, Math. Ann. {\bf 279} (1987) 193-216.
\medskip
\noindent
[V] J. F. Voloch, {\it Integrality of torsion points on abelian varieties 
over $p$-adic fields}, Math. Research Letters, to appear.
\medskip
\noindent
[Vo] P. Vojta, {\it Diophantine approximations and value distribution theory},
Lecture Notes in Math. {\bf 1239}, Springer, Berlin 1987.
\medskip
\noindent
Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA
\smallskip
\noindent
e-mail: voloch@math.utexas.edu



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