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\centerline{\bf 
 Diophantine Approximation in characteristic $p$}
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\centerline{Jos\'e Felipe Voloch}
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{\it Abstract:} We study diophantine approximations to algebraic functions
in characteristic $p$. We precise a theorem of Osgood and give two classes
of examples showing that this result is nearly sharp. One of these classes
exhibits a new phenomenon.
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  In this note we will be concerned about the approximation of 
functions, algebraic over a global field $K$ of positive characteristic by 
elements of $K$ with respect to a valuation $v$ of $K$.
 We define, for $y \in K_v \setminus K$ (although we will consider only $y$ 
algebraic over $K$ in what follows): 
$$\alpha(y)=\limsup_{r \in K}v(y-r)/h(r),$$
where $h(r) = [K:k(r)]$, where $k$ is the constant field of $K$.
We will give some examples that exhibit pathological behaviour. Recall
that $2 \le \alpha(y) \le d(y) := [K(y):K]$, which are analogues of
the classical theorems of Dirichlet and Liouville.
Osgood [O] has shown that $\alpha(y) \le [(d(y)+3)/2]$ if $y$ does not
satisfy a Riccati equation and we will prove the same bound
if the cross ratio of any four conjugates of $y$ over $K$ is non
constant.  There are some results on $\alpha(y)$ if $y$
satisfies $y^q = (ay + b)/(cy + d)$ where $a,b,c,d \in K, ad-bc \ne 0$ and
$q$ is a power of $p$,
due to the author [V1] and others([BS],[dM],[MR]). One may conjecture that 
these are actually the only functions not satisfying Osgood's bound.
We shall give examples that show that Osgood's bound is close to being best
possible.\par
Take $K = k(x)$ and $y$ satisfying $y^p -y = x$ and $z = y^2$ ($y$ is a classical example
of Mahler's). We have $\alpha(y)= d(y) = d(z) = p$. Also, whenever
$v(y-r)/h(r)$ is near $p$ we have $v(z-r^2)/h(r^2)$ near $p/2$. It 
follows (see below) that $\alpha(z)= p/2$. Note that $z$
does not satisfy a Riccati equation. This example can be generalized as
follows: Given $y$ and $R(Y) \in K(Y)$ a rational function of degree $d$
in $Y$, then $d(R(y)) \le d(y)$ and $\alpha(R(y)) \ge \alpha(y)/d$. So if
$\alpha(y)$ is large we get new examples of well approximated functions which
in general do not satisfy Riccati equations. We shall also produce a very
different class of examples with "large" $\alpha(y)$.\par
Define, for $y$ as above and $\alpha$ a real number,
$$b(y,\alpha) = \limsup_{r \in K} v(y-r) - \alpha h(r).$$  
(Compare [dM], but note that our definitions are
minus the logarithms of those there).
We have that $b(y,\alpha) = + \infty$ (resp. $- \infty$) if
 $\alpha < \alpha(y)$ (resp. $ > \alpha(y)$).
For example, Osgood actually showed that $b(y,[(d(y)+3)/2]) \ne + \infty$
if $y$ does not satisfy a Riccati equation. 
We need the following

\proclaim Lemma 1. Let $y \in K_v, y \notin K$. Suppose $r_n \in K$ satisfy
$$\lim_{n \to \infty} v(y-r_n)/h(r_n) = \alpha,
 \lim_{n \to \infty} h(r_{n+1})/h(r_n) = \beta,$$
where $\alpha > {\beta}^{1/2} + 1$. Then $\alpha(y) = \alpha$ and
$b(y,\alpha) = \limsup_n  v(y-r_n) - \alpha h(r_n)$.

{\it Proof:} Except for the last statement, this is proposition 5 of [V1],
and the last statement also follows easily from the proof given there.
All the results in [V1] are stated for $K = k(x)$ but they all immediately
generalize with their proofs for general $K$.

We can now state

\proclaim Theorem 1. Let $y \in K_v$ satisfy $y^q = (ay + b)/(cy + d)$ 
where $a,b,c,d \in K, ad-bc \ne 0$ and $q$ is a power of the characteristic
of $K$. Let $R(Y) \in K(Y)$ be a rational function of degree $d$ in $Y$.
Assume that $\alpha(y) > d(q^{1/2} + 1)$, Then $$\alpha(R(y)) = \alpha(y)/d$$
and $b(R(y),\alpha(R(y))) \ne \pm \infty$.

{\it Proof:} If $R(y) = y$ this is proved in [V1] and [dM]. We may then assume
$d > 1$. It follows from Theorems 1 and 2 of [V1] and the above lemma that
there is a sequence
$r_n \in K$ as in the lemma with $\alpha(y)= \alpha$ and $\beta = q$.
If we consider the sequence $R(r_n)$, then
we can apply the lemma with $\alpha = \alpha (y)/d$ and $\beta = q$. Finally,
it is clear that $$v(R(y) - R(r_n)) - (\alpha(y)/d) h(R(r_n)) =
v(y-r_n) - \alpha(y) h(r_n) + O(1).$$
This completes the proof.

By taking $y$ as in the theorem with $\alpha(y) = d(y)$ (see above or [V1] for
specific examples) and $R$ as in the theorem with $d = 2$, we get examples
$R(y)$ such that $\alpha(R(y)) = d(y)/2$ and 
$b(R(y),\alpha(R(y))) \ne \pm \infty$. In general, $R(y)$ will not satisfy
a Riccati equation which shows that Osgood's theorem is nearly sharp.
Our next examples will also show that Osgood's theorem is nearly sharp but
will be of a different nature.\par

Suppose that $k$ is a finite field with $q$ elements and let $E$ be an 
elliptic curve defined over $k$. Let $K = k(E)$ be its function field.
A point in $E(K)$ corresponds to a rational map $E \to E$ defined over $k$.
Let $P_0 \in E(K)$ correspond to the identity $I$ and $P_n \in E(K)$ 
correspond to the $n$-th iterate of the $k$-Frobenius map $F$. Note that $P_n$
belong to the subgroup of $E(K)$ generated by $P_0$ and $P_1$, which is of
finite index on $E(K)$ if and only if $E$ is ordinary. For example 
$P_2 + aP_1 -qP_0 =0$, where $a = q + 1 - \#E(k)$. The N\'eron-Tate height
of a point of $E(K)$ is the degree of the corresponding map. For example,
$P_n - P_0$ correspond to $F^n - I$ hence $h(P_n - P_0) = q^n + 1 - a_n =
\#E(k_n)$, where $[k_n:k] = n$ and $ |a_n| \le 2q^{n/2}$.\par
Fix now an integer $m \ge 2, (m,q) = 1$ and assume that 
$E(k)$ contains the $m$-torsion on $E$.
Then $P_n - P_0 = mQ_n, Q_n \in E(K)$. Note that $P_0$ is
not divisible by $m$ in $E(K)$ but let $Q$ be the point on $E$ defined over
the algebraic closure of $K$ which satisfies $mQ = -P_0$ and $K(Q)/K$ 
corresponds to the isogeny multiplication by $m$. Choose a Weierstrass
equation for $E$ and let $s$ be the $x$-coordinate of $Q$. Let $v$ be the
place of $K$ corresponding to the point at infinity of $E$.

\proclaim Theorem 2. Notation as above. The function $s$ belongs to $K_v$
and is algebraic of degree $d(s) = m^2$ over $K$. Moreover, if $m^2 > 2(q^{1/2} 
+ 1)$, then $\alpha(s) = d(s)/2$ and $b(s,\alpha(s)) = + \infty$.

{\it Proof:} The first claim of the theorem is standard. Let $r_n$ be the
$x$-coordinate of $Q_n$ as above. Note that $P_n \to 0$ $v$-adically so
$Q_n \to Q$. Moreover, $h(r_n) = 2h(Q_n) = (2/m^2)h(P_n - P_0)$ and since
multiplication by $m$ is an \'etale map, it follows easily that $v(r_n - s) =
q^n$. The theorem now follows from lemma 1 and the (well-known) fact that
$a_n/2q^{n/2}$ gets arbitrarily close to $1$ as $n \to \infty$.\par

Note that the examples given by theorem 2 are genuinely different from those
in theorem 1, as attested by the behaviour of "$b$". The conditions above
impose some restrictions on $m,q$, namely $m^2 \le q + 1 + 2q^{1/2}, m^2 > 2(q^
{1/2} + 1), m | (q-1)$ (see [V2]) but these conditions are satisfied by some
values of $q$ as soon as $m > 2$. For example $m = 3, q = 4,7, m =4, q = 9,
13, 17, 25, 29, 37, 41$. Another interesting remark is that these examples
seem to be the only known algebraic functions $s$ with $b(s,\alpha(s)) = \pm
\infty$.
Finally note that the
above examples can be modified to work over $k(x)$ as follows. If $E$ has
equation $Y^2 = f(X)$ (assume $q$ odd), consider the elliptic curve $E'$
defined over $k(x)$ by the equation $f(x)Y^2 = f(X)$. $E'$ is a twist of $E$
and the $K$-rational points of $E$ considered above will give points on 
$E'(k(x))$ to which one can apply the same arguments and get the examples over
$k(x)$. This trick already occurs in Manin's elementary proof of the Riemann
hypothesis for elliptic curves over finite fields.\par

As for the promised improvement on Osgood's result we have

\proclaim Theorem 3. Suppose that $y \in K_v$ is algebraic over $K$ of degree 
$d$. 
If $b(y,[(d+3)/2]) = + \infty$ then  the cross ratio of any four 
conjugates of $y$ lies in $k$.

By definition, the cross ratio of $x_1, \ldots ,x_4$ is
$$[x_1,x_2,x_3,x_4] = (x_4 - x_1)(x_3 - x_2)/(x_4 - x_2)(x_3 - x_1).$$

{\it Proof:} By Osgood's theorem [O], $y$ satisfies a Riccati differential
equation $dy/dx = ay^2 + by + c$ where $a, b, c, x \in K$ and $x$ a separating
variable (Osgood only states the result for $K = k(x)$ but it is true in general
).
Let ${\cal D}(Y) = dY/dx - (aY^2 + bY + c)$. Suppose $r_n \in K$
are such that $\lim_{n \to \infty} v(y - r_n) - [(d+3)/2]h(r_n) = + \infty$. Then
$v({\cal D}(r_n)) = v({\cal D}(r_n) - {\cal D}(y)) =  v(y - r_n) + O(1)$ and
$h({\cal D}(r_n)) \le 2h(r_n) + O(1)$. On the other hand $v({\cal D}(r_n)) \le
h({\cal D}(r_n))$ unless ${\cal D}(r_n) = 0$. It follows that ${\cal D}(r_n) = 0$
for $n$ sufficiently large. 
We may assume that $1,2,3$ are "sufficiently large" after renumbering and it
follows from classical properties of Riccati equations that
$$d/dx [y,r_1,r_2,r_3] = d/dx [r_n,r_1,r_2,r_3] = 0$$ for all $n$.
$[Y,r_1,r_2,r_3] = \gamma Y$ is a fractional 
linear transformation with coefficients in $K$ and from the above we have that
$\gamma y = y_2^p, \gamma r_n = s_n^p, y_2 \in K(y), s_n \in K$, where $p$ is
the characteristic of $K$. It follows readily that
$\lim_{n \to \infty} v(y_2 - s_n) - [(d+3)/2]h(s_n) = + \infty$ and it follows
that $y_2$ also satisfies a Riccati differential equation. We can then iterate
this procedure and find fractional linear transformations $\gamma_n$ with
 coefficients in $K$ such that $\gamma_n y = y_n^{p^n}, y_n \in K(y)$. If 
$y, y',y'',y'''$ are any four conjugates of $y$ then $$[y,y',y'',y'''] =
[\gamma_n y,\gamma_n y',\gamma_n y'',\gamma_n y'''] \in K^{p^n}$$ and this
implies the theorem.\par
{\it Remark:} Let $D$ be the divisor on ${\bf P}^1$ formed by the conjugates
of $y$ over $K$, so $D$ is of degree $d$ and is defined over $K$. Let $X$ be
the affine curve ${\bf P}^1 \setminus D$. It can be checked that $y$ satisfies
a Riccati equation if and only if the Kodaira-Spencer class of $X$, in the
sense of [K], vanishes. It can also be checked that the cross ratio of any four
conjugates of $y$ lies in $k$ if and only if $X$ is isotrivial, that is,
isomorphic to an affine curve defined over $k$ perhaps after field extension.
It then follows from theorem 3 that, when $X$ is non-isotrivial, it has only
finitely many integral points.\par
{\it Acknowledgments:} This research was partly supported by the NSF and by the
Univ. of Texas's URI. 
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\centerline{\bf References.}
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\noindent
[BS] L. E. Baum and M. M. Sweet, {\it Badly approximable power series in
characteristic 2}, Ann. Math. {\bf 105} (1977), 573-580.
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[K] N. M. Katz, {\it Algebraic solutions of differential equations (p-curvature
and the Hodge filtration)},
Inventions Math. {\bf 18}, (1972) 1-118.
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[dM] B. de Mathan, {\it Approximation exponents for algebraic functions in
positive characteristic}, Acta Arith. {\bf LX} (1992), 359-370.
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[MR] W. H. Mills and D. P. Robbins, {\it Continued fractions of certain
algebraic power series}, J. Number Theory {\bf 23} (1986), 388-404.
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\noindent
[O] C. F. Osgood, {\it Effective bounds on the "diophantine approximation"
of algebraic functions over fields of arbitrary characteristics and
applications to differential equations}, Indag. Math. {\bf 37}, (1975), 105-
119.
\medskip
\noindent
[V1] J. F. Voloch, {\it Diophantine approximation in positive characteristic},
Periodica Math. Hungarica {\bf 19} (1988), 217-225.
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[V2] J. F. Voloch, {\it A note on elliptic curves over finite fields},
Bull. Soc. Math. France {\bf 116} (1988), 455-458.
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Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, U.S.A.



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