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\centerline{\bf Ramified covers of abelian varieties over function fields}
\medskip
\centerline{\bf Jos\'e Felipe Voloch}
\bs

Lang has conjectured that the set of rational points of a variety of general
type defined over a number field is not Zariski dense (see [L]). This has been
proved by Faltings ([F1,2]) in the special case that the variety is a subvariety of an abelian
variety. The same conjecture can be made for function fields if one excludes
isotrivial varieties, i.e. those defined over the constant field. In positive
characteristic the conjecture is false since there are non-isotrivial
unirational varieties of general type. In characteristic zero, however, the
conjecture is plausible. It has also been proved when the variety is a
subvariety of an abelian variety ([R],[B1],[H]) and when the variety has an
ample $\Omega^1$ ([N],[MD]). We add further evidence to the conjecture by proving
it for certain ramified covers of abelian varieties over function fields of
characteristic zero. Our result is the following:

\proclaim Theorem. Let $K$ be a function field of characteristic zero and
$k$ its constant field. Let $A$ be an abelian variety defined over $K$ with
$K/k$-trace zero and $f$ a non-constant
rational function on $A$ defined over $K$. Then
there exists an integer $n_0$ such that, for $n > n_0$, the set of $K$-rational
points of the cover $X$ of $A$ defined by $z^n = f$ is not Zariski dense in $X$.

{\it Proof:} [B2], theorem 2, states that there exists $n_0$ such that, for
any point $P \in A(K)$ and place $v$ of $K/k$, if $f(P)$ is defined and not
zero then $|v(f(P)| \le n_0$. To prove the theorem it suffices to show that
the rational points of $A$ that lift to rational points of $X$ are not 
Zariski dense in $A$ and we may consider only those points where $f$ is defined
and does not vanish. If $P \in A(K)$ is such a point, lifting to a rational
point on $X$, then $f(P)$ is an $n$-th power, so $n | v(f(P))$, for any place
$v$ of $K/k$ and since $n > n_0$, we get, by the above, that $v(f(P)) = 0$
for all places $v$ of $K/k$, therefore $f(P) \in k$. The theorem now follows
from the following proposition, which has independent interest.

\proclaim Proposition. Let $K$ be a function field of characteristic zero and
$k$ its constant field. Let $A$ be an abelian variety defined over $K$ with
$K/k$-trace zero and $f$ a non-constant
rational function on $A$ defined over $K$. Then
the set $\{ P \in A(K) | f(P) \in k \}$ is not Zariski dense in $A$.

{\it Proof:} Let $G$ be the semi-abelian variety  $A \times \Gm$ and $Y \subset
G$ be the graph of $f$. Let $\Gamma = \{ (P,2^n) \in G | P \in A(K), n \in \Z\}$
and $\Gamma^*$ its Kolchin closure. As the Kolchin closure of the group 
generated by $2$ in $\Gm$ is $k^*$, we get that $\Gamma^*$ contains the set
$\{ (P,x) \in G | P \in A(K), x \in k \}$ and to prove the proposition we need
to show that $\Gamma^* \cap Y$ is not Zariski dense in $Y$. As $\Gamma$ is 
finitely generated, this follows from the main result of [H]. This completes
the proofs of the proposition and the theorem.

{\it Remarks:} 1. The proposition is also true in positive characteristic
and a proof can be given along the lines of [V], theorem 2, which basically
treats the case of elliptic curves.\par
2. It is not hard to show that covers $X$ of $A$ as in the theorem are of 
general type for $n$ sufficiently large, provided the support of the 
divisor of $f$ is ample, which will always be the case if $A$ is simple.\par
3. I owe the following remark to A. Buium. If $A$ is an abelian variety of
dimension at least two and $X$ is a cyclic cover of $A$, branched over a
ample divisor with normal crossings, then the irregularity of $A$ and $X$ are
the same. This follows from a calculation using the Leray spectral sequence
and the Kodaira vanishing theorem. In particular, under these conditions,
the Albanese of $X$ is isogenous to $A$ and thus $X$ does not embed in an
abelian variety. This apply to most $X$ as in the theorem.\par
{\it Acknowledgements:} The author would like to thank A. Buium and E. 
Hrushovski for comments and the NSF for support through the grant DMS-9301157.
\bs
\centerline{\bf References.}
\bigskip
\noindent
[B1] A. Buium, {\it Intersections in jet spaces and a conjecture of S.Lang},
Annals of Math. {\bf 136} (1992), 557-567.
\medskip
\noindent
[B2] A. Buium, {\it The abc conjecture for abelian varieties}, Int. Math.
Research Notices, to appear.
\medskip
\noindent
[F1] G. Faltings, {\it Diophantine approximation on abelian varieties},
Ann. Math. {\bf 133} (1991) 549-576.
\medskip
\noindent
[F2] G. Faltings,  {\it  The general case of  S. Lang's
conjecture}, Proceedings of the Barsotti conference, to appear.
\medskip
\noindent
[H] E. Hrushovski, {\it The Mordell-Lang conjecture for function fields},
preprint, 1993.
\medskip
\noindent
[L] S. Lang, Number theory III: Diophantine Geometry.
Encyclopedia of Mathematical Sciences vol. 60. Springer Berlin - New York 1991.
\medskip
\noindent
[MD] M. Martin-Deschamps, {\it Proprietes de descente des varietes
a fibre cotangent ample}, Ann.Inst.Fourier, {\bf 34} (1984),39-64.
\medskip
\noindent
[N] J. Noguchi, {\it A higher dimensional analogue of Mordell's conjecture
over function fields}, Math. Annalen {\bf 258} (1981) 207-212.
\medskip
\noindent
[R] M. Raynaud, {\it Around the Mordell conjecture for
function fields and a conjecture of Serge Lang.} Algebraic Geometry (Tokyo /
Kyoto 1982) 1-19. Lecture Notes in Math. 1016, Springer Berlin - New York,
1983.
\medskip
\noindent
[V] J. F. Voloch, {\it A diophantine problem on algebraic curves over function
fields of positive characteristic}, Bull. Soc. Math. France {\bf 119} (1991)
121-126.
\medskip
\noindent
Dept. of Mathematics, Univ. of Texas, Austin, TX 78712, USA
\smallskip
\noindent
e-mail: voloch@math.utexas.edu





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