The following theorem was proved by Abramovich and the author [AV1], under restrictive hypotheses and then by Hrushovski [H], in general. It is the characteristic p analogue of a conjecture of Lang.
Remark. The condition on the subgroup 
holds if, e.g.,
it is contained in the prime-to-p division saturation
of a finitely generated group. We conjecture
that in this case we can include p-division.
In this direction, Boxall [Bo] has shown that a similar
result holds if 
is a torsion group for which the orders of all
elements are divisible only by a finite set of primes, in particular it holds
for the p-power torsion subgroup. In a different vein, Denis [D1] has
proposed some conjectures, analogous to the above, replacing abelian varieties
by Drinfeld modules and their higher dimensional generalizations.
The case where X is a curve defined over a function field K, A its
Jacobian and 
, the above theorem reduces to
the Mordell conjecture which was proven by Grauert and
Samuel (see [Sa1,2]).
Let 
be an abelian variety of dimension n.
For any closed subscheme 
there is a
function 
which satisfies
the following property:
for any affine open set 
and any system of generators
of the ideal defining 
in U, we may write
with b
bounded on any bounded subset of 
. The function
is uniquely determined by the above property
up to the addition of a bounded function and is called
the local height
function associated to X. This notion is developed in detail
in [Si].
The characteristic p analogue of a conjecture of Lang predicts that
if A is an abelian variety over a function field K of
characteristic p > 0, the 
-trace of A is zero and
X is an ample divisor on A then the set of integral points of
is finite.
In trying to prove this
conjecture, I was led to formulate an "infinitesimal"
analogue of the Mordell-Lang conjecture. This was proved by Hrushovski.
The following result is theorem 6.3 of [H].
Using the above result and some estimates on distances between points on abelian varieties I managed to prove the following results in [V5]:
Remark: The hypotheses of the theorem hold for sufficiently general A, see the appendix.
Proof of the corollary : In this case, 
, for an S-integral point
of 
, is the sum of 
over the elements of S,
and it follows that the height is bounded, which proves the corollary.
Example:
Let A be a supersingular abelian variety, then
for P near 
,
therefore, considering the sequence 
for suitable P near 
,
we get infinitely many points on 
with 
.
It follows that for any subvariety X of A containing 
, we get
. Note that we can choose A to be nonisotrivial,
although it is always going to be isogenous to isotrivial (see the appendix).
However, we can choose X suitably so that 
is not isotrivial.
