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.