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The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer.

Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ${\bf F}_p$ (where $p>0$ is a prime number).

Let $A$ be an abelian variety over $K$ and suppose that the $K|k$-image of $A$ is trivial (ie there are no non-vanishing $K$-homomorphisms from $A$ to an abelian variety over $K$, which has a model over $k$).

Question : is it true that $\#{\rm Tor}(A(K^{\rm perf}))<\infty$ ? (*)

Here $K^{\rm perf}$ is the maximal purely inseparable extension of $K$ and ${\rm Tor}(A(K^{\rm perf}))$ is the subgroup of $A(K^{\rm perf})$ consisting of elements of finite order.

To put things in context, recall that by the Lang-Néron theorem, we have $\#{\rm Tor}(A(K))<\infty$.

Furthermore, one can show using a specialization argument that $\#{\rm Tor}(A(K^{\rm perf}))<\infty$ if $k$ is replaced by a finite extension of ${\bf F}_p$; in this case, the assumption on the $K|k$-image can actually be dropped.

Notice also that the inequality in question (*) is actually equivalent to the inequality $\#{\rm Tor}_p(A(K^{\rm perf}))<\infty$, where ${\rm Tor}_p(A(K^{\rm perf}))$ is the subgroup of $A(K^{\rm perf})$ consisting of the elements, whose order is a power of $p$. This follows from the fact the multiplication by $n$ morphism is étale if $p\not|n$.

Question (*) has a positive answer if $A$ is an elliptic curve by the work of M. Levin, who proves a much stronger result (see "On the group of rational points...", Amer. J. Math. 90 (1968)).

The question (*) is in part complementary to the following other question in MO :

Etale endomorphisms of abelian varieties in positive characteristic

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