TITLE: Torsion points of abelian varieties in the perfect closure of a function field
QUESTION [18 upvotes]: 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

REPLY [11 votes]: The answer to question (*) is yes. It is Theorem 1.2.2 in the following preprint.