  Let $C$ and $D$ be smooth, proper and geometrically integral curves over a
finite field $F$. Any morphism from $D$ to $C$ induces a morphism of their
\'etale fundamental groups. The anabelian philosophy proposed by Grothendieck
suggests that, when $C$ has genus at least $2$, all open homomorphisms between
the \'etale fundamental groups should arise in this way from a nonconstant
morphism of curves. We relate this expectation to the arithmetic of the curve
$C_K$ over the global function field $K = F(D)$. Specifically, we show that
there is a bijection between the set of conjugacy classes of well-behaved
morphism of fundamental groups and locally constant adelic points of $C_K$ that
survive \'etale descent. We use this to provide further evidence for the
anabelian conjecture by relating it to another recent conjecture by Sutherland
and the second author.
