  We develop a criterion for a point of global function field to be a unique
wild point of some self-equivalence of this field. We show that this happens if
and only if the class of the point in the Picard group of the field is
$2$-divisible. Moreover, given a finite set of points, whose classes are
$2$-divisible in the Picard group, we show that there is always a
self-equivalence of the field for which this is precisely the set of wild
points. Unfortunately, for more than one point this condition is no longer a
necessary one.
