Wild and even points in global function fields

We develop a criterion for a point of a 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 necessary.