Wild and even points in global function fields

TL;DR

This paper characterizes when points in global function fields are wild points of self-equivalences, linking this property to 2-divisibility in the Picard group, and constructs such self-equivalences for finite sets of points.

Contribution

It establishes a criterion based on Picard group divisibility for points to be wild points of self-equivalences in global function fields.

Findings

A point is a wild point iff its class is 2-divisible in the Picard group.

For any finite set of points with 2-divisible classes, a self-equivalence exists with exactly these wild points.

The 2-divisibility condition is necessary for a single point but not for multiple points.

Abstract

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.