Purely inseparable points on curves

TL;DR

This paper establishes explicit upper bounds on the number of purely inseparable points on non-isotrivial curves over function fields of positive characteristic, depending on geometric and reduction properties.

Contribution

It provides the first effective bounds relating purely inseparable points to the genus and reduction data of the curve and function field.

Findings

Bounds depend on genus of the curve and function field

Bounds depend on the number of points of bad reduction

Results are effective and explicit

Abstract

We give effective upper bounds for the number of purely inseparable points on non isotrivial curves over function fields of positive characteristic and of transcendence degree one. These bounds depend on the genus of the curve, the genus of the function field and the number of points of bad reduction of the curve.