Picard group of the forms of the affine line and of the additive group

TL;DR

This paper provides bounds and conditions for the Picard group of forms of the affine line and additive group, including explicit bounds on torsion and examples of trivial Picard groups.

Contribution

It offers explicit upper bounds on torsion, conditions for non-trivial Picard groups, and examples of trivial Picard groups in forms of the affine line.

Findings

Explicit upper bounds on torsion of Picard groups

Conditions for non-trivial Picard groups

Examples of trivial Picard groups in non-trivial forms

Abstract

We obtain an explicit upper bound on the torsion of the Picard group of the forms of the affine line and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of the affine line to be non trivial and we give examples of non trivial forms of the affine line with trivial Picard groups.