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

We obtain an explicit upper bound on the torsion of the Picard group of the forms of Ak1 and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of Ak1 to be nontrivial and we give examples of nontrivial forms of Ak1 with trivial Picard groups.