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 A k 1 and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of A k 1 to be nontrivial and we give examples of nontrivial forms of A k 1 with trivial Picard groups. [ABSTRACT FROM AUTHOR]