We obtain an explicit upper bound on the torsion of the Picard group of the forms of the affine line and of the additive group, and a sufficient condition for this Picard group to be non trivial. We also give examples of non trivial forms of the affine line with trivial Picard groups.
The class of the affine line is a zero divisor in the Grothendieck ring: via $G_2$-Grassmannians
Fibrations by affine lines on rational affine surfaces with irreducible boundaries
Topological computation of some Stokes phenomena on the affine line
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