Picard groups in $p$-adic Fourier theory

Let $L$ be a proper finite extension of the field of $p$-adic numbers and let $o\subset L$ be its integers, viewed as an abelian locally $L$-analytic group. Let $\hat{o}$ be the rigid analytic group variety parametrizing the locally analytic characters of o. We study the Picard group and the Grothendieck group of this variety. This preprint is in final version, replaces the former preprint 'Vector bundles over analytic character varieties' and appears in Manuscripta Mathematica.