Picard groups in p-adic Fourier theory

Let be a proper finite field extension of and its ring of integers viewed as an abelian locally L-analytic group. Let be the rigid L-analytic group parametrizing the locally analytic characters of o constructed by Schneider-Teitelbaum. Let K/L be a finite extension field. We show that the base change (K) has a Picard group Pic( (K) ) which is profinite and that the unit section in (K) provides a divisor class of infinite order. In particular, the abelian group Pic( (K) ) is not finitely generated and is not a torsion group. On the way we show that (K) is a nontrivial ,tale covering of the affine line over K realized via the logarithm map of a Lubin-Tate formal group. We finally prove that rank and determinant mappings induce an isomorphism between K (0)( (K) ) and .