Picard groups in p-adic Fourier theory
Let $${L\neq\mathbb{Q}_p}$$L≠Qp be a proper finite field extension of $${\mathbb{Q}_p}$$Qp and $${o \subset L}$$o⊂L its ring of integers viewed as an abelian locally L-analytic group. Let ô 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 ôK has a Picard group Pic(ôK) which is profinite and that the unit section in ôK provides a divisor class of infinite order. In particular, the abelian group Pic(ôK) is not finitely generated and is not a torsion group. On the way we show that ôK is a nontrivial étale 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 K0(ôK) and $${\mathbb{Z} \oplus {Pic}(\hat{o}_K)}$$Z⊕Pic(o^K).