Picard groups in p-adic Fourier theory

Abstract

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