On the Kernel and the Image of the Rigid Analytic Regulator in Positive Characteristic

Abstract

We will formulate and prove a certain reciprocity law relating certain residues of the differential symbol _dlog_2 from the _K_2 of a Mumford curve to the rigid analytic regulator constructed by the author in a previous paper. We will use this result to deduce some consequences on the kernel and image of the rigid analytic regulator analogous to some old conjectures of Beilinson and Bloch on the complex analytic regulator. We also relate our construction to the symbol defined by Contou-Carrère and to Kato’s residue homomorphism, and we show that Weil’s reciprocity law directly implies the reciprocity law of Anderson and Romo.