On the kernel and the image of the rigid analytic regulator in positive characteristic

TL;DR

This paper establishes a reciprocity law connecting residues of differential symbols in K_2 of Mumford curves to the rigid analytic regulator, with implications for understanding its kernel and image in positive characteristic.

Contribution

It formulates and proves a new reciprocity law linking residues of differential symbols to the rigid analytic regulator, extending classical conjectures to positive characteristic.

Findings

Proves a reciprocity law relating residues of dlog^2 to the rigid analytic regulator.

Deduces consequences for the kernel and image of the regulator, analogous to Beilinson and Bloch conjectures.

Connects the construction to Contou-Carrere's symbol, Kato's residue, and Weil's reciprocity law.

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-Carrere and to Kato's residue homomorphism, and we show that Weil's reciprocity law directly implies the reciprocity law of Anderson and Romo.