Toric regulators

TL;DR

This paper introduces the toric regulator, a new map from motivic cohomology to p-adic tori for varieties with totally degenerate reduction, linking étale regulators and Galois cohomology in mixed characteristic.

Contribution

It generalizes previous work by defining the toric regulator for a broader class of varieties and explores its relation to log-syntomic regulators and specific examples.

Findings

Relation between the toric regulator and rigid analytic regulator for Mumford curves

Conjectured formula for the toric regulator on products of Mumford curves

Connection of the toric regulator with étale regulators and Galois cohomology

Abstract

Let $X$ be a variety defined over a local field $K$ of mixed characteristic $(0,p)$ with a totally degenerate reduction in the sense of Raskind and Xarles. Generalizing earlier work of Raskind and Xarles and relying on some conjectures we define a map, which we call the toric regulator, from the various motivic cohomology groups of $X$ to certain $p$-adically uniformized tori over $K$. This construction captures the part of the \'etale regulators on $X$ that land in the Galois cohomology of the submodules of cohomology which are extensions of $\mathbb{Z}_l$ by $\mathbb{Z}_l(1)$, simultaneously for all $l$. We also discuss the relation with the log-syntomic regulator and study a number of examples. In particular, for $K_2$ of a Mumford curve we find a relation with the rigid analytic regulator of \'Pal and for $K_1$ of the product of Mumford curves we conjecture a formula for the toric regulator.