Toric regulators

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.