Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$

Martí Roset; Peter Xu

Paper

Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$

Abstract

Starting from a topological treatment of the Eisenstein class of a torus bundle, we define log-rigid analytic classes for

. These are group cohomology classes for

valued on log-rigid analytic functions on Drinfeld's

-adic symmetric domain. Such classes can be evaluated at points attached to totally real fields of degree

where

is inert. We conjecture that these values are

-adic logarithms of Gross--Stark units in the narrow Hilbert class field of totally real fields. We provide evidence for the conjecture by comparing our constructions to

-adic

-functions. In addition, we prove it in certain situations where the totally real field is Galois over

, as a consequence of the fact that in this case there is a conjugate of a Gross--Stark unit in