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

TL;DR

This paper introduces log-rigid analytic classes for $ ext{SL}_n( ext{Z})$ derived from topological Eisenstein classes of torus bundles, linking them to $p$-adic logarithms of Gross--Stark units and providing evidence through $p$-adic $L$-functions.

Contribution

It defines new log-rigid analytic classes for $ ext{SL}_n( ext{Z})$ and connects them to Gross--Stark units, extending the understanding of $p$-adic properties of these units.

Findings

Conjecture that these classes evaluate to $p$-adic logarithms of Gross--Stark units.

Proven cases where the totally real field is Galois over $ ext{Q}$.

Comparison with $p$-adic $L$-functions supports the conjecture.

Abstract

Starting from a topological treatment of the Eisenstein class of a torus bundle, we define log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$. These are group cohomology classes for $\mathrm{SL}_n(\mathbb{Z})$ valued on log-rigid analytic functions on Drinfeld's $p$-adic symmetric domain. Such classes can be evaluated at points attached to totally real fields of degree $n$ where $p$ is inert. We conjecture that these values are $p$-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 $p$-adic $L$-functions. In addition, we prove it in certain situations where the totally real field is Galois over $\mathbb{Q}$, as a consequence of the fact that in this case there is a conjugate of a Gross--Stark unit in $\mathbb{Q}_p$.