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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.11514 |
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Table of Contents:
- 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 $\mathbb{Q}_p$.