Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915671178215424 |
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| author | Roset, Martí Xu, Peter |
| author_facet | Roset, Martí Xu, Peter |
| 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 $\mathbb{Q}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_11514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$ Roset, Martí Xu, Peter Number Theory 11F75, 11G15, 11S40, 11S80 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 $\mathbb{Q}_p$. |
| title | Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$ |
| topic | Number Theory 11F75, 11G15, 11S40, 11S80 |
| url | https://arxiv.org/abs/2512.11514 |