Log \(p\)-divisible groups and semi-stable representations

Let \(\mathscr{O}_K\) be a henselian DVR with field of fractions \(K\) and residue field of characteristic \(p>0\). Let \(S\) denote \(\mathop{\mathrm{Spec}} \mathscr{O}_K\) endowed with the canonical log structure. We show that the generic fiber functor \(\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\to...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: Bertapelle, Alessandra, Wang, Shanwen, Heer Zhao
Format: Article
Language:English
Subjects:
Equivalence
Reduction
Representations
Online Access:Get full text
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Summary:Let \(\mathscr{O}_K\) be a henselian DVR with field of fractions \(K\) and residue field of characteristic \(p>0\). Let \(S\) denote \(\mathop{\mathrm{Spec}} \mathscr{O}_K\) endowed with the canonical log structure. We show that the generic fiber functor \(\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\to \mathbf{BT}^{\mathrm{st}}_K\) between the category of dual representable log \(p\)-divisible groups over \(S\) and the category of \(p\)-divisible groups with semistable reduction over \(K\) is an equivalence. If \(\mathscr{O}_K\) is further complete and of mixed characteristic, we show that \(\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\) is also equivalent to the category of semistable Galois \(\mathbb{Z}_p\)-representations with Hodge-Tate weights in \(\{0,1\}\). Finally, we show that the above equivalences respect monodromies.
ISSN:2331-8422