Abstract
For an arbitrary non-archimedean local field we classify reductive group schemes over the corresponding Fargues–Fontaine curve by group schemes over the category of isocrystals. We then classify torsors under such reductive group schemes by a generalization of Kottwitz’ set B(G). In particular, we extend a theorem of Fargues on torsors under constant reductive groups to the case of equal characteristic.
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Notes
Later we call this set \(B({\mathbb {G}})\).
which is not a restriction as fiber functors take their image in vector bundles.
In [17] Saavedra–Rivano calls filtered fiber functors admissible if they are fpqc-locally splittable. By [19] this notion is obsolete and thus we think that our terminology is not very confusing. Also our admissible fiber functors are not equipped with a filtration as would be the case in Saavedra–Rivano’s notation.
We mention the following possible cause of confusion. If G is a reductive group over E with associated group scheme \({\mathbb {G}}\) over \(\varphi -\mathrm {Mod}_L\), then the category of finite dimensional E-representations \(\mathrm {Rep}_E(G)\) of G is the full subcategory of \(\mathrm {Rep}_E({\mathbb {G}})\) given by representations of \({\mathbb {G}}\) whose underlying isocrystal is semistable of slope 0.
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Acknowledgements
As the reader will have noticed this article owes a lot to Laurent Fargues’ work in [6]. Therefore the author wants to thank Laurent Fargues heartily. The author also thanks Jochen Heinloth, Michael Rapoport, Peter Scholze and Torsten Wedhorn for answering several questions related to this paper. Especially, the hint of Michael Rapoport to [4, Theorem 5.3.1.] lead to the full proof of Theorem 10 in the equal characteristic case.
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Communicated by Toby Gee.
