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Abstract We determine the mod $$p$$ cohomological invariants for several affine group schemes $$G$$ in characteristic $$p$$. These are invariants of $$G$$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $$p$$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $$p$$ Milnor K-theory of fields.
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Diagrams in the mod p cohomology of Shimura curves
Abstract We prove a local–global compatibility result in the mod $$p$$ Langlands program for $$\mathrm {GL}_2(\mathbf {Q}_{p^f})$$ . Namely, given a global residual representation $$\bar {r}$$ appearing in the mod $$p$$ cohomology of a Shimura curve that is sufficiently generic at $$p$$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $$p$$ completed cohomology is determined by the restrictions of $$\bar {r}$$ to decomposition groups at $$p$$ . If these restrictions are moreover semisimple, we show that the $$(\varphi ,\Gamma )$$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $$\bar {r}$$ to decomposition groups at $$p$$ .
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- Award ID(s):
- 1703182
- PAR ID:
- 10294554
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 157
- Issue:
- 8
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 1653 to 1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X21007375
