Geodesic
Potentially diagonalisable lifts with controlled Hodge-Tate weights
Documenta mathematica, Tome 26 (2021), pp. 795-827
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Motivated by the weight part of Serre's conjecture we consider the following question. Let K/Qp​ be a finite extension and suppose ρ​:GK​→GLn​(Fp​) admits a crystalline lift with Hodge-Tate weights contained in the range [0,p]. Does ρ​ admit a potentially diagonalisable crystalline lift of the same Hodge-Tate weights? We answer this question in the affirmative when K=Qp​ and n≤5, and ρ​ satisfies a mild 'cyclotomic-free' condition. We also prove partial results when K/Qp​ is unramified and n is arbitrary.
DOI : 10.4171/dm/830
Classification : 11F33, 11F80
Mots-clés : congruences between crystalline representations, integral p-adic Hodge theory, breuil-kisin modules 
@article{10_4171_dm_830,
     author = {Robin Bartlett},
     title = {Potentially diagonalisable lifts with controlled {Hodge-Tate} weights},
     journal = {Documenta mathematica},
     pages = {795--827},
     year = {2021},
     volume = {26},
     doi = {10.4171/dm/830},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/830/}
}
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Robin Bartlett. Potentially diagonalisable lifts with controlled Hodge-Tate weights. Documenta mathematica, Tome 26 (2021), pp. 795-827. doi: 10.4171/dm/830

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