Bartlett, Robin

Potentially diagonalisable lifts with controlled Hodge-Tate weights

Doc. Math. 26, 795-827 (2021)
DOI: 10.25537/dm.2021v26.795-827
Communicated by Don Blasius

Summary

Motivated by the weight part of Serre's conjecture we consider the following question. Let \(K/\mathbb{Q}_p\) be a finite extension and suppose \(\overline{\rho} : G_K \rightarrow \operatorname{GL}_n(\overline{\mathbb{F}}_p)\) admits a crystalline lift with Hodge-Tate weights contained in the range \([0,p]\). Does \(\overline{\rho}\) admit a potentially diagonalisable crystalline lift of the same Hodge-Tate weights? We answer this question in the affirmative when \(K = \mathbb{Q}_p\) and \(n \leq 5\), and \(\overline{\rho}\) satisfies a mild 'cyclotomic-free' condition. We also prove partial results when \(K/\mathbb{Q}_p\) is unramified and \(n\) is arbitrary.

Mathematics Subject Classification

11F80, 11F33

Keywords/Phrases

congruences between crystalline representations, integral \(p\)-adic Hodge theory, breuil-kisin modules

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Affiliation

Bartlett, Robin
Faculty of Mathematics and Computer Science, University Münster, Einsteinstraße 62, 48149 Münster, Germany

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