# Combinatorics of Serre weights in the potentially Barsotti-Tate setting

2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : Let $F$ be a finite unramified extension of $\mathbb{Q}_p$ and $\bar\rho$ be a absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let also $t$ be a tame inertial type of $F$. We relate the Kisin variety associated to these data to the set of Serre weights $\mathcal{D}(t,\bar\rho) = \mathcal{D}(t) \cap \mathcal{D}(\bar\rho)$. We prove that the Kisin variety enriched with its canonical embedding into $(\mathbb{P}^1)^f$ and its shape stratification are enough to determine the cardinality of $\mathcal{D}(t,\bar\rho)$. Moreover, we prove that this dependance is nondecreasing (the smaller is the Kisin variety, the smaller is the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights). These results provide new evidences towards the conjectures in our previous paper.
Document type :
Preprints, Working Papers, ...

https://hal-cnrs.archives-ouvertes.fr/hal-03221168
Contributor : Xavier Caruso <>
Submitted on : Monday, May 10, 2021 - 9:12:14 AM
Last modification on : Wednesday, May 12, 2021 - 3:47:55 PM

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combiSerre.pdf
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### Identifiers

• HAL Id : hal-03221168, version 2
• ARXIV : 2105.04147

### Citation

Xavier Caruso, Agnès David, Ariane Mézard. Combinatorics of Serre weights in the potentially Barsotti-Tate setting. 2021. ⟨hal-03221168v2⟩

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