Abstract

Let $F$ be a finite unramified extension of ${\mathbb{Q}}_p$ of degree $f$ and $\overline{\rho }$ be an absolutely irreducible mod $p$ $2$-dimensional representation of the absolute Galois group of $F$. Let t be a tame inertial type of level $f$ of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti–Tate liftings of $\overline{\rho }$ having type t depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into ${({\mathbb{P}}^1)}^f$ and its shape stratification. We give evidence towards this conjecture by proving that the Kisin variety determines the cardinality of the set of common Serre weights $\mathcal{𝒟}(t<!--FUNCTION\; APPLICATION-->,\overline{\rho })=\mathcal{𝒟}(t<!--FUNCTION\; APPLICATION-->)\cap \mathcal{𝒟}(\overline{\rho })$. Additionally, we prove that this dependence is nondecreasing (the smaller the Kisin variety, the smaller the number of common Serre weights) and compatible with products (if the Kisin variety splits as a product, so does the number of weights).

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