This article deals with the Galois representation attached to the torsion points of an elliptic curve defined over a number field. We first determine explicit uniform criteria for the irreducibility of this Galois representation for elliptic curves varying in some infinite families, characterised by their reduction type at some fixed places of the base field. Then, we deduce from these criteria an explicit form for a bound that appear in a theorem of Momose. Finally, we use these results to precise a previous theorem of the author about the homotheties contained in the image of the Galois representation.

Let $F$ be a unramified finite extension of $\mathbb{Q}_{p}$ and $\overline{\unicode[STIX]{x1D70C}}$ be an irreducible mod $p$ two-dimensional representation of the absolute Galois group of $F$. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil–Kisin modules associated to certain families of potentially Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$. We prove that this variety is a finite union of products of $\mathbb{P}^{1}$. Moreover, it appears as an explicit closed connected subvariety of $(\mathbb{P}^{1})^{[F:\mathbb{Q}_{p}]}$. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped with its stratification may help in determining the ring of Barsotti–Tate deformations of $\overline{\unicode[STIX]{x1D70C}}$.

Let F be a finite unramified extension of Qp and ρ 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 D(t, ρ) = D(t) ∩ D(ρ). We prove that the Kisin variety enriched with its canonical embedding into (P 1 ) f and its shape stratification are enough to determine the cardinality of D(t, ρ). 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 of [CDM2].

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