Reduction of certain crystalline representations and local constancy in the weight space

Bhattacharya, Shalini

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 1, pp. 25-47.

Nous étudions la réduction mod $p$ des représentations galoisiennes cristallines de dimension 2. Berger a montré que lorsque la trace de l’endomorphisme de Frobenius est fixée non nulle, la réduction, sous certaines conditions, est localement constante par rapport au poids. Ici, nous donnons une estimation du rayon de constance de la réduction autour de certains points spéciaux dans l’espace de poids en calculant une majoration pour la valuation $p$-adique du rayon. Notre borne supérieure se révèle être une fonction linéaire de la pente de la représentation cristalline considérée.

We study the mod $p$ reduction of crystalline local Galois representations of dimension 2. Berger showed that for a fixed non-zero trace of the Frobenius, the reduction process is locally constant for varying weights under certain conditions. Here we give an estimate of the radius of this local constancy around some special points in the weight space by computing an upper bound for the exponent of $p^{-1}$ in the radius. Our upper bound turns out to be a linear function of the slope of the crystalline representation under consideration.

Classification : 11F80, 11F70, 13F20
Mots-clés : Crystalline representations, mod $p$ reductions, local Langlands correspondence

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