Determination of certain mod $p$ Galois representations using local constancy
Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 111-161

Let $p \ge 5$ be a prime and $k$ be an integer in $[2p+2, p^2 - p +3]$. Let $b$ be in $[2, p]$ such that $k-2 \equiv b \bmod {p-1}$ and $c := \frac{k-2-b}{p-1}$. We give an explicit radius of local constancy in the weight space of the mod $p$ reduction of the two dimensional crystalline representations $V_{k,a_p}$ of $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, where the slope $\nu (a_p)$ is constrained to be in $(1, c)$ and non-integral. We use the mod $p$ local Langlands correspondence for $\mathrm{GL}_{2} (\mathbb{Q}_{p})$ to compute the mod $p$ reductions explicitly under additional conditions on the slope. We show that the reduction in the disk depends only on $k$ and $\lfloor \nu (a_p)\rfloor $. As an application, we obtain explicit mod $p$ reductions at many new values of $k$ and $a_p$. We also obtain an explicit radius of local constancy in the $a_p$ space (for a fixed $k$ as above) which is bigger than the explicit radius given in a result of Berger.

Soit $p \ge 5$ un nombre premier et $k$ un entier dans $[2p+2,p^{2}-p+3]$. Soit $b$ dans $[2,p]$ tel que $k-2 \equiv b \bmod {p-1}$ et $c := \frac{k-2-b}{p-1}$. Nous donnons un rayon explicite de constance locale dans l’espace des poids de la réduction modulo $p$ des représentations cristallines de dimension deux $V_{k,a_p}$ de $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, où la pente $\nu (a_p)$ est supposée dans $(1,c)$ et non entière. Nous utilisons la correspondance de Langlands locale modulo $p$ pour $\mathrm{GL}_{2} (\mathbb{Q}_{p})$ afin de calculer explicitement les réductions modulo $p$ sous des conditions additionnelles sur la pente. Nous montrons que la réduction dans le disque dépend uniquement de $k$ et $\lfloor \nu (a_p)\rfloor $. En application, nous obtenons des réductions modulo $p$ explicites pour de nombreuses nouvelles valeurs de $k$ et $a_p$. Nous obtenons également un rayon explicite de constance locale dans l’espace $a_p$ (pour un $k$ fixé comme ci-dessus) qui est plus grand que le rayon explicite donné dans un résultat de Berger.

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DOI : 10.5802/jtnb.1357
Classification : 11F80, 11F70, 11F33
Keywords: Reduction of crystalline representations of $\mathrm{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$, Local constancy, mod $p$ local Langlands correspondence for $\mathrm{GL}_{2} (\mathbb{Q}_{p})$
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Abhik Ganguli; Suneel Kumar. Determination of certain mod $p$ Galois representations using local constancy. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 1, pp. 111-161. doi: 10.5802/jtnb.1357
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