Asymptotic values of modular multiplicities for ${GL}_2$

Rozensztajn, Sandra

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 465-482.

Valeurs asymptotiques de multiplicités modulaires pour ${GL}_2$.

Nous étudions les constituants irréductibles de la réduction modulo $p$ d’une représentation algébrique irréductible $V$ du groupe ${Res}_{K/{\mathbb{Q}}_p}{GL}_2$ pour une extension finie $K$ de ${\mathbb{Q}}_p$. Nous montrons qu’asymptotiquement, la multiplicité de chaque constituant ne dépend que de la dimension de $V$ et du caractère central de sa réduction modulo $p$. Nous appliquons ce résultat au calcul de la valeur asymptotique de multiplicités qui sont l’objet de la conjecture de Breuil-Mézard.

We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K/{\mathbb{Q}}_p}{GL}_2$ for $K$ a finite extension of ${\mathbb{Q}}_p$. We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$. As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

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