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

Sandra Rozensztajn

doi:10.5802/jtnb.875

Asymptotic values of modular multiplicities for

{GL}_{2}

Sandra Rozensztajn ¹

¹ UMPA, ENS de Lyon UMR 5669 du CNRS 46, allée d’Italie 69364 Lyon Cedex 07 France

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

Abstract (VO)
Résumé

We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . 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.

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 ${Res}_{K / ℚ_{p}} {GL}_{2}$ pour une extension finie $K$ de $ℚ_{p}$ . 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.

MR

DOI : 10.5802/jtnb.875

Affiliations des auteurs :

Sandra Rozensztajn ¹

¹ UMPA, ENS de Lyon UMR 5669 du CNRS 46, allée d’Italie 69364 Lyon Cedex 07 France

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     author = {Sandra Rozensztajn},
     title = {Asymptotic values of modular multiplicities for $\operatorname{GL}_2$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {465--482},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.875},
     mrnumber = {3320488},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.875/}
}

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Sandra Rozensztajn. Asymptotic values of modular multiplicities for $\operatorname{GL}_2$. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 465-482. doi : 10.5802/jtnb.875. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.875/

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