Asymptotic values of modular multiplicities for GL 2
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/ p GL 2 pour une extension finie K de 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/ p GL 2 for K a finite extension of 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.

Reçu le : 2012-12-26
Accepté le : 2013-02-18
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.875
@article{JTNB_2014__26_2_465_0,
     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 = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_465_0/}
}
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/item/JTNB_2014__26_2_465_0/

[BDJ10] K. Buzzard, F. Diamond, and F. Jarvis, On Serre’s conjecture for mod Galois representations over totally real fields, Duke Math. J., 155, 1 (2010), 105–161. | MR 2730374 | Zbl 1227.11070

[BL94] L. Barthel and R. Livné, Irreducible modular representations of GL 2 of a local field, Duke Math. J., 75, 2 (1994), 261–292. | MR 1290194 | Zbl 0826.22019

[BLGG] T. Barnet-Lamb, T. Gee, and D. Geraghty, Serre weights for rank two unitary groups, to appear in Math Ann. | MR 3072811

[BM02] C. Breuil and A. Mézard, Multiplicités modulaires et représentations de GL 2 (Z p ) et de Gal (Q ¯ p /Q p ) en l=p, Duke Math. J., 115(2), (2002), 205–310. With an appendix by Guy Henniart. | MR 1944572 | Zbl 1042.11030

[BM12] C. Breuil and A. Mézard, Multiplicités modulaires raffinées, Bull. Soc. Math. France 142, (2014), 127–175. | MR 3248725

[Bon11] C. Bonnafé, Representations of SL 2 (𝔽 q ), Algebra and Applications, 13, (2011), Springer-Verlag London Ltd., London. | MR 2732651 | Zbl 1203.22001

[BP12] C. Breuil and V. Paškūnas, Towards a modulo p Langlands correspondence for GL 2 , Mem. Amer. Math. Soc., 216(1016):vi+114, (2012). | MR 2931521 | Zbl 1245.22010

[Dav13] A. David, Calculs de multiplicités dans la conjecture de Breuil-Mézard, preprint, (2013).

[Dia07] F. Diamond, A correspondence between representations of local Galois groups and Lie-type groups, L-functions and Galois representations, London Math. Soc. Lecture Note Ser. 320 (2007), 187–206. Cambridge Univ. Press, Cambridge. | MR 2392355 | Zbl 1230.11069

[EG11] M. Emerton and T. Gee, A geometric perspective on the Breuil-Mézard conjecture 09, (2011).

[GK12] T. Gee and M. Kisin, The Breuil-Mézard conjecture for potentially Barsotti-Tate representations, preprint (2012). | MR 3292675

[Glo78] D.J. Glover, A study of certain modular representations, J. Algebra, 51(2), (1978), 425–475. | MR 476841 | Zbl 0376.20008

[Kis08] M. Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc., 21(2), (2008), 513–546. | MR 2373358 | Zbl 1205.11060

[Kis09] M. Kisin, The Fontaine-Mazur conjecture for GL 2 , J. Amer. Math. Soc., 22(3), (2009), 641–690. | MR 2505297 | Zbl 1251.11045

[Kis10] M. Kisin, The structure of potentially semi-stable deformation rings, Proceedings of the International Congress of Mathematicians, New Delhi, Hindustan Book Agency II, (2010), 294–311. | MR 2827797 | Zbl 1273.11090

[Paš12] V. Paškūnas, On the Breuil-Mézard conjecture, preprint (2012). | MR 3306557

[Sch08] M.M. Schein, Weights in Serre’s conjecture for Hilbert modular forms, the ramified case, Israel J. Math., 166, (2008), 369–391. | MR 2430440 | Zbl 1197.11063