Serre weights and the Breuil–Mézard conjecture for modular forms

Hanneke Wiersema

doi:10.5802/jtnb.1154

Hanneke Wiersema ¹

¹ King’s College London, Strand London, WC2R 2LS, United Kingdom

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 107-124.

Abstract (VO)
Résumé

Serre’s strong conjecture, now a theorem of Khare and Wintenberger, states that every two-dimensional continuous, odd, irreducible mod $p$ Galois representation $ρ$ arises from a modular form of a specific minimal weight $k (ρ)$ , level $N (ρ)$ and character $ϵ (ρ)$ . In this short paper we show that the minimal weight $k (ρ)$ is equal to a notion of minimal weight inspired by the recipe for weights introduced by Buzzard, Diamond and Jarvis in [4]. Moreover, using the Breuil–Mézard conjecture we show that both weight recipes are equal to the smallest $k \geq 2$ such that $ρ$ has a crystalline lift of Hodge–Tate type $(0, k - 1)$ .

La forme forte de la conjecture de Serre, démontrée par Khare et Wintenberger, assure que toute représentation galoisienne $ρ$ modulo $p$ , de dimension $2$ , continue, irréductible et impaire provient d’une forme modulaire de poids minimal $k (ρ)$ , de niveau $N (ρ)$ et de caractère $ϵ (ρ)$ prescrits. Dans ce court article, nous démontrons que le poids minimal $k (ρ)$ coïncide avec une autre notion de poids minimal, qui est inspirée par la recette pour les poids de $ρ$ introduite par Buzzard, Diamond et Jarvis dans [4]. De plus, en utilisant la conjecture de Breuil–Mézard, nous démontrons que le poids défini par ces recettes équivalentes est égal au plus petit entier $k \geq 2$ tel que $ρ$ possède un relèvement cristallin de type de Hodge–Tate $(0, k - 1)$ .

Reçu le : 2020-04-03
Accepté le : 2020-12-01
Publié le : 2021-05-21

DOI : 10.5802/jtnb.1154

Classification : 11F80, 20C20
Mots-clés : Galois representations, Serre’s modularity conjecture, Breuil–Mézard conjecture

Affiliations des auteurs :

Hanneke Wiersema ¹

¹ King’s College London, Strand London, WC2R 2LS, United Kingdom

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Hanneke Wiersema},
     title = {Serre weights and the {Breuil{\textendash}M\'ezard} conjecture for modular forms},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {107--124},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
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     year = {2021},
     doi = {10.5802/jtnb.1154},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/}
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Hanneke Wiersema. Serre weights and the Breuil–Mézard conjecture for modular forms. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 107-124. doi : 10.5802/jtnb.1154. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/

Bibliographie
Cité par

[1] Avner Ash; Glenn Stevens Modular forms in characteristic $l$ and special values of their $L$ -functions, Duke Math. J., Volume 53 (1986) no. 3, pp. 849-868 | DOI | MR | Zbl

[2] Laurent Berger An introduction to the theory of p-adic representations, Geometric Aspects of Dwork Theory. Vol. I, Walter de Gruyter, 2004, pp. 255-292 | DOI | MR | Zbl

[3] Christophe Breuil; Ariane Mézard Multiplicités modulaires et représentations de ${GL}_{2} (Z_{p})$ et de $Gal ({\bar{Q}}_{p} / Q_{p})$ en $l = p$ , Duke Math. J., Volume 115 (2002) no. 2, pp. 205-310 (With an appendix by Guy Henniart) | DOI | MR | Zbl

[4] Kevin Buzzard; Fred Diamond; Frazer Jarvis On Serre’s conjecture for mod $ℓ$ Galois representations over totally real fields, Duke Math. J., Volume 155 (2010) no. 1, pp. 105-161 | DOI | MR | Zbl

[5] Fred Diamond A correspondence between representations of local Galois groups and Lie-type groups, $L$ -functions and Galois representations (London Mathematical Society Lecture Note Series), Volume 320, Cambridge University Press, 2007, pp. 187-206 | DOI | MR | Zbl

[6] Fred Diamond; Davide A. Reduzzi Crystalline lifts of two-dimensional mod $p$ automorphic Galois representations, Math. Res. Lett., Volume 25 (2015) no. 1, pp. 43-73 | DOI | MR | Zbl

[7] Bas Edixhoven The weight in Serre’s conjectures on modular forms., Invent. Math., Volume 109 (1992) no. 3, pp. 563-594 | DOI | MR | Zbl

[8] Toby Gee Automorphic lifts of prescribed types, Math. Ann., Volume 350 (2011) no. 1, pp. 107-144 | DOI | MR | Zbl

[9] Toby Gee; Florian Herzig; David Savitt General Serre weight conjectures, J. Eur. Math. Soc., Volume 20 (2018) no. 12, pp. 2859-2949 | DOI | MR | Zbl

[10] Toby Gee; Mark Kisin The Breuil–Mézard conjecture for potentially Barsotti–Tate representations, Forum Math. Pi, Volume 2 (2014), e1, 56 pages | DOI | MR | Zbl

[11] Toby Gee; Tong Liu; David Savitt The Buzzard–Diamond–Jarvis conjecture for unitary groups, J. Am. Math. Soc., Volume 27 (2014) no. 2, pp. 389-435 | DOI | MR | Zbl

[12] Yongquan Hu; Fucheng Tan The Breuil–Mézard conjecture for non-scalar split residual representations, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 6, pp. 1383-1421 | DOI | MR | Zbl

[13] Chandrashekhar Khare; Jean-Pierre Wintenberger Serreś modularity conjecture (II), Invent. Math., Volume 178 (2009) no. 3, p. 505 | DOI | MR | Zbl

[14] Mark Kisin The Fontaine–Mazur conjecture for ${GL}_{2}$ , J. Am. Math. Soc., Volume 22 (2009) no. 3, pp. 641-690 | DOI | MR | Zbl

[15] James Newton Serre weights and Shimura curves, Proc. Lond. Math. Soc., Volume 108 (2014) no. 6, pp. 1471-1500 | DOI | MR | Zbl

[16] Vytautas Paškūnas On the Breuil–Mézard conjecture, Duke Math. J., Volume 164 (2015) no. 2, pp. 297-359 | DOI | MR | Zbl

[17] Fabian Sander Hilbert–Samuel multiplicities of certain deformation rings, Math. Res. Lett., Volume 21 (2014) no. 3, pp. 605-615 | DOI | MR | Zbl

[18] Jean-Pierre Serre Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1971) no. 4, pp. 259-331 | DOI | Zbl

[19] Jean-Pierre Serre Linear representations of finite groups, Graduate Texts in Mathematics, 42, Springer, 1977, 172 pages (translated from the second French edition by Leonard L. Scott) | MR | Zbl

[20] Jean-Pierre Serre Sur les représentations modulaires de degré $2$ de $Gal (\bar{Q} / Q)$ , Duke Math. J., Volume 54 (1987) no. 1, pp. 179-230 | DOI | MR | Zbl

[21] Jean-Pierre Serre Lettre à Mme Hamer, 2 Juillet 2001

[22] Shen-Ning Tung On the automorphy of 2-dimensional potentially semi-stable deformation rings of $G_{ℚ_{p}}$ (2018) (https://arxiv.org/abs/1803.07451)

[23] Shen-Ning Tung On the modularity of 2-adic potentially semi-stable deformation rings (2019) (https://arxiv.org/abs/1908.06174)

Cité par Sources :