Serre weights and the Breuil–Mézard conjecture for modular forms
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 107-124.

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 k2 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 k2 tel que ρ possède un relèvement cristallin de type de Hodge–Tate (0,k-1).

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1154
Classification: 11F80, 20C20
Keywords: Galois representations, Serre’s modularity conjecture, Breuil–Mézard conjecture

Hanneke Wiersema 1

1 King’s College London, Strand London, WC2R 2LS, United Kingdom
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JTNB_2021__33_1_107_0,
     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},
     number = {1},
     year = {2021},
     doi = {10.5802/jtnb.1154},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/}
}
TY  - JOUR
AU  - Hanneke Wiersema
TI  - Serre weights and the Breuil–Mézard conjecture for modular forms
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 107
EP  - 124
VL  - 33
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/
DO  - 10.5802/jtnb.1154
LA  - en
ID  - JTNB_2021__33_1_107_0
ER  - 
%0 Journal Article
%A Hanneke Wiersema
%T Serre weights and the Breuil–Mézard conjecture for modular forms
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 107-124
%V 33
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/
%R 10.5802/jtnb.1154
%G en
%F JTNB_2021__33_1_107_0
Hanneke Wiersema. Serre weights and the Breuil–Mézard conjecture for modular forms. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 107-124. doi : 10.5802/jtnb.1154. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1154/

[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 (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 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 (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 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)

Cited by Sources: