Reduction of certain crystalline representations and local constancy in the weight space
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 25-47.

We study the mod p reduction of crystalline local Galois representations of dimension 2. Berger showed that for a fixed non-zero trace of the Frobenius, the reduction process is locally constant for varying weights under certain conditions. Here we give an estimate of the radius of this local constancy around some special points in the weight space by computing an upper bound for the exponent of p -1 in the radius. Our upper bound turns out to be a linear function of the slope of the crystalline representation under consideration.

Nous étudions la réduction mod p des représentations galoisiennes cristallines de dimension 2. Berger a montré que lorsque la trace de l’endomorphisme de Frobenius est fixée non nulle, la réduction, sous certaines conditions, est localement constante par rapport au poids. Ici, nous donnons une estimation du rayon de constance de la réduction autour de certains points spéciaux dans l’espace de poids en calculant une majoration pour la valuation p-adique du rayon. Notre borne supérieure se révèle être une fonction linéaire de la pente de la représentation cristalline considérée.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1110
Classification: 11F80,  11F70,  13F20
Keywords: Crystalline representations, mod p reductions, local Langlands correspondence
Shalini Bhattacharya 1

1 Indian Institute of Science Education and Research (IISER) Tirupati, Tirupati, Andhra Pradesh, India-517507
@article{JTNB_2020__32_1_25_0,
     author = {Shalini Bhattacharya},
     title = {Reduction of certain crystalline representations and local constancy in the weight space},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {25--47},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {32},
     number = {1},
     year = {2020},
     doi = {10.5802/jtnb.1110},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1110/}
}
TY  - JOUR
TI  - Reduction of certain crystalline representations and local constancy in the weight space
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2020
DA  - 2020///
SP  - 25
EP  - 47
VL  - 32
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1110/
UR  - https://doi.org/10.5802/jtnb.1110
DO  - 10.5802/jtnb.1110
LA  - en
ID  - JTNB_2020__32_1_25_0
ER  - 
%0 Journal Article
%T Reduction of certain crystalline representations and local constancy in the weight space
%J Journal de Théorie des Nombres de Bordeaux
%D 2020
%P 25-47
%V 32
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1110
%R 10.5802/jtnb.1110
%G en
%F JTNB_2020__32_1_25_0
Shalini Bhattacharya. Reduction of certain crystalline representations and local constancy in the weight space. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 25-47. doi : 10.5802/jtnb.1110. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1110/

[1] Bodan Arsovski Reduction modulo p of two-dimensional crystalline representations of G p of slope less than three (2015) (https://arxiv.org/abs/1503.08309)

[2] Laure Barthel; Ron Livné Irreducible modular representations of GL 2 of a local field, Duke Math. J., Volume 75 (1994) no. 2, pp. 261-292 | Article | MR: 1290194 | Zbl: 0826.22019

[3] Laure Barthel; Ron Livné Modular representations of GL 2 of a local field: the ordinary, unramified case, J. Number Theory, Volume 55 (1995) no. 1, pp. 1-27 | Article | MR: 1361556 | Zbl: 0841.11026

[4] Laurent Berger Errata for my articles (perso.ens-lyon.fr/laurent.berger/articles.php)

[5] Laurent Berger Représentations modulaires de GL 2 ( p ) et représentations galoisiennes de dimension 2, Représentations p-adiques de groupes p-adiques II: Représentations de GL 2 ( p ) et (φ,Γ)-modules (Astérisque) Volume 330, Société Mathématique de France, 2010, pp. 263-279 | MR: 2642408 | Zbl: 1233.11060

[6] Laurent Berger Local constancy for the reduction mod p of 2-dimensional crystalline representations, Bull. Lond. Math. Soc., Volume 44 (2012) no. 3, pp. 451-459 | Article | Zbl: 1279.11046

[7] Laurent Berger; Hanfeng Li; Hui June Zhu Construction of some families of 2-dimensional crystalline representations, Math. Ann., Volume 329 (2004) no. 2, pp. 365-377 | Article | MR: 2060368 | Zbl: 1085.11028

[8] Shalini Bhattacharya; Eknath Ghate Reductions of Galois representations for slopes in (1,2), Doc. Math., Volume 20 (2015), pp. 943-987 | MR: 3404215 | Zbl: 1376.11047

[9] Shalini Bhattacharya; Eknath Ghate; Sandra Rozensztajn Reductions of Galois representations of slope 1, J. Algebra, Volume 508 (2018), pp. 98-156 | Article | MR: 3810291 | Zbl: 06890928

[10] Christophe Breuil Sur quelques représentations modulaires et p-adiques de GL 2 ( p ). II, J. Inst. Math. Jussieu, Volume 2 (2003) no. 1, pp. 23-58 | Zbl: 1165.11319

[11] Christophe Breuil Sur quelques représentations modulaires et p-adiques de GL 2 ( p )0. I, Compos. Math., Volume 138 (2003) no. 2, pp. 165-188 | Article | Zbl: 1044.11041

[12] Kevin Buzzard; Toby Gee Explicit reduction modulo p of certain two-dimensional crystalline representations, Int. Math. Res. Not., Volume 2009 (2009) no. 12, pp. 2303-2317 | MR: 2511912 | Zbl: 1189.11054

[13] Kevin Buzzard; Toby Gee Explicit reduction modulo p of certain two-dimensional crystalline representations. II, Bull. Lond. Math. Soc., Volume 45 (2013) no. 4, pp. 779-788 | Article | Zbl: 1305.11046

[14] Pierre Colmez; Jean-Marc Fontaine Construction des représentations p-adiques semi-stables, Invent. Math., Volume 140 (2000) no. 1, pp. 1-43 | Article | Zbl: 1010.14004

[15] Bas Edixhoven The weight in Serre’s conjectures on modular forms, Invent. Math., Volume 109 (1992) no. 3, pp. 563-594 | Article | MR: 1176206 | Zbl: 0777.11013

[16] Abhik Ganguli; Eknath Ghate Reductions of Galois representations via the mod p Local Langlands Correspondence, J. Number Theory, Volume 147 (2015), pp. 250-286 | Article | MR: 3276325 | Zbl: 1383.11075

[17] Eknath Ghate A zigzag conjecture and local constancy for Galois representations (2019) (https://arxiv.org/abs/1903.08996v1)

[18] Eknath Ghate; Vivek Rai Reductions of Galois representations of slope 3/2 (2019) (https://arxiv.org/abs/1901.01728)

[19] D. J. Glover A study of certain modular representations, J. Algebra, Volume 51 (1978), pp. 425-475 | Article | MR: 476841 | Zbl: 0376.20008

[20] George S. Kazandzidis Congruences on binomial coefficients, Bull. Soc. Math. Grèce, N. Ser., Volume 9 (1968) no. 1, pp. 1-12 | MR: 265271 | Zbl: 0179.06601

[21] Sandra Rozensztajn An algorithm for computing the reduction of 2-dimensional crystalline representations of Gal( ¯ p | p ), Int. J. Number Theory, Volume 14 (2018) no. 7, pp. 1857-1894 | Article | MR: 3831397 | Zbl: 06912228

[22] Sandra Rozensztajn On the locus of 2-dimensional crystalline representations with a given reduction modulo p, Algebra Number Theory, Volume 14 (2020) no. 3, pp. 655-720 | MR: 4113777 | Zbl: 07213911

Cited by Sources: