Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 575-607.

We show that the Eigenvariety attached to Hilbert modular forms over a totally real field F is smooth at the points corresponding to certain classical weight one theta series and we give a precise criterion for etaleness over the weight space at those points. In the case where the theta series has real multiplication, we construct a non-classical overconvergent generalised eigenform and compute its Fourier coefficients in terms of p-adic logarithms of algebraic numbers. Our approach uses deformations of Galois representations.

On montre que la variété de Hecke associée aux formes de Hilbert sur un corps totalement réel F est lisse aux points correspondant à certaines séries thêta de poids 1 et on donne aussi un critère pour que le morphisme poids soit étale en ces points. Lorsque les séries thêta sont à multiplication réelle, on construit des formes surconvergentes propres généralisées qui ne sont pas classiques et on exprime leurs coefficients de Fourier à l’aide de logarithmes p-adiques de nombres algébriques. Notre approche utilise la théorie des déformations galoisiennes.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1040
Classification: 11F80,  11F33,  11R23
Keywords: Déformations de représentations galoisiennes p-adiques, familles de Hida de formes de Hilbert et formes modulaires de Hilbert de poids 1.
Adel Betina 1

1 School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom
@article{JTNB_2018__30_2_575_0,
     author = {Adel Betina},
     title = {Les vari\'et\'es de {Hecke{\textendash}Hilbert} aux points classiques de poids parall\`ele 1},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {575--607},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1040},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1040/}
}
TY  - JOUR
TI  - Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2018
DA  - 2018///
SP  - 575
EP  - 607
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1040/
UR  - https://doi.org/10.5802/jtnb.1040
DO  - 10.5802/jtnb.1040
LA  - fr
ID  - JTNB_2018__30_2_575_0
ER  - 
%0 Journal Article
%T Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1
%J Journal de Théorie des Nombres de Bordeaux
%D 2018
%P 575-607
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1040
%R 10.5802/jtnb.1040
%G fr
%F JTNB_2018__30_2_575_0
Adel Betina. Les variétés de Hecke–Hilbert aux points classiques de poids parallèle 1. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 575-607. doi : 10.5802/jtnb.1040. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1040/

[1] Fabrizio Andreatta; Adrian Iovita; Vincent Pilloni On overconvergent Hilbert modular cusp forms, p-adic arithmetic of Hilbert modular forms (Astérisque) Volume 382, Société Mathématique de France, 2016, pp. 163-193 | Zbl: 06720833

[2] Joël Bellaïche; Gaëtan Chenevier Lissité de la courbe de Hecke de GL 2 aux points Eisenstein critiques, J. Inst. Math. Jussieu, Volume 5 (2006) no. 2, pp. 333-349 | Zbl: 1095.11025

[3] Joël Bellaïche; Gaëtan Chenevier Families of Galois representations and Selmer groups, Astérisque, Volume 324, Société Mathématique de France, 2009, xxu+314 pages | Zbl: 1192.11035

[4] Joël Bellaïche; Mladen Dimitrov On the eigencurve at classique weight one points, Duke Math. J., Volume 165 (2016) no. 2, pp. 245-266 | Zbl: 06556667

[5] Stéphane Bijakowski Classicité de formes modulaires de Hilbert, p-adic arithmetic of Hilbert modular forms (Astérisque) Volume 382, Société Mathématique de France, 2016, pp. 49-71 | Zbl: 1353.11003

[6] Armand Brumer On the units of algebraic number fields, Mathematika, Volume 14 (1967), pp. 121-124 | Zbl: 0171.01105

[7] Kevin Buzzard Eigenvarieties, L-functions and Galois representations (Durham, 2004) (London Mathematical Society Lecture Note Series) Volume 320, Cambridge University Press, 2007, pp. 59-120 | Zbl: 1230.11054

[8] Henri Carayol Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Supér., Volume 19 (1986) no. 3, pp. 409-468 | Zbl: 0616.10025

[9] Gaëtan Chenevier Familles p-adiques de formes automorphes pour GL n , J. Reine Angew. Math., Volume 570 (2004), pp. 143-217 | Zbl: 1093.11036

[10] S. Cho; Vinayak Vatsal Deformations of induced Galois representations, J. Reine Angew. Math., Volume 556 (2003), pp. 79-98 | Zbl: 1041.11039

[11] Robert Coleman; Barry Mazur The eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996) (London Mathematical Society Lecture Note Series) Volume 254, Cambridge University Press, 1996, pp. 1-113 | Zbl: 0932.11030

[12] Henri Darmon; Alan Lauder; Victor Rotger Overconvergent generalised eigenforms of weight one and class fields of real quadratic field, Adv. Math., Volume 283 (2015), pp. 130-142 | Zbl: 1393.11038

[13] Henri Darmon; Alan Lauder; Victor Rotger Stark points and p-adic iterated integrals attached to modular forms of weight one, Forum Math. Pi, Volume 3 (2015), e8, 95 pages (Art. ID e8, 95 p.)

[14] Pierre Deligne; Jean-Pierre Serre Formes modulaires de poids 1, Ann. Sci. Éc. Norm. Supér., Volume 7 (1974), pp. 507-530 | Zbl: 0321.10026

[15] Shaunak V. Deo On the eigenvariety of Hilbert modular form at classique parallel weight one point with dihedral projective image, Trans. Am. Math. Soc., Volume 370 (2018) no. 6, pp. 3885-3912 | Article

[16] Hansheng Diao; Ruochuan Liu The eigencurve is proper, Duke Math. J., Volume 165 (2016) no. 7, pp. 1381-1395 | Zbl: 06591243

[17] Mladen Dimitrov; Eknath Ghate On classical weight one forms in Hida families, J. Théor. Nombres Bordx, Volume 24 (2012) no. 3, pp. 669-690 | Zbl: 1271.11060

[18] Mladen Dimitrov; Gabor Wiese Unramifiedness of Galois representations attached to Hilbert modular eigenforms mod p of weight 1 (2018) (à paraître dans J. Inst. Math. Jussieu) | Article

[19] Kazumasa Fujiwara Deformation ring and Hecke algebra in the totally real case (2006) (https://arxiv.org/abs/math/0602606)

[20] Shin Hattori On a properness of the Hilbert eigenvariety at integral weights : the case of quadratic residue fields (2016) (https://arxiv.org/abs/1601.00775)

[21] Haruzo Hida On p-adic Hecke algebras for mathrmGL 2 over totally real fields, Ann. Math., Volume 128 (1988) no. 2, pp. 295-384 | Zbl: 0658.10034

[22] Haruzo Hida Nearly ordinary Hecke algebras and Galois representations of several variables, Algebraic analysis, geometry, and number theory (JAMI, Baltimore, 1988)) (Supplement to the American Journal of Mathematics), Hopkins, 1989, pp. 115-134 | Zbl: 0782.11017

[23] Haruzo Hida On nearly ordinary Hecke algebras for GL(2) over totally real fields, Algebraic number theory (Advanced Studies in Pure Mathematics) Volume 17, Academic Press, 1989, pp. 139-169 | Zbl: 0742.11026

[24] Haruzo Hida; Yoshitaka Maeda Non-abelian base change for totally real fields, Pac. J. Math., Volume Spec. Issue (1998), pp. 189-217 | Zbl: 0942.11026

[25] Frazer Jarvis On Galois representations associated to Hilbert modular forms, J. Reine Angew. Math., Volume 491 (1997), pp. 199-216 | Zbl: 0914.11025

[26] Mark Kisin Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math., Volume 153 (2003) no. 2, pp. 373-454 | Zbl: 1045.11029

[27] Mark Kisin Moduli of finite flat group schemes and modularity, Ann. Math., Volume 170 (2009) no. 3, pp. 1085-1180 | Zbl: 1201.14034

[28] Mark Kisin; King Fai Lai Overconvergent Hilbert modular forms, Am. J. Math., Volume 127 (2005) no. 4, pp. 735-783 | Zbl: 1129.11020

[29] Louise Nyssen Pseudo-représentations, Math. Ann., Volume 306 (1996) no. 2, pp. 257-283 | Zbl: 0863.16012

[30] Masami Ohta Hilbert modular forms of weight one and Galois representations, Automorphic forms of several variables (Katata, 1983) (Progress in Mathematics) Volume 46, Birkhäuser, 1983, pp. 333-352 | Zbl: 0549.12006

[31] Vincent Pilloni; Benoît Stroh Surconvergence et classicité : le cas Hilbert, J. Ramanujan Math. Soc., Volume 32 (2017) no. 4, pp. 355-396

[32] Jonathan D. Rogawski; Jerrold B. Tunnell On Artin L-functions associated to Hilbert modular forms of weight one, Invent. Math., Volume 74 (1983) no. 1, pp. 1-42 | Zbl: 0523.12009

[33] Raphaël Rouquier Caractérisation des caractères et pseudo-caractères, J. Algebra, Volume 180 (1996) no. 2, pp. 571-586 | Zbl: 0857.16013

[34] Richard Taylor On Galois representations associated to Hilbert modular forms, Invent. Math., Volume 98 (1989) no. 2, pp. 265-280 | Zbl: 0705.11031

[35] Andrew John Wiles On ordinary λ-adic representations associated to modular forms, Invent. Math., Volume 94 (1988) no. 3, pp. 529-573 | Zbl: 0664.10013

Cited by Sources: