Local Indecomposability of Hilbert Modular Galois Representations  [ Indécomposabilité locale des représentations modulaires galoisiennes de Hilbert ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1521-1560.

Nous prouvons l’indécomposabilité de la représentation galoisienne restreinte au groupe de p-décomposition attaché à une forme modulaire quasi-ordinaire de Hilbert sans multiplication complexe de poids 2 sous certainess hypothèses.

We prove the indecomposability of the Galois representation restricted to the p-decomposition group attached to a non CM nearly p-ordinary weight two Hilbert modular form over a totally real field F under the assumption that either the degree of F over is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of F.

DOI : https://doi.org/10.5802/aif.2889
Classification : 11F80,  11G18,  14K22
Mots clés : Représentation galoisienne, formes modulaires de Hilbert, multiplication complexe
@article{AIF_2014__64_4_1521_0,
     author = {Zhao, Bin},
     title = {Local Indecomposability of Hilbert Modular Galois Representations},
     journal = {Annales de l'Institut Fourier},
     pages = {1521--1560},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     doi = {10.5802/aif.2889},
     mrnumber = {3329672},
     zbl = {06387316},
     language = {en},
     url = {www.numdam.org/item/AIF_2014__64_4_1521_0/}
}
Zhao, Bin. Local Indecomposability of Hilbert Modular Galois Representations. Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1521-1560. doi : 10.5802/aif.2889. http://www.numdam.org/item/AIF_2014__64_4_1521_0/

[1] Balasubramanyam, B.; Ghate, E.; Vatsal, V. On local Galois representations associated to ordinary Hilbert modular forms (Preprint)

[2] Brakočević, M. Anticyclotomic p-adic L-function of central critical Rankin-Selberg L-value (IMRN. doi:10.1093/imrn/rnq275)

[3] Carayol, H. Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4), Volume 19 (1986), pp. 409-468 | Numdam | MR 870690 | Zbl 0616.10025

[4] Coleman, R. Classical and overconvergent modular forms, Invent. Math., Volume 124 (1996) no. 1-3, pp. 215-241 | Article | MR 1369416 | Zbl 0851.11030

[5] Conrad, B.; Chai, C.-L.; Oort, F. CM Liftings, 2011

[6] de Jong, A.J.; Noot, R.; G. van der Geer, F. Oort and J.Steenbrink Jacobians with complex multiplication, Arithmetic Algebraic Geometry (Progress in Math.) Volume 89, Brikhäuser, Boston, 1991, pp. 177-192 | Zbl 0732.14014

[7] Deligne, P.; Pappas, G. Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compositio Math., Volume 90 (1994), pp. 59-79 | Numdam | MR 1266495 | Zbl 0826.14027

[8] Emerton, M. A p-adic variational Hodge conjecture and modular forms with complex multiplication (preprint available at Emerton’s homepage: http://www.math.uchicago.edu/~emerton/pdffiles/cm.pdf)

[9] Faltings, G.; G. Cornell and J. Silverman Finiteness theorems for abelian varieties over number fields, Arithmetic geometry, Springer-Verlag, New York, 1986, pp. 9-27 | MR 861971 | Zbl 0602.14044

[10] Ghate, E. Ordinary forms and their local Galois representations, Algebra and number theory, Hindustan Book Agency, Delhi, 2005, pp. 226-242 | MR 2193355 | Zbl 1085.11029

[11] Ghate, E.; Vatsal, V. On the local behaviour of ordinary Λ-adic representations, Ann. Inst. Fourier (Grenoble), Volume 54 (2004), pp. 2143-2162 | Article | Numdam | MR 2139691 | Zbl 1131.11341

[12] Goren, E. Lectures on Hilbert Modular Varieties and Modular Forms, CRM monograph series, American Mathematical Soc., 2001 no. 14 | MR 1863355 | Zbl 0986.11037

[13] Gouvêa, F.; Mazur, B. Families of modular eigenforms, Math. Comp., Volume 58 (1992) no. 198, pp. 793-805 | Article | MR 1122070 | Zbl 0773.11030

[14] Hida, H. Elliptic Curves and Arithmetic Invariant (book manuscript to be published by Springer)

[15] Hida, H. Local indecomposability of Tate modules of non CM abelian varieties with real multiplication (to appear in J. Amer. Math. Soc., preprint available at Hida’s homepage: http://www.math.ucla.edu/~hida/AbNSS.pdf) | MR 3037789 | Zbl 1284.14033

[16] Hida, H. On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves, Amer. J. Math., Volume 103 (1981), pp. 727-776 | Article | MR 623136 | Zbl 0477.14024

[17] Hida, H. On p-adic Hecke algebras for GL 2 over totally real fields, Ann. of Math., Volume 128 (1988), pp. 295-384 | Article | MR 960949 | Zbl 0658.10034

[18] Hida, H. Nearly ordinary Hecke algebras and Galois representations of several variables, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 115-134 | MR 1463699 | Zbl 0782.11017

[19] Hida, H. On nearly ordinary Hecke algebras for GL(2) over totally real fields, Algebraic number theory, Adv. Stud. Pure Math., 17, Academic Press, Boston, MA, 1989, pp. 139-169 | MR 1097614 | Zbl 0742.11026

[20] Hida, H. p-adic automouphism forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, 2004 | MR 2055355 | Zbl 1055.11032

[21] Hida, H. Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs, Oxford University Press, 2006 | MR 2243770 | Zbl 1122.11030

[22] Hida, H. The Iwasawa μ-invariant of p-adic Hecke L-functions, Ann. of Math., Volume 172 (2010), pp. 41-137 | Article | MR 2680417 | Zbl 1223.11131

[23] Hida, H. Geometric Modular Forms and Elliptic Curves, World Scientific, Singapore, 2012 | MR 1794402 | Zbl 1254.11055

[24] Katz, N. p-adic properties of modular schemes and modular forms, Modular functions of one variable III (Lecture Notes in Math.) Volume 350, Springer, Berlin, 1973, pp. 69-190 | MR 447119 | Zbl 0271.10033

[25] Katz, N. M. p-adic L-functions for CM fields, Invent. Math., Volume 49 (1978), pp. 199-297 | Article | MR 513095 | Zbl 0417.12003

[26] Katz, N. M. Serre-Tate local moduli, Algebraic surfaces (Orsay, 1976-78) (Lecture Notes in Math.) Volume 868, Springer, Berlin-New York, 1981, pp. 138-202 | MR 638600 | Zbl 0477.14007

[27] Mumford, D. Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Volume 34, Springer-Verlag, Berlin-New York, 1965 | MR 214602 | Zbl 0797.14004

[28] Mumford, D. An analytic construction of degenerating abelian varieties over complete rings, Compositio Math., Volume 24 (1972), pp. 239-272 | Numdam | MR 352106 | Zbl 0241.14020

[29] Rapoport, M. Compactifications de l’espace de modules de Hilbert-Blumenthal, Compositio Math., Volume 36 (1978) no. 3, pp. 255-335 | Numdam | MR 515050 | Zbl 0386.14006

[30] Ribet, K. A. Galois action on division points of abelian varieties with real multiplications, Amer. J. Math., Volume 98 (1976), pp. 751-804 | Article | MR 457455 | Zbl 0348.14022

[31] Serre, J-P.; Tate, J. Good reduction of abelian varieties, Ann. of Math., Volume 88 (1965), pp. 492-517 | Article | MR 236190 | Zbl 0172.46101

[32] Shimura, G. On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math., Volume 78 (1963), pp. 149-192 | Article | MR 156001 | Zbl 0142.05402

[33] Tate, J. Number theoretic background, Automorphic forms, representations and L-functions (1977), pp. 3-26 (part 2) | MR 546607 | Zbl 0422.12007

[34] Wiles, A. On p-adic representations for totally real fields, Ann. of Math., Volume 123 (1986), pp. 407-456 | Article | MR 840720 | Zbl 0613.12013

[35] Wiles, A. On ordinary λ-adic representations associated to modular forms, Invent. math., Volume 94 (1988), pp. 529-573 | Article | MR 969243 | Zbl 0664.10013