Local Indecomposability of Hilbert Modular Galois Representations
Annales de l'Institut Fourier, Volume 64 (2014) no. 4, p. 1521-1560

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.

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.

DOI : https://doi.org/10.5802/aif.2889
Classification:  11F80,  11G18,  14K22
Keywords: Galois representation, Hilbert modular forms, complex multiplication
@article{AIF_2014__64_4_1521_0,
     author = {Zhao, Bin},
     title = {Local Indecomposability of Hilbert Modular Galois Representations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     pages = {1521-1560},
     doi = {10.5802/aif.2889},
     mrnumber = {3329672},
     zbl = {06387316},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2014__64_4_1521_0}
}
Zhao, Bin. Local Indecomposability of Hilbert Modular Galois Representations. Annales de l'Institut Fourier, Volume 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), Tome 19 (1986), pp. 409-468 | Numdam | MR 870690 | Zbl 0616.10025

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

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

[6] De Jong, A.J.; Noot, R.; G. Van Der Geer, F. Oort And J.Steenbrink Jacobians with complex multiplication, Arithmetic Algebraic Geometry, Brikhäuser, Boston (Progress in Math.) Tome 89 (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., Tome 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), Tome 54 (2004), pp. 2143-2162 | Article | Numdam | MR 2139691 | Zbl 1131.11341

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

[13] Gouvêa, F.; Mazur, B. Families of modular eigenforms, Math. Comp., Tome 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., Tome 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., Tome 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-Verlag, Springer Monographs in Mathematics (2004) | MR 2055355 | Zbl 1055.11032

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

[22] Hida, H. The Iwasawa μ-invariant of p-adic Hecke L-functions, Ann. of Math., Tome 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, Springer, Berlin (Lecture Notes in Math.) Tome 350 (1973), pp. 69-190 | MR 447119 | Zbl 0271.10033

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

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

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

[28] Mumford, D. An analytic construction of degenerating abelian varieties over complete rings, Compositio Math., Tome 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., Tome 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., Tome 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., Tome 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., Tome 78 (1963), pp. 149-192 | Article | MR 156001 | Zbl 0142.05402

[33] Tate, J. Number theoretic background, Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore. (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., Tome 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., Tome 94 (1988), pp. 529-573 | Article | MR 969243 | Zbl 0664.10013