Local Indecomposability of Hilbert Modular Galois Representations

Zhao, Bin

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 $\mathbb{Q}$ is odd or the automorphic representation attached to the Hilbert modular form is square integrable at some finite place of $F$.

Classification : 11F80, 11G18, 14K22
Keywords: Galois representation, Hilbert modular forms, complex multiplication
Mot clés : Représentation galoisienne, formes modulaires de Hilbert, multiplication complexe

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | DOI | MR | Zbl

[17] Hida, H. On $p$-adic Hecke algebras for $G{L}_2$ over totally real fields, Ann. of Math., Volume 128 (1988), pp. 295-384 | DOI | MR | Zbl

[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 | Zbl

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

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

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

[22] Hida, H. The Iwasawa $\mu$-invariant of $p$-adic Hecke $L$-functions, Ann. of Math., Volume 172 (2010), pp. 41-137 | DOI | MR | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

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

[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 | Zbl

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

[35] Wiles, A. On ordinary $\lambda$-adic representations associated to modular forms, Invent. math., Volume 94 (1988), pp. 529-573 | DOI | MR | Zbl