Several-variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 32 (1999) no. 4, p. 499-574
@article{ASENS_1999_4_32_4_499_0,
     author = {Tilouine, Jacques and Urban, Eric},
     title = {Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Elsevier},
     volume = {Ser. 4, 32},
     number = {4},
     year = {1999},
     pages = {499-574},
     doi = {10.1016/s0012-9593(99)80021-4},
     zbl = {0991.11016},
     mrnumber = {2000j:11064},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_1999_4_32_4_499_0}
}
Tilouine, J.; Urban, E. Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 32 (1999) no. 4, pp. 499-574. doi : 10.1016/s0012-9593(99)80021-4. http://www.numdam.org/item/ASENS_1999_4_32_4_499_0/

[1] A. Ash and G. Stevens, p-adic deformations of cohomology classes of subgroups of GL(n, Z), Coll. Math. 48, Proc. Journées Arithmétiques, 1995, Barcelone. | MR 98h:11064 | Zbl 0866.20038

[2] A. Borel and J-P. Serre, Corners and arithmetic groups, Comment. Math. Helv., 48, 1973, pp. 436-491. | MR 52 #8337 | Zbl 0274.22011

[3] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups and representations of reductive groups, Ann. of Math. Stud., 94, Princeton Univ. Press, Princeton, 1980. | MR 83c:22018 | Zbl 0443.22010

[4] N. Bourbaki, Commutative algebra, Chapters 1-7, Addison Wesley, 1972. | MR 50 #12997

[5] K. Buecker, Congruences between Siegel modular forms on the level of group cohomology, Ann. Inst. Fourier (Grenoble), 46, 1996. | Numdam | MR 98f:11038 | Zbl 0853.11038

[6] K. Buecker, On the control theorem for the symplectic group, Compositio Math., 113, 1998. | MR 99m:11056 | Zbl 0923.11076

[7] C.-L. Chai and G. Faltings, Degeneration of abelian varieties, Erg. Math. 3. Folge Band 22, Springer-Verlag, 1990. | MR 92d:14036 | Zbl 0744.14031

[8] J. Franke, Harmonic Analysis in weighted L²-spaces, Ann. Éc. Norm. Sup., 4ième Série, t. 31, 1998, p. 181-279. | Numdam | MR 2000f:11065 | Zbl 0938.11026

[9] G. Faltings, Crystalline cohomology and Galois representations in Algebraic Analysis, Geometry and Number Theory, Proceedings of JAMI Inaugural Conference, John Hopkins Univ. Press, 1989. | MR 98k:14025

[10] J.M Fontaine, Représentations p-adiques semi-stables, Séminaire de Bures sur les périodes p-adiques, Astérisque, 223, SMF, Paris, 1994. | Zbl 0865.14009

[11] R. Godement, Topologie algébrique et théorie des faisceaux, Hermann Paris (1964).

[12] H. Hida, On congruence divisors of cusp forms as factors of the special values of their zeta functions, Inv. Math., 64, 1981, pp. 221-262. | MR 83h:10066 | Zbl 0472.10028

[13] H. Hida, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Inv. math., 85, 1986, pp. 545-613. | MR 87k:11049 | Zbl 0612.10021

[14] H. Hida, Modules of Congruences of Hecke algebras and L-functions associated with cusp forms, Amer. J. of Math., 110, 1988, pp. 323-382. | MR 89i:11058 | Zbl 0645.10029

[15] H. Hida, Nearly ordinary Hecke algebras and several variables Galois representations, pp. 115-134, in Algebraic Analysis, Geometry and Number Theory, Proc. of the JAMI inaugural conference, ed. J.-I. Igusa, Johns Hopkins Univ. Press, Baltimore, 1990. | Zbl 0782.11017

[16] H. Hida, p-Ordinary cohomology groups for SL(2) for number fields. Duke Math. J., Vol. 69, No. 2, 1993. | MR 94g:11031 | Zbl 0941.11024

[17] H. Hida, Control Theorems of p-nearly ordinary Cohomology Groups for SL(n), Bull. Soc. Math. France, 123, 1995, pp. 425-475. | Numdam | MR 97c:11053 | Zbl 0852.11023

[18] H. Hida, Automorphic induction for GLn and the Leopoldt conjecture, preprint 1994.

[19] H. Hida, Control theorems of coherent sheaves on Shimura varieties of PEL-type, preprint.

[20] H. Hida, J. Tilouine and E. Urban, Adjoint Modular Galois representations and their Selmer groups, Proc. Conf. Nat. Acad. Sci. USA, vol. 94, pp. 111121-11124, Oct. 1997. | MR 98m:11034 | Zbl 0909.11025

[21] N. Iwahori and H. Matsumoto, On some Bruhat decompositions and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. IHES, 1965, pp. 237-280. | Numdam | MR 32 #2486 | Zbl 0228.20015

[22] J.C. Jantzen, Representations of Algebraic Groups, Academic Press, 1987. | MR 89c:20001 | Zbl 0654.20039

[23] G. Laumon, Sur la cohomologie à supports compacts des variétés de Shimura pour GSp(4)Q, Comp. Math., vol. 105, 1996, pp. 267-359. | MR 2000a:11097 | Zbl 0877.11037

[24] A. Mokrane and J. Tilouine, Modulo p crystalline BGG and freeness of the cohomology modules, preprint.

[25] L. Nyssen, Pseudo-représentations, Math. Ann., 306, 1996, pp. 257-283. | MR 98a:20013 | Zbl 0863.16012

[26] B. Perrin-Riou, Représentations galoisiennes ordinaires, Séminaire de Bures sur les périodes p-adiques, Astérisque, 223, SMF, Paris 1994.

[27] J. Schwermer, On arithmetic quotients of the Siegel upper helf space of degree two, Comp. Math., 58, 1986, pp. 233-258. | Numdam | MR 87j:11040 | Zbl 0596.10029

[28] J. Schwermer, letter to the authors, Sept. 28, 1995.

[29] J. Schwermer, Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lect. Notes 988. Berlin-Heidelberg-New York-Tokyo, Springer, 1983. | MR 87i:22034 | Zbl 0506.22015

[30] S. Sen, Continuous cohomology and p-Adic Galois Representations, Inv. Math., 62, 1980, pp. 89-116. | MR 82e:12018 | Zbl 0463.12005

[31] S. Sen, An Infinite dimensional Hodge-Tate theory, Bull. Soc. Math. France, 121, 1993, pp. 13-34. | Numdam | MR 94e:11121 | Zbl 0786.11067

[32] G. Shimura, On modular correspondences for Sp(N, Z) and their congruence relations, Proc. Nat. Acad. Sci., 49, 1963, pp. 824-828. | MR 28 #250 | Zbl 0122.08803

[33] R. Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J., 63, 1991, pp. 281-332. | MR 92j:11044 | Zbl 0810.11033

[34] R. Taylor, On the l-adic cohomology of Siegel threefolds, Inv. Math., 114, 1993, pp. 289-310. | MR 95j:11054 | Zbl 0810.11034

[35] R. Taylor and A. Wiles, Ring-theoretical properties of certain Hecke algebras, Annals of Math., 141, 1995, pp. 553-572. | MR 96d:11072 | Zbl 0823.11030

[36] J. Tilouine, Deformations of Galois representations and Hecke algebras, Publ. Mehta Res. Inst., Narosa Publ., New Delhi, 1996. | MR 99i:11038 | Zbl 1009.11033

[37] J. Tilouine, Deformations of Siegel-Hilbert Hecke eigensystems and their Galois representations, in Proc. of the Tiruchirapalli Conference, eds. K. Murty and M. Waldschmidt, Publ. AMS, Cont. Math. 210, 1998. | MR 99a:11067 | Zbl 1020.11036

[38] J. Tilouine and E. Urban, Familles p-adiques à trois variables de formes de Siegel et de représentations galoisiennes, C.R.A.S. Paris, t. 321, Série I, 1995, pp. 5-10. | MR 96c:11048 | Zbl 1019.11502

[39] E. Urban, Module de congruences pour GL2 d'un corps quadratique imaginaire et théorie d'Iwasawa d'un corps CM biquadratique, Duke Math. J., vol. 92, No 1, 1998, pp. 179-220. | MR 98m:11035 | Zbl 0973.11095

[40] E. Urban, Selmer group and the Eisenstein-Klingen Ideal, preprint, 1997.

[41] E. Urban, Letter to J. Tilouine, Nov. 1997.

[42] E. Urban, Sur les représentations p-adiques associées aux représentations cuspidales de GSp4Q, in preparation.

[43] D. Vogan, Representations of Real Reductive Lie Groups, Progress in Mathematics 15, Birkhäuser, 1981. | MR 83c:22022 | Zbl 0469.22012

[44] D. Vogan and G. Zuckerman, Unitary representations with non-zero cohomology, Comp. Math., 53, 1984, pp. 51-90. | Numdam | MR 86k:22040 | Zbl 0692.22008

[45] J.-L. Waldspurger, Cohomologie des espaces de formes automorphes d'après Franke, Séminaire Bourbaki Novembre 1995, exp.809. | Numdam | Zbl 0883.11025

[46] R. Weissauer, A special case of the fundamental lemma : the case GSp4, I, II, III, preprints.

[47] A. Wiles, On p-adic representations for totally real fields, Ann. of Math., 123, 1986, pp. 407-456. | MR 87g:11142 | Zbl 0613.12013

[48] A. Wiles, Modular curves and Fermat's Last Theorem, Ann. of Math., 123, 1995.