Cohomology of Siegel varieties with $p$-adic integral coefficients and applications

Cohomology of Siegel varieties with

p

-adic integral coefficients and applications

Mokrane, Abdellah ; Tilouine, Jacques

Cohomology of Siegel varieties, Astérisque, no. 280 (2002), pp. 1-95.

MR Zbl | 8 citations dans Numdam

@incollection{AST_2002__280__1_0,
     author = {Mokrane, Abdellah and Tilouine, Jacques},
     title = {Cohomology of {Siegel} varieties with $p$-adic integral coefficients and applications},
     booktitle = {Cohomology of Siegel varieties},
     editor = {Mokrane, Abdellah and Polo, Patrick and Tilouine, Jacques},
     series = {Ast\'erisque},
     pages = {1--95},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {280},
     year = {2002},
     mrnumber = {1944174},
     zbl = {1078.11037},
     language = {en},
     url = {http://www.numdam.org/item/AST_2002__280__1_0/}
}

TY  - CHAP
AU  - Mokrane, Abdellah
AU  - Tilouine, Jacques
TI  - Cohomology of Siegel varieties with $p$-adic integral coefficients and applications
BT  - Cohomology of Siegel varieties
AU  - Collectif
ED  - Mokrane, Abdellah
ED  - Polo, Patrick
ED  - Tilouine, Jacques
T3  - Astérisque
PY  - 2002
SP  - 1
EP  - 95
IS  - 280
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2002__280__1_0/
LA  - en
ID  - AST_2002__280__1_0
ER  -

%0 Book Section
%A Mokrane, Abdellah
%A Tilouine, Jacques
%T Cohomology of Siegel varieties with $p$-adic integral coefficients and applications
%B Cohomology of Siegel varieties
%A Collectif
%E Mokrane, Abdellah
%E Polo, Patrick
%E Tilouine, Jacques
%S Astérisque
%D 2002
%P 1-95
%N 280
%I Société mathématique de France
%U http://www.numdam.org/item/AST_2002__280__1_0/
%G en
%F AST_2002__280__1_0

Mokrane, Abdellah; Tilouine, Jacques. Cohomology of Siegel varieties with $p$-adic integral coefficients and applications, dans Cohomology of Siegel varieties, Astérisque, no. 280 (2002), pp. 1-95. http://www.numdam.org/item/AST_2002__280__1_0/

Bibliographie
Cité par

[1] A. Ash, D. Mumford, M. Rapoport, Y. S. Tai: Smooth compactification of locally symmetric varieties, Math. Sci. Press., Brookline, 1975. | MR | Zbl

[2] A. A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers, Astérisque 100 (1982). | Numdam | MR | Zbl

[3] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand: Differential operators on the base affine space and a study of the $g$ -modules, in Lie Groups and their Representations, ed. I.M. Gelfand, Proc. Conf. Budapest 1971, Adam Hilger Publ., London, 1975. | MR | Zbl

[4] P. Berthelot, A. Ogus: Notes on crystalline cohomology, P.U.P., Princeton, 1978. | MR | Zbl

[5] A. Borel: Stable real cohomology of arithmetic groups, Ann. Sci. Éc. Norm. Sup. (4) 7 (1974) 235-272. | DOI | EuDML | Numdam | MR | Zbl

[6] A. Borel: Stable and $L^{2}$ -cohomology of arithmetic groups, Bull. Am. Math. Soc., (N.S.) 3, (1980), 1025-1027. | DOI | MR | Zbl

[7] A. Borel: Regularization theorems in Lie algebra cohomology, applications, Duke Math. J. 50 (1983), 605-623. | DOI | MR | Zbl

[8] A. Borel, N. Wallach: Continuous cohomology, discrete subgroups, and representations of reductive groups, P.U.P. 1980. | MR | Zbl

[9] D. Blasius, M. Harris, D. Ramakrishnan: Coherent cohomology, limits of discrete series and Galois conjugation, Duke Math. J. 73, 1994, 647-684. | MR | Zbl

[10] D. Blasius, J. D. Rogawski: Motives for Hilbert modular forms, Inv. Math. 114, (1993) 55-87. | DOI | EuDML | MR | Zbl

[11] C. Breuil: Une remarque sur les représentations locales $p$ -adiques et les congruences entre formes modulaires de Hilbert, Bull. Soc. Math. France 127 (1999), no. 3, 459-472. | DOI | EuDML | Numdam | MR | Zbl

[12] N. Bourbaki: Groupes et Algèbres de Lie, Chap. 7, 8, Hermann 1975. | Zbl

[13] C. L. Chai, G. Faltings: Degeneration of Abelian Varieties, Erg. Math. Wiss. 3. folge, 22, Springer Verlag, 1990. | MR | Zbl

[14] L. Clozel: Motifs et formes automorphes: application du principe de fonctorialité, in Proc. Ann Arbor Conf. (Automorphic Forms, Shimura Varieties and $L$ -functions, vol. I and II, Eds. L. Clozel, J.S. Milne), vol. I, 77-159., Persp. in Math., Acad. Press 1990. | MR | Zbl

[15] P. Deligne: Travaux de Griffiths, Sém. Bourbaki exp. 376, LN 180, Springer Verlag 1972. | Numdam | Zbl

[16] F. Diamond: The Taylor-Wiles construction and multiplicity one, Inv. Math. 128, (1997) 379-391. | DOI | MR | Zbl

[17] M. Demazure, A. Grothendieck: Schémas en groupes III (SGA 3, 1962/64), Springer Lect. Notes 153, Springer Verlag, 1970. | MR | Zbl

[18] R. Deheuvels: Formes quadratiques et groupes classiques, PUF, Paris 1981. | MR | Zbl

[19] M. Dimitrov: Valeur critique de la fonction $L$ adjointe d'une forme modulaire de Hilbert et arithmétique du motif correspondant, thèse Paris 13, 2003.

[20] F. El Zein: Complexe dualisant et applications à la classe fondamentale d'un cycle, Mémoire de la SMF 58, 1978. | EuDML | Numdam | MR | Zbl

[21] G. Faltings: On the cohomology of locally symmetric hermitian spaces, in Séminaire d'Algèbre, pp. 349-366, Springer Lecture Notes 1029, 1983. | MR | Zbl

[22] G. Faltings: Crystalline cohomology and $p$ -adic Galois representations, in Algebraic Analysis, ed. J.-I. Igusa, Proc JAMI inaugural Conference, the Johns Hopkins Univ. Press, 1990. | MR | Zbl

[23] G. Faltings: $F$ -isocrystals on open varieties. Results and conjectures, in Grothendieck's Festschrift, Birkhäuser, 1990, Boston. pp. 219-248. | MR | Zbl

[24] J.-M. Fontaine, G. Laffaille: Construction de représentations $p$ -adiques, Ann. Sci. Éc. Norm. Sup. (4), 15 (1982) 547-608. | DOI | EuDML | Numdam | MR | Zbl

[25] J. Franke: Harmonic analysis in weighted $L^{2}$ -spaces, Ann. Sci. Éc. Norm. Sup. (4), 31 (1998) 181-279. | DOI | EuDML | Numdam | MR | Zbl

[26] E. Freitag: Siegelsche Modulfunktionen, Grundl. Math. Wiss. 254, Springer Verlag 1983. | MR | Zbl

[27] E. Freitag, R. Kiehl: Etale cohomology and the Weil Conjecture, Springer Verlag 1988. | DOI | MR | Zbl

[28] K. Fujiwara: Level optimization in the totally real case, preprint 1999.

[29] K. Fujiwara: Deformation rings and Hecke algebras in the totally real case, version 2.0, preprint 1999.

[30] K. Fujiwara: Arithmetical compactifications of Shimura varieties, preprint.

[31] W. Fulton: Intersection Theory, Erg. Math., Springer Verlag 1984. | MR | Zbl

[32] W. Fulton, J. Harris: Representation Theory, Springer Verlag 1991. | MR | Zbl

[33] A. Grothendieck, J. Dieudonné: Eléments de Géométrie Algébrique IV, Etude locale des schémas et des morphismes de schémas (seconde partie), Pub. Math. IHES 24 (1965). | EuDML | Numdam | MR | Zbl

[34] G. Harder: unpublished notes, 1992.

[35] G. Harder: Eisensteinkohomologie und die Konstruktion gemischter motive, SLN 1562, Springer Verlag 1993. | MR | Zbl

[36] M. Harris: Automorphic forms of $\bar{\partial}$ -cohomology as coherent cohomology classes, J. Diff. Geom. 32, (1990) 1-63. | DOI | MR | Zbl

[37] M. Harris: Functorial properties of toroidal compactifications of locally symmetric varieties, Proc. London Math. Soc 59, (1989) 1-22. | DOI | MR | Zbl

[38] M. Harris, R. Taylor: Deformations of automorphic Galois representations, preprint 1998.

[39] M. Harris, S. Zucker: Boundary Cohomology of Shimura varieties: Eisenstein classes in coherent cohomology, Ann. Sci. Éc. Norm. Sup. (4), 27 (1994), no. 3, 249-344. | DOI | EuDML | Numdam | MR | Zbl

[40] H. Hida: On $p$ -adic Hecke algebras for $G L_{2}$ over totally real fields, Ann. of Math. 128 (1988), 295-384. | DOI | MR | Zbl

[41] H. Hida: Nearly ordinary Hecke algebras and Galois representations of several variables, in Algebraic Analysis, ed. J.-I. Igusa, Proc JAMI inaugural Conference, the Johns Hopkins Univ. Press, 1990. | MR | Zbl

[42] H. Hida: Control Theorems of $p$ -nearly Ordinary Cohomology Groups, Bull. Soc. Math. France 123, (1995) 425-475. | DOI | EuDML | Numdam | MR | Zbl

[43] H. Hida: Control theorems for coherent sheaves on Shimura varieties of $P E L$ type, preprint. | DOI | MR | Zbl

[44] L. Illusie: Réduction semi-stable et décomposition de complexes de de Rham à coefficients, Duke Math. J. 60, (1990) 139-185. | MR | Zbl

[45] L. Illusie: Cohomologie de de Rham et cohomologie étale $p$ -adique (d'après G. Faltings, J.-M. Fontaine et al.), Séminaire Bourbaki, Exp. No. 726, vol. 1989/90 Astérisque No. 189-190, (1990) 325-374. | EuDML | Numdam | MR | Zbl

[46] J. C. Jantzen: Representations of algebraic groups, Acad. Press, 1987. | MR | Zbl

[47] J. C. Jantzen: letter to the authors, May 11, 1998.

[48] K. Kato: Logarithmic structures of Fontaine-Illusie, Algebraic Analysis, ed. J.-I. Igusa, Proc JAMI inaugural Conference, the Johns Hopkins Univ. Press, 1990. | MR | Zbl

[49] N. Katz: Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publ. Math. I.H.E.S. 39 (1971), 175-232. | DOI | EuDML | Numdam | MR | Zbl

[50] N. Katz, W. Messing: Some consequences of the Riemann hypothesis for varieties over finite fields, Inv. Math. 23, (1974) 73-77. | DOI | EuDML | MR | Zbl

[51] S. Kleiman: The Standard Conjectures, p. 3-20, in Motives, vol. I, PSPM 55, AMS 1994. | DOI | MR | Zbl

[52] G. Laumon: Sur la cohomologie à supports compacts des variétés de Shimura pour $G S p {(4)}_{/ ℚ}$ , Comp. Math. 105, (1996) 267-359. | DOI | MR | Zbl

[53] G. Laumon: Fonctions zeta des variétés de Siegel de dimension trois, to appear.

[54] B. Mazur, J. Tilouine: Représentations galoisiennes, différentielles de Kähler et conjectures principales, Publ. Math. IHES 71, 1990. | DOI | EuDML | Numdam | MR | Zbl

[55] D. Mumford: Abelian Varieties, Oxford Univ. Press 1970. | MR | Zbl

[56] D. Mumford: Hirzebruch's proportionality theorem in the non-compact case, Inv. Math. 42 (1977), 239-272. | DOI | EuDML | MR | Zbl

[57] A. Mokrane: Cohomologie cristalline des variétés ouvertes, Maghreb Math. Rev., 2, (1993) 161-175. | MR

[58] R. Pink: Arithmetical compactification of mixed Shimura varieties, Dissertation, Bonn 1989. | MR | Zbl

[59] R. Pink: On $ℓ$ -adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification, Math. Ann. 292, (1992) 197-240. | DOI | EuDML | MR | Zbl

[60] B. Perrin-Riou: Representations $p$ -adiques ordinaires, in Périodes $p$ -adiques, Astérisque 223, 1994. | Numdam | MR | Zbl

[61] P. Polo, J. Tilouine: Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $ℤ_{p}$ , this volume. | Numdam | Zbl

[62] K. Ribet: On $ℓ$ -adic representations attached to modular forms, Inv. Math. 28 (1975) 245-275. | DOI | EuDML | MR | Zbl

[63] B. Roberts: Non-vanishing of global theta-lifts from orthogonal groups, preprint, 1999. | MR | Zbl

[64] B. Roberts: letter to the authors, 21/07/99.

[65] B. Roberts: Global $L$ -packets for $G S p (2)$ and theta lifts, Documenta Mathematica, 6 (2001), 247-314. | EuDML | MR | Zbl

[66] T. Saito: Hilbert modular forms and p-adic Hodge theory, preprint, April 1999. | MR | Zbl

[67] J. Schwermer: On Euler products and residual Eisenstein cohomology classes for Siegel modular varieties, Forum Math. 7 (1995), 1-28. | DOI | EuDML | MR | Zbl

[68] G. Shimura: A reciprocity law in non-solvable extensions, J. reine ang. Math., 221 (1966), 209-220. | EuDML | MR | Zbl

[69] G. Shimura: On the Eisenstein series of Hilbert Modular Groups, Rev. Matematica Iberoamericana, 1, (1985) no. 3, 1-42. | EuDML | MR | Zbl

[70] T. A. Springer: Reductive groups, in Automorphic Forms, Representations, and Lfunctions, Corvallis, PSPM XXXIII, t. 1, AMS 1979. | DOI | MR | Zbl

[71] R. Taylor: Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63, (1991) 281-332. | MR | Zbl

[72] R. Taylor: On the $ℓ$ -adic cohomology of Siegel threefolds, Inv. Math. 114, (1993) 289-310. | DOI | EuDML | MR | Zbl

[73] R. Taylor, A. Wiles: Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141, (1995) 553-572. | DOI | MR | Zbl

[74] SGA4: M. Artin, A. Grothendieck, J.-L. Verdier: Théorie des topos et cohomologie étale des schémas, Lect. Notes in Math. 305, Springer Verlag 1972-73. | MR | Zbl

[75] SGA41/2: P. Deligne Et Al.: Cohomologie Étale, Lect. Notes in Math. 569, Springer Verlag 1977. | MR | Zbl

[76] J. Tilouine: Deformations of Galois representations and Hecke algebras, Narosa Publ., New Delhi 1996. | MR | Zbl

[77] J. Tilouine, E. Urban: Several variable $p$ -adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, Ann. Sci. Ec. Norm. Sup. (4), 32, (1999) 499-574. | DOI | EuDML | Numdam | MR | Zbl

[78] T. Tsuji: Syntomic complexes and $p$ -adic vanishing cycles, J. reine angew. Math. 472, (1996) 69-138. | EuDML | MR | Zbl

[79] E. Urban: On the torsion of the cohomology of the Siegel threefold, preprint 1998.

[80] E. Urban: Sur les représentations $p$ -adiques associées aux représentations cuspidales de $G S p {(4)}_{ℚ}$ , preprint 1998.

[81] M.-F. Vignéras: Correspondances entre représentations automorphes de $G L (2)$ sur une extension quadratique et $G S p (4)$ sur $ℚ$ , Conjecture locale de Langlands pour $G S p (4)$ in Proc. Bowdoin Conf. (1984): "The Selberg Trace Formula and Related Topics", eds. D. Hejhal, P. Sarnak, A. Terras, Cont. Math. Ser. 53, 1986, AMS Publ., Providence RI. | MR | Zbl

[82] D. Vogan, G. Zuckermann: Unitary representations with non-zero cohomology, Comp. Math. 53 (1984), 51-90. | EuDML | Numdam | MR | Zbl

[83] N. Wach: Représentations cristallines de torsion, Comp. Math. 108, (1997) 185-240. | DOI | MR | Zbl

[84] J.-L. Waldspurger: Cohomologie des espaces de formes automorphes, d'après J. Franke, Séminaire Bourbaki, Exp. 809, Astérisque 241, 1997. | Numdam | MR | Zbl

[85] N. R. Wallach: On the constant term of a square integrable automorphic form, in Proc. Conf on Operator Algebras and Group Representations, held in Neptun (Romania), Eds. Arsene, Stratila, Verona, Voiculescu; Monographs and Studies in Mathematics, Pitman Publ., London 1984. | MR | Zbl

[86] R. Weissauer: An application of the hard Lefschetz theorem, preprint 1996.

[87] R. Weissauer: Four dimensional Galois representations, preprint, 1996.

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

[89] A. Wiles: On $λ$ -adic representations associated to modular forms, Inv. Math. 94, (1988), 529-573. | DOI | EuDML | MR | Zbl

[90] H. Yoshida: Siegel's modular forms and the arithmetic of quadratic forms, Inv. Math. 60, (1980) 193-248. | DOI | EuDML | MR | Zbl

[91] H. Yoshida: Letter to the authors, August 3, 1999.