Congruences between Siegel modular forms on the level of group cohomology
Annales de l'Institut Fourier, Volume 46 (1996) no. 4, p. 877-897

Vector-valued Siegel modular forms may be found in certain cohomology groups with coefficients lying in an irreducible representation of the symplectic group. Using functoriality in the coefficients, we show that the ordinary components of the cohomology are independent of the weight parameter. The meaning of ordinary depends on a choice of parabolic subgroup of GSp(4), giving a particular direction in the change of weight. Our results complement those of Taylor and Tilouine-Urban for the two other possible classes of parabolic subgroups.

Les formes modulaires de Siegel à valeurs vectorielles se trouvent dans certains groupes de cohomologie dont les coefficients sont des représentations irréductibles de groupes symplectiques. En utilisant la fonctorialité par rapport aux coefficients, on montre que les composantes ordinaires de la cohomologie ne dépendent pas du poids. Le sens de ordinaire dépend d’un choix de sous-groupe parabolique de GSp(4), ce qui donne une certaine direction au changement de poids. Nos résultats complètent ceux de Taylor et Tilouine-Urban pour les autres classes de sous-groupes paraboliques.

     author = {Buecker, Karsten},
     title = {Congruences between Siegel modular forms on the level of group cohomology},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {4},
     year = {1996},
     pages = {877-897},
     doi = {10.5802/aif.1533},
     zbl = {0853.11038},
     mrnumber = {98f:11038},
     language = {en},
     url = {}
Buecker, Karsten. Congruences between Siegel modular forms on the level of group cohomology. Annales de l'Institut Fourier, Volume 46 (1996) no. 4, pp. 877-897. doi : 10.5802/aif.1533.

[A] A.N. Andrianov, Quadratic forms and Hecke operators, Grundlehren d. Math. Wiss., 286, Springer, 1987. | MR 88g:11028 | Zbl 0613.10023

[AS] A. Ash and G. Stevens, Modular forms in characteristic l and special values of their L-functions, Duke Math. Jour., 53, n° 3 (1986), 849-868. | MR 88h:11036 | Zbl 0618.10026

[B] K.S. Brown, Cohomology of groups, Grad. Texts in Math., 87, Springer, 1982. | MR 83k:20002 | Zbl 0584.20036

[Buck] K. Buecker, Congruences between Siegel modular forms, Ph.D. Thesis, Cambridge, 1995.

[Cole] R.F. Coleman, p-adic Banach spaces and families of modular forms, preprint. | Zbl 0918.11026

[Falt] G. Faltings, On the cohomology of locally symmetric Hermitian spaces, in LNM 1029, Springer (1983), 55-98. | MR 85k:22028 | Zbl 0539.22008

[Frei] E. Freitag, Siegelsche Modulfunktionen, Grundlehren d. Math. Wiss., 254 Springer, 1983. | MR 88b:11027 | Zbl 0498.10016

[Hi1] H. Hida, Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms, Invent. Math., 85 (1986), 545-613. | MR 87k:11049 | Zbl 0612.10021

[Hi2] H. Hida, On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math., 128 (1988), 295-384. | MR 89m:11046 | Zbl 0658.10034

[Hi3] H. Hida, Elementary theory of L-functions and Eisenstein series, LMS Student Texts 26, CUP, 1993. | MR 94j:11044 | Zbl 0942.11024

[Hmph] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9, Springer, 1972. | MR 48 #2197 | Zbl 0254.17004

[KPS] M. Kuga, W. Parry, C. Sah, Group cohomology and Hecke operators, in Manifolds and Lie groups, Eds. J. Hano et al., Birkhäuser, 1981. | MR 83m:10041 | Zbl 0479.10019

[MW] B. Mazur, A. Wiles, On p-adic analytic families of Galois representations, Comp. Math., 59 (1986), 231-264. | Numdam | MR 88e:11048 | Zbl 0654.12008

[Pan] A.A. Panchishkin, Non-Archimedean L-functions of Siegel and Hilbert modular forms, LNM 1471, Springer, 1991. | MR 93a:11044 | Zbl 0732.11026

[Shim] G. Shimura, On certain reciprocity laws for theta functions and modular forms, Acta Math., 141 (1978), 35-71. | MR 58 #10757 | Zbl 0402.10030

[St] R. Steinberg, Lectures on Chevalley Groups, Lecture Notes, Yale (1968).

[Tayl] R.L. Taylor, On congruences between modular forms, Ph.D. Thesis, Princeton, 1988.

[TU] J. Tilouine, E. Urban, Nearly ordinary cohomology for Sp4, preprint.

[W] A. Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math., 94 (1988), 529-573. | MR 89j:11051 | Zbl 0664.10013