Congruences between Siegel modular forms on the level of group cohomology

Buecker, Karsten

Annales de l'Institut Fourier, Tome 46 (1996) no. 4, pp. 877-897.

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 $GSp\left(4\right)$, 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.

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 $GSp\left(4\right)$, 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.

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