On the Eisenstein cohomology of arithmetic groups

Eigenvarieties for reductive groups

“…The question has been settled in fairly good generality thanks to the work of J. Franke [Fra98] and works built on it. See, for example, the results of J.-S. Li and J. Schwermer [LS04]. It also results from the works of J. Franke that there is a trace formula for the cohomology…”

Cohomologie cohérente et représentations Galoisiennes

“…-Nous allons utiliser l'analogue des résultats de [LSt] dans le cas Hodge sur C p , l'article [LSt] se plaçant dans le cas PEL. La même démonstration reste valable dans le cas Hodge si l'on dispose de l'analogue des théorèmes d'annulation de cohomologie cohérente complexe de [LSu] (lesénoncés d'annulation en cohomologie de Betti sont vrais pour toute variété de Shimura par [LSw,cor.5.6]). Remarquons que lorsque κ est une puissance tensorielle de 1, la formule de projection et [LSt,par.4] montrent directement le résultat voulu dans le cas Hodge.…”

On Residual Cohomology Classes Attached to Relative Rank One Eisenstein Series for the Symplectic Group

“…Suppose π ∈ φ P r is an irreducible cuspidal automorphic representation of the Levi component L r (A) on the subspace V π of the space of cusp forms on L r (A). By carrying through the construction of residues or derivatives of Eisenstein series attached to (π, V π ) (as in [24], Section 3), the corresponding contribution to H * (sp n , K R , A E,{P r },φ ⊗ C E) is embodied in the cohomology…”

“…The existence of these classes, in particular, their arithmetic nature and close relation to the theory of L-functions are the core issues of these investigations. There are some results for groups G of small Q-rank other than GL 2 or very specific types of Eisenstein series [12], [8], [24], [30], [32], [33]. In describing in the case of the symplectic group Sp n that part in the cohomology which is attached to relative rank one Eisenstein series this work concerns an algebraic group of an arbitrary Q-rank, the case n = 2 being already treated in [31], [34].…”

“…They give the remaining conditions in the theorems, and as already mentioned reduce the study of the poles of the Eisenstein series to the region where the poles of the automorphic L-functions in question are known. The geometric conditions are derived from the study of the way in which the Eisenstein series may give rise to a non-trivial cohomology class [24,Section 3], [33,Remark 4.12]. These are reduced to certain abstract equations involving the Weyl group and the root system.…”