Cohomologie cohérente et représentations Galoisiennes

Abstract

Dans cet article, on montre comment les idées introduites dans l'article [S4] s'appliquentà l'étude de la cohomologie cohérente des variétés de Siegel, et plus généralement des variétés de Shimura de type Hodge. Le résultat principal affirme que les classes de cohomologie cohérente supérieure sont des limites p-adique de formes modulaires cuspidales. Ceci permet dans certains cas d'associer des représentations Galoisiennesà des formes automorphes apparaissant dans la cohomologie cohérente.

Strata Hasse invariants, Hecke algebras and Galois representations

“…I.3.1 and Th. I.3.2 were obtained simultaneously and independently by Pilloni-Stroh [86]. Their method is completely different: It is based on Scholze's theory of perfectoid Shimura varieties.…”

Chern classes of automorphic vector bundles

“…The results of the present section are entirely due to Scholze [Sch15], then to Caraiani-Scholze [CS17] and Pilloni-Stroh [PS16].…”

“…The following theorem is essentially due to Pilloni and Stroh. In [PS16] this is proved for the Siegel modular variety, although it is not stated in this form. In Section 5 we explain their result and show how to obtain Theorem 2.22 for general Shimura varieties of Hodge type, under Hypothesis 2.18.…”

“…Proof. This is essentially equivalent to Corollaire 1.16 of [PS16]. More precisely, the generic fiber of the bundle denoted ω mod A in [PS16] is exactly [E St ] dR,Σ .…”

Oddness of residually reducible Galois representations

“…Remark 2.7. In certain cases, Galois representations have been constructed for irregular π (see, in particular, the recent papers of [15,23] in the case that the descent of π to a unitary group has a non-degenerate limit of discrete series as archimedean component). In Theorem 4.1, we consider representations that are residually reducible and not necessarily Hodge-Tate regular (for an example see [4, Theorem 5.1]).…”