*p*-Adic Automorphic Forms on Shimura Varieties
pp 225-249 |
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# Generalized Eichler-Shimura Map

## Abstract

Let G/ℤ_{(p)} for ℤ(*p*)=ℚ∩ℤ_{ p } be a connected reductive group split over ℤ_{ p } (see Section 4.4.3 for the definition of split algebraic groups). We first prove semi- simplicity of the commutative Hecke algebra acting on the nearly ordinary cohomology group *H*^{ q }_{ n.ord }(*X*(*S*), *L*) inside the interior cohomology *H*_{ ! }^{ q }(*X*(*S*),*L*)for a modular variety X(S) associated with an arbitrary *p*-power level open compact subgroup *S* of *G*(A^{∞}), where *H*_{ ! }^{ q } is the image of the compactly supported cohomology group inside the standard cohomology group. Here the locally constant or coherent sheaf *L* of ℚ-vector spaces on *X*(*S*) is associated with a rational representation of *G* twisted by a finite-order character. After dealing with topological cohomology groups, we relate as Hecke modules the topological and the coherent cohomology groups via the generalized Eichler-Shimura map (which shows the semi-simplicity of the Hecke algebra acting on the coherent cohomology and topological cohomology). Although we have assumed that *G* is split over ℤ_{ p }, the argument works equally well for a connected reductive *G* smooth quasi-split over ℤ_{ p }.

## Keywords

Cohomology Group Compact Subgroup Hermitian Form Congruence Subgroup Maximal Compact Subgroup## Preview

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