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Generalized Eichler-Shimura Map

  • Haruzo Hida
Part of the Springer Monographs in Mathematics book series (SMM)

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, LLC 2004

Authors and Affiliations

  • Haruzo Hida
    • 1
  1. 1.Mathematics DepartmentUCLALos AngelesUSA

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