Israel Journal of Mathematics

, Volume 135, Issue 1, pp 355–380 | Cite as

Cohomology of discrete groups in harmonic cochains on buildings



Modules of harmonic cochains on the Bruhat-Tits building of the projective general linear group over ap-adic field were defined by one of the authors, and were shown to represent the cohomology of Drinfel’d’sp-adic symmetric domain. Here we define certain non-trivial natural extensions of these modules and study their properties. In particular, for a quotient of Drinfel’d’s space by a discrete cocompact group, we are able to define maps between consecutive graded pieces of its de Rham cohomology, which we show to be (essentially) isomorphisms. We believe that these maps are graded versions of the Hyodo-Kato monodromy operatorN.


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  1. [A-dS]
    G. Alon and E. de Shalit,On the cohomology of Drinfel’d’s p-adic symmetric domain, Israel Journal of Mathematics129 (2002), 1–20.MATHCrossRefMathSciNetGoogle Scholar
  2. [C-I]
    R. Coleman and A. Iovita,The Frobenius and the monodromy operators for curves and abelian varieties, Duke Mathematical Journal97 (1999), 171–215.MATHCrossRefMathSciNetGoogle Scholar
  3. [dS]
    E. de Shalit,Residues on buildings and de Rham cohomology of p-adic symmetric domains, Duke Mathematical Journal106 (2000), 123–191.CrossRefGoogle Scholar
  4. [dS2]
    E. de Shalit,Eichler cohomology and periods of modular forms on p-adic Schottky groups, Journal für die reine und angewandte Mathematik400 (1989), 3–31.MATHCrossRefGoogle Scholar
  5. [G]
    H. Garland,p-adic curvature and the cohomlogy of discrete subgroups of p-adic groups, Annals of Mathematics (2)97 (1973), 375–423.CrossRefMathSciNetGoogle Scholar
  6. [H]
    O. Hyodo,On the de Rham-Witt complex attached to a semi-stable family, Compositio Mathematica78 (1991), 241–260.MATHMathSciNetGoogle Scholar
  7. [I-S]
    A. Iovita and M. Spiess,Logarithmic differential forms on p-adic symmetric spaces, Duke Mathematical Journal110 (2000), 253–278.CrossRefMathSciNetGoogle Scholar
  8. [M]
    A. Mokrane,La suite spectrale des poids en cohomologie de Hyodo-Kato, Duke Mathematical Journal72 (1993), 301–337.MATHCrossRefMathSciNetGoogle Scholar
  9. [S]
    P. Schneider,The cohomology of local systems on p-adically uniformized varieties, Mathematische Annalen293 (1992), 623–650.MATHCrossRefMathSciNetGoogle Scholar
  10. [S-S]
    P. Schneider and U. Stuhler,The cohomology of p-adic symmetric spaces, Inventiones Mathematicae105 (1991), 47–122.MATHCrossRefMathSciNetGoogle Scholar
  11. [S-S2]
    P. Schneider and U. Stuhler,Resolutions for smooth representations of the general linear group over a local field, Journal für die reine und angewandte Mathematik436 (1993), 19–32.MATHMathSciNetGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2003

Authors and Affiliations

  1. 1.Institute of MathematicsThe Hebrew University of JerusalemGivat Ram, JerusalemIsrael

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