JournalscmhVol. 73, No. 1pp. 45–70

A vanishing theorem for modular symbols on locally symmetric spaces

  • Toshiyuki Kobayashi

    University of Tokyo, Japan
  • Tadao Oda

    University of Tokyo, Japan
A vanishing theorem for modular symbols on locally symmetric spaces cover
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Abstract

A modular symbol is the fundamental class of a totally geodesic submanifold Γ\G/K\Gamma'\backslash G'/K' embedded in a locally Riemannian symmetric space Γ\G/K\Gamma \backslash G / K , which is defined by a subsymmetric space G/KG/KG'/ K' \hookrightarrow G / K . In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the π\pi -component (πG^)(\pi \in \widehat {G}) in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction πG\pi |_{G'} . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV.

Cite this article

Toshiyuki Kobayashi, Tadao Oda, A vanishing theorem for modular symbols on locally symmetric spaces. Comment. Math. Helv. 73 (1998), no. 1, pp. 45–70

DOI 10.1007/S000140050045