# A vanishing theorem for modular symbols on locally symmetric spaces

### Toshiyuki Kobayashi

University of Tokyo, Japan### Tadao Oda

University of Tokyo, Japan

## Abstract

A modular symbol is the fundamental class of a totally geodesic submanifold $\Gamma'\backslash G'/K'$ embedded in a locally Riemannian symmetric space $\Gamma \backslash G / K$ , which is defined by a subsymmetric space $G'/ K' \hookrightarrow G / K$ . 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 $\pi$ -component $(\pi \in \widehat {G})$ in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction $\pi |_{G'}$ . 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