A vanishing theorem for modular symbols on locally symmetric spaces
type: Item
instance of: scholarly article
title: A vanishing theorem for modular symbols on locally symmetric spaces
zbMATH Open document ID: 0933.11027
author: Toshiyuki Kobayashi
author: Takayuki Oda
published in: Commentarii Mathematici Helvetici
publication date: 1998-06-14T00:00:00Z
review text: The theory of modular symbols was introduced and extensively studied for elliptic modular curves \(C = \Gamma \backslash \mathfrak h\) in the work of Manin and Drinfeld and Mazur and Swinnerton-Dyer. A modular symbol is defined to be a chain or cycle on the compactification \(C^{*}\) of \(C\), obtained as the images of arcs on \(\mathfrak h\) joining two cusps with respect to \(\Gamma \subset SL_2(\mathbb Z)\). This theory has been applied to understanding the special values of modular \(L\)-functions and to the construction of \(p\)-adic \(L\)-functions of modular forms. These same cycles also appear in the case of half-integral weight forms (see for example the work of \textit{T. Shintani} [Nagoya Math. J. 58, 83-126 (1975; Zbl 0316.10016)]. Since the late 70's, there have been attempts to push forward the notion of generalized modular symbols for higher dimensional arithmetic quotients. The definitions are as follows. Let \(G\) be a semisimple Lie group and \(\Gamma\) a discrete subgroup such that the volume vol\((\Gamma\backslash G) < \infty\). Let \(G'\) be a closed subgroup of \(G\) such that \(\Gamma' := G' \cap \Gamma\) is also discrete and co-volume finite in \(G'\). In this paper the authors further restrict themselves to the cocompact case. Then the inclusion map induces a map of real analytic varieties of double cosets \(\iota: \Gamma'\backslash G' /K' \rightarrow \Gamma\backslash G/K\), where \(K\) is a maximal compact subgroup of \(G\) and \(K' := K\cap G'\). The image of \(\iota\) defines a chain \(C_{(G',\iota)}\) on \(\Gamma\backslash G/K\), and when \(\Gamma\backslash G\) is compact, \(C_{(G',\iota)}\) is a cycle on \(\Gamma\backslash G/K\). When \(\Gamma\backslash G\) is non-compact, \(C_{(G',\iota)}\) can be compactified to give a cycle on the natural compactification of \(\Gamma\backslash G/K\). In some special cases, the periods of automorphic forms along generalized modular symbols are represented as the special values of automorphic \(L\)-fun…
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 1146262
zbMATH Keywords: vanishing theorem
zbMATH Keywords: locally symmetric spaces
zbMATH Keywords: modular symbols
zbMATH Keywords: generalized modular symbols
zbMATH Keywords: higher-dimensional arithmetic quotients
reviewed by: Kevin James
MaRDI profile type: Publication
full work available at URL: s000140050045
OpenAlex ID: W2028605356
DOI: 10.1007/S000140050045
description: scientific article; zbMATH DE number 1146262