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**A vanishing theorem for modular symbols on locally symmetric spaces.**
*(English)*
Zbl 0933.11027

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 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 \(L\)-functions (see for example the work of G. van der Geer [Hilbert modular surfaces, Berlin, Springer (1988; Zbl 0634.14022)] and that of the second author [Periods of Hilbert modular surfaces, Progress in Mathematics, Vol. 19. Boston, Birkhäuser (1982; Zbl 0489.14014)]). In general, however, our knowledge of these generalized modular symbols is still quite limited. In fact, much of the literature is devoted to showing the nonvanishing of these generalized modular symbol themselves.

In this paper, the authors give sufficient conditions that the modular symbol \(\iota_*[\Gamma'\backslash G' /K']\) is annihilated by the \(\pi\)-component \(H^{d(G')}(\pi:\Gamma)\) (\(\subset H^{d(G')}_{\text{de Rham}}(\Gamma\backslash G/K,\mathbb C)\)) for \(\pi \in \hat G\) where \(d(G') = \dim(G'/K')\). Here \(\iota_*\) denotes the induced map of homologies \(\iota_* : H_{d(G')}(\Gamma'\backslash G' /K',\mathbb C) \rightarrow H_{d(G')}(\Gamma\backslash G/K, \mathbb C)\) and we think of \(\iota_*[\Gamma'\backslash G' /K']\) as sitting inside \(H^{d(G')}_{\text{de Rham}} (\Gamma\backslash G/K,\mathbb C)\) via the Poincaré duality map.

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 \(L\)-functions (see for example the work of G. van der Geer [Hilbert modular surfaces, Berlin, Springer (1988; Zbl 0634.14022)] and that of the second author [Periods of Hilbert modular surfaces, Progress in Mathematics, Vol. 19. Boston, Birkhäuser (1982; Zbl 0489.14014)]). In general, however, our knowledge of these generalized modular symbols is still quite limited. In fact, much of the literature is devoted to showing the nonvanishing of these generalized modular symbol themselves.

In this paper, the authors give sufficient conditions that the modular symbol \(\iota_*[\Gamma'\backslash G' /K']\) is annihilated by the \(\pi\)-component \(H^{d(G')}(\pi:\Gamma)\) (\(\subset H^{d(G')}_{\text{de Rham}}(\Gamma\backslash G/K,\mathbb C)\)) for \(\pi \in \hat G\) where \(d(G') = \dim(G'/K')\). Here \(\iota_*\) denotes the induced map of homologies \(\iota_* : H_{d(G')}(\Gamma'\backslash G' /K',\mathbb C) \rightarrow H_{d(G')}(\Gamma\backslash G/K, \mathbb C)\) and we think of \(\iota_*[\Gamma'\backslash G' /K']\) as sitting inside \(H^{d(G')}_{\text{de Rham}} (\Gamma\backslash G/K,\mathbb C)\) via the Poincaré duality map.

Reviewer: Kevin L.James (University Park/PA)

### MSC:

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

22E40 | Discrete subgroups of Lie groups |