Eichler-Shimura homology, intersection numbers and rational structures on spaces of modular forms. (English) Zbl 0618.10020
Let \(S_{2m+1}(\Gamma)\) denote the space of holomorphic cusp forms of weight \(2m+2\) relative to a Fuchsian group \(\Gamma \subset SL_ 2({\mathbb{R}})\) of finite covolume and Sh: \(S_{2m+2}(\Gamma)\to H^ 1(\Gamma,E)\), \(E=Sym^ 2{\mathbb{C}}^ 2\), the embedding of Shimura.
In this paper it is shown as main result that the relative (hyperbolic) Poincaré series \(\theta_{m+1,\gamma}\) span \(S_{2m+2}(\Gamma)\) as a real vector space if and only if the decomposable classes span \(H_ 1(\Gamma,E^*_{{\mathbb{R}}})\), \(E^*_{{\mathbb{R}}}\) the dual vector space of the real vector space \(E_{{\mathbb{R}}}\). In order to prove this you have to reformulate the definition of Eichler-Shimura periods and to compare this version with more or less classically defined periods.
This paper refers to an earlier paper of the first author [Invent. Math. 80, 469-480 (1985; Zbl 0566.10018)] and some of the results obtained here are stronger versions or reformulations of results obtained in that earlier paper.
In this paper it is shown as main result that the relative (hyperbolic) Poincaré series \(\theta_{m+1,\gamma}\) span \(S_{2m+2}(\Gamma)\) as a real vector space if and only if the decomposable classes span \(H_ 1(\Gamma,E^*_{{\mathbb{R}}})\), \(E^*_{{\mathbb{R}}}\) the dual vector space of the real vector space \(E_{{\mathbb{R}}}\). In order to prove this you have to reformulate the definition of Eichler-Shimura periods and to compare this version with more or less classically defined periods.
This paper refers to an earlier paper of the first author [Invent. Math. 80, 469-480 (1985; Zbl 0566.10018)] and some of the results obtained here are stronger versions or reformulations of results obtained in that earlier paper.
Reviewer: M.Heep
MSC:
11F11 | Holomorphic modular forms of integral weight |
11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
55N25 | Homology with local coefficients, equivariant cohomology |
22E40 | Discrete subgroups of Lie groups |
30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
11F06 | Structure of modular groups and generalizations; arithmetic groups |
Keywords:
period of cusp form; closed geodesic; Kronecker index; cohomology class; local coefficients; decomposable cycles; Eichler-Shimura periodsCitations:
Zbl 0566.10018References:
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