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:
| [1] | Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. · Zbl 0052.29502 |
| [2] | Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. · Zbl 0443.22010 |
| [3] | K. Brown, Cohomology of groups, Graduate Texts in Math., vol. 87, Springer-Verlag, Berlin and New York, 1972. |
| [4] | William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200 – 225. · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9 |
| [5] | William M. Goldman and John J. Millson, Eichler-Shimura homology and the finite generation of cusp forms by hyperbolic Poincaré series, Duke Math. J. 53 (1986), no. 4, 1081 – 1091. · Zbl 0618.10021 · doi:10.1215/S0012-7094-86-05353-6 |
| [6] | Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. · Zbl 0408.14001 |
| [7] | D. Johnson and J. Millson, Deformation spaces associated to compact hyperbolic manifolds, Discrete groups in geometry and analysis, Birkhäuser (to appear). · Zbl 0613.53032 |
| [8] | Svetlana Katok, Closed geodesics, periods and arithmetic of modular forms, Invent. Math. 80 (1985), no. 3, 469 – 480. · Zbl 0566.10018 · doi:10.1007/BF01388727 |
| [9] | Stephen S. Kudla and John J. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979), no. 3, 193 – 211. · Zbl 0429.30038 · doi:10.1007/BF01390229 |
| [10] | Yozô Matsushima and Shingo Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric riemannian manifolds, Ann. of Math. (2) 78 (1963), 365 – 416. · Zbl 0125.10702 · doi:10.2307/1970348 |
| [11] | S. Murakami, Cohomology groups of vector-valued forms on symmetric spaces, Lecture Notes, University of Chicago, 1966. |
| [12] | Дискретные подгруппы групп Ли., Издат. ”Мир”, Мосцощ, 1977 (Руссиан). Транслатед фром тхе Енглиш бы О. В. Šварцман; Едитед бы Ѐ. Б. Винберг; Щитх а супплемент ”Аритхметициты оф ирредуцибле латтицес ин семисимпле гроупс оф ранк греатер тхан 1” бы Г. А. Маргулис. |
| [13] | Goro Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291 – 311 (French). · Zbl 0090.05503 · doi:10.2969/jmsj/01140291 |
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