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Correspondences for Hecke rings and (co-)homology groups on smooth compactifications of Siegel modular varieties. (English) Zbl 0716.11021

This paper is devided in two parts: in the first one the author uses a previously introduced representation \(f_{\bullet}\) of the Hecke ring \(HR(\Gamma,GSp^+(g,{\mathbb{Z}}))\) on the homology \(H_{\bullet}(M,{\mathbb{Z}})\) of a simultaneous smooth projective toroidal compactification M of \(\Gamma \rtimes {\mathbb{Z}}^{2g}\setminus {\mathbb{H}}_ g\times {\mathbb{C}}^ g,\) where \({\mathbb{H}}_ g\) is Siegel’s upper half plane of degree g and \(\Gamma\) a principal congruence subgroup of level \(\geq 3\) in Sp(g,\({\mathbb{Z}})\), to extend Lefschetz fixed point theory. In the second part an action of this ring on \(\ell\)-adic cohomology on a smooth compactification of the Siegel modular variety \(\Gamma \setminus {\mathbb{H}}_ g\) is constructed, compatible to the former \(f_{\bullet}\) through the comparison theorem of \(\ell\)-adic cohomology \(H^{\bullet}(M,{\mathbb{C}})\cong H^{\bullet}(M,{\mathbb{Q}}_ 1)\otimes_{{\mathbb{Q}}_ 1}{\mathbb{C}}.\)
To be more precise, first any nonzero element \({\mathcal A}\) of the Hecke ring induces in a natural way a certain locally analytic subvariety \({\mathcal Z}_ M({\mathcal A})\) of dimension \(d=\dim M\) in \(M\times M\). Furthermore for \({\mathcal A}_ 1,{\mathcal A}_ 2\) of this ring define coincidence numbers \(L_ M({\mathcal A}_ 1,{\mathcal A}_ 2)\) by \[ \sum^{2d}_{n=0}(-1)^ nTr\{D_ n\circ^ t(f_{2d-n}({\mathcal A}_ 2)\otimes id)\circ D_ n^{-1}\circ (f_ n({\mathcal A}_ 1)\otimes id)\} \] and coincidence indices \(I_ M({\mathcal A}_ 1,{\mathcal A}_ 2)\in H_{2d}(M\times M,M\times M-\Delta,{\mathbb{Z}})\cong {\mathbb{Z}}\) respectively as usual via the fundamental and Thom class of M and a certain natural \({\mathbb{Z}}\)-bilinear pairing \(F_ n\) on the Hecke ring with values in \(Hom_{{\mathbb{Z}}}(H_{\bullet}(M,{\mathbb{Z}}),H_{\bullet}(M\times M,{\mathbb{Z}}))\). Of course, \(D_ n\) is the \({\mathbb{Q}}\)-linear isomorphisms given by Poincaré duality. Then these numbers coincide, i.e. \(L_ M({\mathcal A}_ 1,{\mathcal A}_ 2)=I_ M({\mathcal A}_ 1,{\mathcal A}_ 2)\) (Th. 2) and \(Z_ M({\mathcal A}_ 1)\) meets \(Z_ M({\mathcal A}_ 2)\) if \(L_ M({\mathcal A}_ 1,{\mathcal A}_ 2)\) does not vanish (Th. 3). The proofs are based on former thorough studies of M by the author, the well-known fact due to Lojasiewicz, that a compact complex manifold may be (analytically) triangulated and more or less standard arguments in homology theory. Concerning the \(\ell\)-adic results the theory of strict quotients is used as well as fundamental results of Deligue, Artin and other people on étale topology. In an additional correction [cf. ibid. 13, No.2, 493 (1990)] several printing errors are listed.
Reviewer: F.W.Knoeller

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F85 \(p\)-adic theory, local fields
14F20 Étale and other Grothendieck topologies and (co)homologies
55M20 Fixed points and coincidences in algebraic topology
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