Correspondences for Hecke rings and \(\ell\)-adic cohomology groups on smooth compactifications of Siegel modular varieties (Q1264175)

For smooth projective complex algebraic varieties M of complex dimension d and maps A, B: \(M\to M\) one can consider the coincidence number \(I_ M(A,B)=Tr((A\times B)^*(\Delta))\), where \(\Delta \in H^{2d}(M\times M, {\mathbb{Q}})\) is the class of the diagonal and Tr:\(H^{2d}(M, {\mathbb{Q}})\simeq {\mathbb{Q}}\) is the trace map. Fix a basis \(e_ i^{(\nu)}\) on \(H^{\nu}(M, {\mathbb{Q}})\) and a dual basis \(f_ i^{(2d-\nu)}\) on \(H^{2d-\nu}(M,{\mathbb{Q}})\) with respect to Poincaré duality. Suppose \(A^*(e_ i^{(\nu)})=\sum a_{ji}^{(\nu)}e_ j^{(\nu)}\) and \(B^*(f_ i^{(2d-\nu)})=\sum b_{ji}^{(\nu)}f_ j^{(2d-\nu)}\), then using Poincaré duality and the Künneth formula one can easily compute the coincidence number as a Lefschetz trace \(\sum_{\nu,i,j}(- 1)^{\nu}a_{ji}^{(\nu)}b_{ji}^{(\nu)}\). The author announces the extension of this to a pair A, B of Hecke correspondences in the special case, where M is a toroidal compactification of (a product of the universal abelian variety over the) Siegel moduli space.