Title: Correspondences for Hecke rings and $l$-adic cohomology groups on smooth compactifications of Siegel modular varieties
Abstract:Introduction.We report that the Hecke rings act on the/-adic coho- mology groups of suitable non-singular projective toroidal compactifica- tions of the higher dimensional modular varieties.We extend ...Introduction.We report that the Hecke rings act on the/-adic coho- mology groups of suitable non-singular projective toroidal compactifica- tions of the higher dimensional modular varieties.We extend the fixed point theory of Lefschetz to the correspondences for the Hecke rings on those compactifications.We treat here the Siegel modular case.For details see Hatada [6], which will appear elsewhere.1. Let g_> 1, w_> 0, ]>_ 1, k_> 1, and N_> 3 be rational integers.Let denote a ring.Write M,() =the set of ] k matrices with coefficients in =M,()1 0 g)=the Siegel upper half plane of degree g {Z e M,(C) Z =Z.Im Z is positive definite.}Sp(g, Z)--the full symplectic modular group M,,(Z) GSp (g, R) {r e GL (2g, R) I"(JrJ is a scalar matrix whose eigenvalue is positive.}GSp (g, Z)=GSp (g, R) M,(Z) r(a)=the eigenvalue of oJoJfor a e GSp (g, R) GSp + (g, R) < R the semi-direct product of GSp (g, R) and R w with R normal such that m).n) + n 0 for all m and n e R " and all and / e GSp (g, R). (In the right side the products are those for matrices.)We let GSp (g, R) R :w act on the com- plex analytic space g) X C w={(Z, x, 2, ..., w) lZ e , e C for any ] e [1, w].) to the left as follows.Write m= (m, n, m2, n2, ., row, n,) e R :w with and neR" for any ]e [1, w], and write =( ))eGSp +(g,R) partitioned into blocks on dimension g Xg.Then (( B)(ml,// m2, n2, rnw, nw)) (Z, ,, :,No. 2Read More