The congruence relation in the non-PEL case (Q2781432)

The so-called Eichler-Shimura congruence relation for certain Shimura varieties is a conjectural, rather involved statement concerning the structure of a naturally associated Galois module. This conjecture is suggested by Langlands's philosophy on global \(L\)-parameters, as it has been thoroughly analyzed in the work of D. Blasius and J. D. Rogawski ten years ago [cf. \textit{D. Blasius} and \textit{J. D. Rogawski}, Zeta functions of Shimura varieties, Proc. Symp. Pure Math. 55, 525--571 Part 2, (1994)]. The paper under review settles the Eichler-Shimura congruence relation (á la Blasius-Rogawski) for certain five-dimensional Hodge-type Shimura varieties by a completely new, highly innovating approach. Based upon results obtained in his 1997 Oxford thesis [cf.: O. Bültel, On the mod \({\mathfrak P}\)-reduction of ordinary CM-points, Ph.D. thesis, Oxford University, UK (1997)], and introducing a stronger hypothesis called (NVC) on the behavior of Hecke correspondences, the author proves the validity of the Eichler-Shimura congruence relation for his particular Shimura 5-folds by verifying the (NVC)-hypothesis for them. As for the proof of the crucial fact that (NVC) implies the conjectural Eichler-Shimura congruence relation, a theorem of R. Noot on CM-lifts of ordinary points in positive characteristic [cf.: R. Noot, J. Alg. Geom. 5, 187--207 (1996; Zbl 0864.14015)] is used as a major tool, along with a refined analysis of the mod \({\mathfrak P}\)-reductions of various Hecke translates of those CM-lifts. In view of the particular algebraic-geometric prerequisites, the author has added two appendices providing relevant facts on the algebra of correspondences and on specializations of algebraic cycles, respectively.