2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116539This article surveys some recent work of the author on Hilbert modular fourfolds X. After some preliminaries on the cohomology and special, codimension 2 cycles Z on X of Hirzebruch-Zagier type, a proof of the Tate conjecture for X over abelian fields is exposed. It is followed by a description of a method to construct classes in the Bloch's Chow group CH^3(X,1) by using Hecke translates of the cycles Z as above with suitable intersections in translates of modular curves. The article ends with the introduction of a modular Gersten complex for a general Shimura variety X and the corresponding groups CH^p_{mod}(X) and CH^p_{mod}(X,1).15 pages, Proceedings of a conference on Algebraic Cycles and Motives (in honor of Jacob Murre) in Leiden, The NetherlandsNumber TheoryAlgebraic Geometry11F41; 11G35; 14C25; 14C35; 14J35text