2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139700These are my notes for a talk at the The Tate Conjecture workshop at the American Institute of Mathematics in Palo Alto, CA, July 23--July 27, 2007, somewhat revised and expanded. The intent of the talk was to review what is known and to suggest directions for research. v2: Revised expanded (24 pages).A pdf file with better fonts, style file, etc. is available at http://www.jmilne.org/math/ v2: Revised expanded (24 pages)Algebraic GeometryNumber Theory14C25; 11G25text