Tate conjecture and mixed perverse sheaves

2006/03/27 by Morihiko Saito, Saito, Morihiko
Mathematics · #14C25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #math.AG #msc:14C25

paper · pdf · doi:10.48550/arxiv.math/0603622

24 pages

openalex publication_date 2006/03/27 · arxiv created 2007/06/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the theory of mixed perverse sheaves, we extend arguments on the Hodge conjecture initiated by Lefschetz and Griffiths to the case of the Tate conjecture, and show that the Tate conjecture for divisors is closely related to the de Rham conjecture for nonproper varieties, finiteness of the Tate-Shafarevich groups, and also to some conjectures in the analytic number theory.

Related

• The Tate and Standard Conjectures for Certain Abelian Varieties
• A sheaf-theoretic reformulation of the Tate conjecture
• A survey around the Hodge, Tate and Mumford-Tate conjectures for abelian varieties
• On the Tate conjecture for divisors on varieties with h^2,0 = 1 in positive characteristics
• A Survey of the Hodge Conjecture for Abelian Varieties