Finiteness and the Tate Conjecture in Codimension 2 for K3 Squares

Abstract: Two years ago, via a refined CM lifting theory, Ito-Ito-Koshikawa proved the Tate Conjecture for squares of K3 surfaces over finite fields by reducing to Tate's theorem on the endomorphisms of abelian varieties. I will explain a different proof, which is based on a twisted version of Fourier-Mukai transforms between K3 surfaces. In particular, I do not use Tate's theorem after assuming some known properties of individual K3's. The main purpose of doing so is to illustrate Tate's insight on the connection between the Tate conjecture and the positivity results in algebraic geometry for codimension 2 cycles, through some "geometry in cohomological degree 2".

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic