A simple intrinsic proof of the Tate conjecture for K3's of finite height

In the past decade, a major triumph for the Tate conjecture (over finite fields) is its resolution for K3 surfaces. However, all known proofs rely crucially on the Kuga-Satake construction—effectively outsourcing certain difficulties to the theory of abelian varieties. There is an alternative approach which seeks to prove the conjecture by linking it to finiteness statements in arithmetic geometry, and using only the geometry of K3 surfaces. The spirit of this approach originated in Artin and Swinnerton-Dyer’s proof for elliptic K3's and has recently been revitalized by Lieblich-Maulik-Snowden and Charles through the moduli theory of (twisted) sheaves. In particular, Charles gave an intrinsic proof assuming the Picard number is at least 2. In this talk, I will explain an intrinsic proof that finite height K3's a priori have an even geometric Picard number, and hence proves one step further in the second approach.