Recent progress on the Tate conjecture

Abstract

Abstract. We survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely related with other big problems in arithmetic and algebraic geometry, including the Hodge and Birch-Swinnerton-Dyer conjectures. We conclude by discussing the recent proof of the Tate conjecture for K3 surfaces over finite fields.

Curves on K3 surfaces

“…Moreover, by [Shi74,§2], the Picard rank of a uniruled K3 surface over an algebraically closed field is equal to its second Betti number, which is equal to 22, i.e., the K3 surface is supersingular (in the sense of Shioda). Since the Tate conjecture for K3 surfaces is by now fully established, Shioda's notion of supersingular coincides with the others (height of the formal Brauer group [Art74] or in terms of the Newton polygon on cohomology), see [Tot17]. We note that supersingular K3 surfaces form 9-dimensional families [Art74] and are thus very special, even in positive characteristic.…”

Noncommutative counterparts of celebrated conjectures

“…Conjecture T l (X): The cycle class map (1.7) is surjective. This conjecture holds when d ≤ 1, when X is an abelian variety and d ≤ 3, and also when X is a K3-surface; consult Totaro's survey [62]. Besides these cases (and some other cases scattered in the literature), it remains wide open.…”

A note on Tate's conjectures for abelian varieties

“…Much less is known when r > 1. We refer to the surveys [Tot17,Mil07,Tat94,Ram89] for a nice summary of known results. The goal of this short note is to present some examples of abelian varieties X over number fields for which Tate's conjectures hold for algebraic cycles in arbitrary codimension r.…”