From dimension four to dimension five in the Tate conjecture for abelian varieties over finite fields

Source abstract

Assume the Tate conjecture for abelian varieties of dimension at most four over finite fields. We prove it in dimension five, in every codimension and for every prime $\ell\ne p$. In dimension five we also prove standard conjecture~$D_\ell$ over $\overline{\mathbf F}_p$ and independence from $\ell$ of rational cycle class kernels over algebraically closed fields of characteristic $p$. An analysis of Künneth summands and Newton polygons reduces the proof to a block formed from an almost ordinary surface, an ordinary surface, and a supersingular elliptic curve. A CM half twist, a Morita decomposition, and Kimura nilpotence identify the Tate classes of this block with endomorphisms of the ordinary surface. Kahn's theorem then gives equality of rational and numerical equivalence and Parshin's conjecture over finite fields in dimension five.