The Tate conjecture for abelian fivefolds over finite fields

Abstract

We prove the Tate conjecture for abelian fivefolds over finite fields. The proof constructs correspondences for a residual motive using a Moret--Bailly family, Gross--Schoen heights, and monodromy. We also prove standard conjecture~$D_{\ell }$ over ${\overline{\mathbf{F}}}_p$ and independence of $\ell$ of rational cycle class kernels over algebraically closed fields of characteristic $p$. Over finite fields, rational and numerical equivalence agree with rational coefficients, and higher algebraic $K$-groups vanish rationally.

arXiv commentWithdrawn by the author following a reassessment of the paper's contribution in light of recent work on the Hodge and Tate conjectures for abelian varieties. The author no longer plans to pursue this manuscript as a standalone publication and intends to develop selected methods in a separate paper