The Tate conjecture for abelian fivefolds over finite fields

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~$D_{\mathrm{\ell }}$ over ${\bar{\mathbf{F}}}_p$ and independence of $\mathrm{\ell }$ 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.