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arXiv · 2609.06265

The Tate conjecture for abelian fivefolds over finite fields

Ningyi Li
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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.

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Ningyi Li. 2026-09-05. The Tate conjecture for abelian fivefolds over finite fields. https://arxiv.org/abs/2609.06265

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