发表机构
Faculty of Mathematics, University of Regensburg（雷根斯堡大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在假设维数至多四的阿贝尔簇满足Tate猜想的前提下，证明了维数五阿贝尔簇在所有余维数和素数下的Tate猜想，并借助Künneth分解、CM半扭转及Kahn定理等工具，同时建立了标准猜想与Parshin猜想。
AI 中文摘要
假设Tate猜想对有限域上维数至多为四的阿贝尔簇成立。我们在维数五中证明该猜想，涵盖所有余维数及每个素数$\ell\ne p$。在维数五中，我们还证明了在$\overline{\mathbf F}_p$上的标准猜想$D_\ell$，以及在特征为$p$的代数闭域上有理环类核的$\ell$无关性。对Künneth求和项和Newton多边形的分析将证明归结为由一个几乎普通曲面、一个普通曲面和一条超奇异椭圆曲线构成的块。一个CM半扭转、一个Morita分解和Kimura幂零性将该块的Tate类与普通曲面的自同态等同起来。随后，Kahn定理给出了有理等价与数值等价的相等性，以及有限域上维数五的Parshin猜想。
英文摘要
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.
CommentsWithdrawn 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
