发表机构
University of Regensburg（雷根斯堡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了有限域上维数至多四的阿贝尔簇的任意幂的Tate猜想，以及复CM阿贝尔四维簇的任意幂的Hodge猜想，并由此推出标准猜想D和zeta函数极点阶数。
AI 中文摘要
我们证明了在有限域上维数至多为四的阿贝尔簇的任意幂的每个余维数上的Tate猜想。对于几何简单的四维簇，我们使用了Broe定理和Frobenius关系的分类。对于阿贝尔曲面的二面体对，我们利用阿贝尔三维簇族、Gross--Schoen高度公式和极化收缩构造了代数类。我们还利用Markman关于Weil类的定理和Milne准则证明了复CM阿贝尔四维簇的任意幂的Hodge猜想。有限域结果蕴含了标准猜想D以及zeta函数在每个幂上的预期极点阶数。
英文摘要
We prove the Tate conjecture in every codimension on every power of an abelian variety of dimension at most four over a finite field. For geometrically simple fourfolds, we use Broe's theorem and the classification of Frobenius relations. For dihedral pairs of abelian surfaces, we construct algebraic classes using families of abelian threefolds, the Gross--Schoen height formula, and polarization contractions. We also prove the Hodge conjecture for every power of a complex CM abelian fourfold, using Markman's theorem on Weil classes and Milne's criterion. The finite field results imply standard conjecture~D and the expected pole orders of the zeta function on every power.
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
