AI 中文总结
该研究证明了有限域上阿贝尔四维簇的Tate猜想，结合相关定理推导出任意域上阿贝尔四维簇的同调与数值等价标准猜想成立，完成了阿贝尔四维簇标准猜想的证明。
AI 中文摘要
我们证明了有限域上阿贝尔四维簇的Tate猜想。这是自20世纪60年代Tate的工作以来，首次解决有限域上固定维数所有阿贝尔簇的该猜想。该证明依赖于Ancona证明阿贝尔四维簇的Hodge型标准猜想的技术，最终归结为Markman关于复阿贝尔簇上Weil类代数性的结果。将上述Tate猜想的情况与Ancona和Kahn的定理结合，我们推导出任意域上阿贝尔四维簇的同调等价与数值等价的标准猜想成立，这完成了阿贝尔四维簇标准猜想的证明。
英文摘要
We prove the Tate conjecture for abelian fourfolds over finite fields. This is the first resolution of the conjecture for all abelian varieties of a fixed dimension over finite fields since the work of Tate in the 1960s. The proof relies on techniques from Ancona's proof of the standard conjecture of Hodge type for abelian fourfolds, and ultimately reduces to Markman's results on the algebraicity of Weil classes on complex abelian varieties. Combining the above case of the Tate conjecture with theorems of Ancona and Kahn, we deduce that the standard conjecture on homological versus numerical equivalence holds for abelian fourfolds over arbitrary fields. This completes the proof of the standard conjectures for abelian fourfolds.
Comments25 pages. Comments welcome
