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The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds

  • Ningyi Li
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{li2026tate,
  title = {{The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds}},
  author = {Ningyi Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.27916},
  eprint = {2609.27916},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27916},
}
RIS
TY  - UNPB
TI  - The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds
AU  - Ningyi Li
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.27916
UR  - https://arxiv.org/abs/2609.27916
AB  - 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.
ER  -