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On the Hodge, Tate, and standard conjectures for abelian varieties

Primarily human-written textHuman understanding: all partsmath.AG — Algebraic Geometry

Contributed by James Milne ↗

Version 1 / Nov 10, 2026 / CC BY 4.0

Abstract

This is a report on two preprints of OpenAI, " Milne's rationality conjecture for abelian varieties" and "The rational Hodge conjecture for CM abelian varieties", placing them in context.

Provenance statement

Totally human written.

References

  1. Bergeron, Nicolas; Millson, John; Moeglin, Colette. The Hodge conjecture and arithmetic quotients of complex balls. Acta Math., vol. 216, no. 1, pp. 1–125. 2016
  2. Deligne, Pierre. Hodge cycles on abelian varieties (Notes by J.S. Milne). Hodge cycles, motives, and Shimura varieties, pp. 9–100. 1982
  3. Hazama, Fumio. Hodge cycles on abelian varieties of SnS_n-type. J. Algebraic Geom., vol. 9, no. 4, pp. 711–753. 2000
  4. Hazama, Fumio. On the general Hodge conjecture for abelian varieties of CM-type. Publ. Res. Inst. Math. Sci., vol. 39, no. 4, pp. 625–655. 2003
  5. Mark Kisin; Rong Zhou. Strongly compatible systems associated to semistable abelian varieties. 2025. arXiv:2505.02165arXiv
  6. Langlands, R. P.; Rapoport, M. Shimuravarietäten und Gerben. J. Reine Angew. Math., vol. 378, pp. 113–220. 1987
  7. Liu, Yifeng. Relative trace formulae toward Bessel and Fourier-Jacobi periods on unitary groups. Manuscripta Math., vol. 145, no. 1-2, pp. 1–69. 2014
  8. Liu, Yifeng. Fourier-Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu). Camb. J. Math., vol. 9, no. 1, pp. 1–147. 2021. Appendix by Chao Li and Yihang Zhu
  9. Madapusi, Keerthi. Milne's rationality conjecture via simultaneous lifting of Weil classes and endomorphisms. 2026. hexagon:2610.00226v1
  10. Milne, J. S. Motives over finite fields. Motives (Seattle, WA, 1991), vol. 55, Part 1, pp. 401–459. 1994
Show all 19 referencesShow fewer references
  1. Milne, J. S. Lefschetz classes on abelian varieties. Duke Math. J., vol. 96, no. 3, pp. 639–675. 1999
  2. Milne, J. S. Lefschetz motives and the Tate conjecture. Compositio Math., vol. 117, no. 1, pp. 45–76. 1999
  3. Milne, J. S. Polarizations and Grothendieck's standard conjectures. Ann. of Math. (2), vol. 155, no. 2, pp. 599–610. 2002
  4. Milne, J. S. Rational Tate classes. Mosc. Math. J., vol. 9, no. 1, pp. 111–141. 2009
  5. Milne, J. S. Addendum to ``Values of zeta functions of varieties over finite fields''. Amer. J. Math., vol. 137, no. 6, pp. 1703–1712. 2015
  6. Milne, J. S. Tannakian categories. Cambridge University Press, Cambridge. 2027. To appear
  7. Serre, Jean-Pierre. Propriétés conjecturales des groupes de Galois motiviques et des représentations ll-adiques. Motives (Seattle, WA, 1991), vol. 55, Part 1, pp. 377–400. 1994
  8. van Hoften,Pol; Sempliner, Jack. The Hodge Conjecture for CM abelian varieties. 2026. hexagon:2610.00259v2
  9. Zink, Thomas. Isogenieklassen von Punkten von Shimuramannigfaltigkeiten mit Werten in einem endlichen Körper. Math. Nachr., vol. 112, pp. 103–124. 1983

Version history

  1. v1 · 10 pagesSubmitted by James MilneInitial depositCurrentNov 09, 2026