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Hodge and Mumford–Tate groups of an Abelian Variety, Complex Multiplication, and Frobenius Elements

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References

  1. B. B. Gordon, A Survey of the Hodge Conjecture, in CRM Monograph Series, Ed. by J. D. Lewis (Amer. Math. Soc., Providence, RI, 1999), Vol. 10, pp. 297–356.

    Google Scholar 

  2. B. Moonen, Notes on Mumford–Tate Groups (Centre Emile Borel, Paris, 1999).

    Google Scholar 

  3. J. Carlson, S. Müller-Stach, and C. Peters, Period Mappings and Period Domains, in Cambridge Stud. in Adv. Math. (Cambridge Univ. Press, Cambridge, 2018), Vol. 168.

    MATH  Google Scholar 

  4. B. Moonen, An Introduction to Mumford–Tate Groups (2004).

    Google Scholar 

  5. P. Deligne, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. (Springer- Verlag, Berlin–New York, 1982), Vol. 900, pp. 9–100.

    Chapter  MATH  Google Scholar 

  6. D. Mumford, Abelian Varieties, in Tata Institute of Fund. Research Stud. in Math. (Oxford University Press, London, 1970), Vol. 5.

    MATH  Google Scholar 

  7. S. Lang, Complex Multiplication (Springer- Verlag, New York, 1983).

    Book  MATH  Google Scholar 

  8. P. van Wamelen, Math. Comp. 68 (225), 307 (1999).

    Article  MathSciNet  Google Scholar 

  9. G. Shimura and Yu. Taniyama, Complex Multiplication of Abelian Varieties and its Applications to Number Theory (Math. Soc. of Japan, Tokyo, 1961).

    MATH  Google Scholar 

  10. D. Mumford, Math. Ann. 181, 345 (1969).

    Article  MathSciNet  Google Scholar 

  11. G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, in Princeton Math. Ser. (Princeton Univ. Press, Princeton, NJ, 1988), Vol. 46.

    MATH  Google Scholar 

  12. J.-P. Serre and J. Tate, Ann. of Math. (2) 88 (3), 492 (1968).

    Article  MathSciNet  Google Scholar 

  13. I. I. Pyatetskii-Shapiro, Math. USSR-Sb. 14 (4), 615 (1971).

    Article  Google Scholar 

  14. M. V. Borovoi, in Problems of Group Theory and Homological Algebra (Yaroslavl Gos. Un-t, Yaroslavl, 1977), Vol. 1, pp. 3–53.

    Google Scholar 

  15. J.-P. Serre, Abelian \(l\)-adic Representations and Elliptic Curves (Benjamin, New York– Amsterdam, 1968).

    Google Scholar 

  16. Yu. Taniyama, J. Math. Soc. Japan 9 (3), 330 (1957).

    Article  MathSciNet  Google Scholar 

  17. J. S. Milne, Complex Multiplication (2020 http://www.jmilne.org/math/CourseNotes/).

    Google Scholar 

  18. F. A. Bogomolov, C. R. Acad. Sci. Paris Ser. A-B 290 (15), 701 (1980).

    MathSciNet  Google Scholar 

  19. W. Ch. Chi, Amer. J. Math. 114 (2), 315 (1992).

    Article  MathSciNet  Google Scholar 

  20. J. Tate, Invent. Math. 2 (2), 134 (1966).

    Article  MathSciNet  Google Scholar 

  21. J. Tate, in Séminaire Bourbaki, v. 352, Lecture Notes in Math. (Springer, Berlin, 1971), Vol. 175, pp. 95–110.

    Google Scholar 

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Correspondence to S. G. Tankeev.

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Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 622–625 https://doi.org/10.4213/mzm13804.

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Tankeev, S.G. Hodge and Mumford–Tate groups of an Abelian Variety, Complex Multiplication, and Frobenius Elements. Math Notes 113, 601–604 (2023). https://doi.org/10.1134/S0001434623030331

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