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Honda–Tate theorem

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In mathematics, the Honda–Tate theorem classifies abelian varieties over finite fields up to isogeny. It states that the isogeny classes of simple abelian varieties over a finite field of order q correspond to algebraic integers all of whose conjugates (given by eigenvalues of the Frobenius endomorphism on the first cohomology group or Tate module) have absolute value q.

Tate (1966) showed that the map taking an isogeny class to the eigenvalues of the Frobenius is injective, and Taira Honda (1968) showed that this map is surjective, and therefore a bijection.

References

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  • Honda, Taira (1968), "Isogeny classes of abelian varieties over finite fields", Journal of the Mathematical Society of Japan, 20 (1–2): 83–95, doi:10.2969/jmsj/02010083, ISSN 0025-5645, MR 0229642
  • Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae, 2 (2): 134–144, Bibcode:1966InMat...2..134T, doi:10.1007/BF01404549, ISSN 0020-9910, MR 0206004, S2CID 245902
  • Tate, John (1971), "Classes d'isogénie des variétés abéliennes sur un corps fini (d'après T. Honda)", Séminaire Bourbaki vol. 1968/69 Exposés 347-363, Lecture Notes in Mathematics, vol. 179, Springer Berlin / Heidelberg, pp. 95–110, doi:10.1007/BFb0058807, ISBN 978-3-540-05356-9