Honda–Tate theorem

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

• 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