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
