Honda–Tate theorem
From HandWiki
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, http://projecteuclid.org/euclid.jmsj/1260463295
- Tate, John (1966), "Endomorphisms of abelian varieties over finite fields", Inventiones Mathematicae 2 (2): 134–144, doi:10.1007/BF01404549, ISSN 0020-9910, Bibcode: 1966InMat...2..134T
- 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, 179, Springer Berlin / Heidelberg, pp. 95–110, doi:10.1007/BFb0058807, ISBN 978-3-540-05356-9, http://www.numdam.org/item?id=SB_1968-1969__11__95_0
 |  |
