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.

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