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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