Isogeny classes of abelian varieties over finite fields
Taira Honda · Journal of the Mathematical Society of Japan · 1968
In the present paper we shall give a complete classification of isogeny classes of abelian varieties over finite fields in terms of Frobenius endomor- phism and indicate some of its applications.Let $p$ be a fixed prime number and $\Omega$ the algebraic closure of the prime field of characteristic $p$ .Let $k_{a}$ denote the finite field with $p^{a}$ elements.We consider $ k_{a}\subset\Omega$ .By an algebraic number field we mean a subfield of the complex number field $C$ which is of finite degree over the rational number field $Q$ .We identify an ideal of an algebraic number field $K$ with its extensions to over-fields of $K$ as usual.We denote by $Z$ the ring of rational integers and by $Z_{l}$ its l-adic completion for a prime number $l$ .We shall say that an algebraic integer $\pi$ (resp.an integral ideal $\mathfrak{a}$ of an algebraic number field) is of type $(A_{0})$ , if we have $\pi^{\sigma}\pi^{\sigma\rho}=p^{a}$