Tate Conjecture And Finiteness of Abelian Varieties over Finite Field

Abstract

In this paper we will prove that Tate conjecture of abelian varieties over finite field is equivalent to the finiteness of isomorphism classes of abelian varieties with a fixed dimension. We give a different approach with Zarhin's result.

I didn't find Zarhin's paper when I prove the result. So I give a corrected version here. This result is first proved by Zarhin. And this note only gives a different method to prove the same result. Zarhin's paper can be found in the reference