Tate Conjecture And Finiteness of Abelian Varieties over Finite Field

TL;DR

This paper demonstrates that the Tate conjecture for abelian varieties over finite fields is equivalent to the finiteness of their isomorphism classes of a fixed dimension, offering a new perspective using Zarhin's results.

Contribution

It establishes the equivalence between the Tate conjecture and finiteness of isomorphism classes, providing a novel approach with existing results.

Findings

Proves the equivalence between Tate conjecture and finiteness of abelian varieties.

Introduces a new approach leveraging Zarhin's results.

Enhances understanding of the structure of abelian varieties over finite fields.

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.