The characteristic polynomials of geometrically split ordinary abelian varieties of dimension $3$

In the paper, we explicitly describe all possible characteristic polynomials of the Frobenius endomorphism for ordinary geometrically decomposable Abelian varieties of dimension $3$ over a finite field. These polynomials encode many arithmetic properties of abelian varieties including number of points. More precisely, if $\chi_{A,{q}}(T)$ is the characteristic polynomial of the Frobenius endomorphism on $A$ over $\mathbb{F}_q$, then the number of points on $A$ is equal to $\chi_{A,{q}}(1)$.