Point counts of abelian varieties over finite fields determining their zeta function

Let $A$ be an abelian variety of dimension $g$ over a finite field ${\mathbf{F}}_q$. We show that if $q$ is sufficiently large relative to $g$, the $g$ point counts $\mathrm{\#}A({\mathbf{F}}_{q^i})$ for $1\le i\le g$ determine the zeta function of $A$, equivalently the characteristic polynomial of its Frobenius endomorphism, and hence the isogeny class of $A$. This count is best possible for $g=2$ and $g=4$, but not in general: for $g=3$ two point counts already determine the zeta function, whereas a single count never does. The proof combines the functional equation of the $L$-polynomial with Newton’s identities and an inductive error analysis that controls the power sums of the inverse Frobenius eigenvalues with enough precision to recover them, as integers, by rounding.