Weil polynomial (awaiting review)

For a prime power $q$, a Weil $q$-polynomial is a monic polynomial with integer coefficients whose complex roots are of absolute value $\sqrt{q}$.

Given $q$ and a nonnegative integer $d$, there are only finitely many Weil $q$-polynomials of degree $d$.

The characteristic polynomial of an abelian variety over $\F_q$ is a Weil $q$-polynomial, but it is not quite true that every Weil $q$-polynomial arises in this way. Every irreducible Weil $q$-polynomial has a unique power that is the characteristic polynomial of a simple abelian variety over $\F_q$; it is the products of these powers that arise from abelian varieties.