Characteristic polynomial of an abelian variety over a finite field (reviewed)

Let $A$ be an abelian variety of dimension $g$ defined over $\F_q$. Fix a prime $\ell \nmid q$. The $q$-power Frobenius endomorphism $F$ of $A$ induces an endomorphism of the Tate module $T_\ell A$. The characteristic polynomial (or Weil polynomial) of $A$ is the characteristic polynomial of the latter endomorphism: $$P_A(t) = \det(t-F|T_\ell A).$$ It is a monic polynomial of degree $2g$ with integer coefficients that are independent of $\ell$. It is the reverse of the $L$-polynomial. By a theorem of Weil, $P_A(t)$ is a Weil $q$-polynomial.

Alternatively, $P_A(t)$ can be defined using $\ell$-adic étale cohomology: If $F_q$ is the inverse of the field automorphism $x \mapsto x^q$ in $\Gal(\overline{\F}_q/\F_q)$, then $$P_A(t) = \det(t-F_q|H^1(A_{\overline{\F}_q}, \Q_\ell)).$$