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Let $A$ be an abelian variety of dimension $g$ defined over $\F_q$. Let $F_q$ be the inverse of the field automorphism $x \mapsto x^q$ in $\Gal(\overline{\F}_q/\F_q)$, which acts on $\ell$-adic étale cohomology. The L-polynomial of $A$ is
$$L_A(t) = \det(1-t F_q|H^1(A_{\overline{\F}_q}, \Q_\ell)).$$ This is a polynomial of degree $2g$ with integer coefficients that are independent of $\ell$. Its constant term is $1$.

The L-polynomial $L_A(t)$ is the reverse of the characteristic polynomial $P_A(t)$, which is a Weil $q$-polynomial. Thus the complex roots of $L_A(t)$ have absolute value $q^{-1/2}$.

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  • Last edited by Bjorn Poonen on 2022-03-26 19:29:35
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