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Let $A/\mathbb{F}_q$ be an abelian variety of dimension $g$ defined over a finite field. Its Weil $q$-polynomial is the polynomial $$P(A/\mathbb{F}_q,t) = \det(t-F_q|H^1((A_{\overline{\mathbb{F}}_q})_{et}, \mathbb{Q}_l)),$$ where $F_q$ is the inverse of Frobenius acting on cohomology. This is a polynomial of degree $2g$ with integer coefficients. By a theorem of Weil, the complex roots of this polynomial all have norm $\sqrt{q}$; this means that there are only finitely many Weil polynomials for any fixed pair $(q,g)$.

The Weil $q$-polynomial of $A$ is the reverse of the $L$-polynomial.

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• Last edited by Kiran S. Kedlaya on 2019-09-04 17:53:52
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