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

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- 2022-03-26 19:16:10 by Bjorn Poonen
- 2022-03-26 19:13:46 by Bjorn Poonen
- 2022-03-26 19:04:10 by Bjorn Poonen
- 2022-03-26 18:56:59 by Bjorn Poonen
- 2019-09-04 17:53:52 by Kiran S. Kedlaya (Reviewed)
- 2019-09-04 17:42:34 by Kiran S. Kedlaya
- 2017-10-11 20:55:56 by Christelle Vincent (Reviewed)

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