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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 polynomial has degree $2g$, and by a theorem of Weil, the complex roots of this polynomial all have norm $\sqrt{q}$. As a consequence, this means that there are only finitely many Weil polynomials for any fixed pair $(q,g)$.

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• Last edited by Christelle Vincent on 2017-10-11 20:55:56
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