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The computation is based of the Honda-Tate theorem which states that isogeny classes of abelian varieties over finite fields are completely determined by the characteristic polynomial of their Frobenius automorphism acting on the first $\ell$-adic cohomology group. For a given dimension $g$ and base field of size $q$, a complete list of all Weil polynomials that do occur can be enumerated using a technique developed by Kedlaya [MR:2459990, arXiv:0608104]. In 2016, Dupuy, Kedlaya, Roe and Vincent improved upon Kedlaya's original code to generate these tables and the data they contain.

See also the article of Kedlaya and Sutherland [MR:3540942, arXiv:1511.06945], where these techniques are used to compute Weil polynomials for K3 surfaces.

The determination of which isogeny classes contain Jacobians includes the constraint that if $X$ is a curve over $\F_q$, then $\#X(\F_q) \geq 0$ and $\#X(\F_{q^{mn}}) \geq \#X(\F_{q^n})$ for all positive integers $m,n$; together with additional constraints from the following papers.

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  • This knowl is being renamed from dq.av.fq.source
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  • Last edited by Kiran S. Kedlaya on 2019-11-19 15:51:09
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