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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:math/0608104 ]. In 2016, Dupuy, Kedlaya, Roe, and Vincent improved upon Kedlaya's original code to generate these tables and the data they contain [arXiv:2003.05380 ].

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 the number of degree-$n$ places of $X$ is nonnegative for all positive integers $n$; together with additional constraints from: Howe and Lauter, New methods for bounding the number of points on curves over finite fields [MR:2987661 , arXiv:1202.6308 , 10.4171/119-1/12 ]. This incorporates prior results from the following papers.

The list of curves with Jacobian in a given isogeny class were computed by Andrew Sutherland (dimensions $2$ and $3$) and Xavier Xarles (dimension $4$ over $\F_2$; [arXiv:2007.07822 ]).

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  • Last edited by Kiran S. Kedlaya on 2021-10-04 19:00:46
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