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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 $\#X(\F_q) \geq 0$ and $\#X(\F_{q^{mn}}) \geq \#X(\F_{q^n})$ for all positive integers $m,n$; together with additional constraints from: 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 $4$).

Knowl status:
  • Review status: beta
  • Last edited by David Roe on 2020-11-06 02:10:39
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