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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 gg and base field of size qq, 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 XX is a curve over Fq\F_q, then the number of degree-nn places of XX is nonnegative for all positive integers nn; 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 22 and 33 except as otherwise indicated);
  • Kiran Kedlaya (hyperelliptics in dimensions 33 and 44, using the methods of Xarles and of Howe [arXiv:2401.15255]);
  • Xavier Xarles (dimension 44 over F2\F_2; [arXiv:2007.07822]);
  • Jonas Bergström, Carel Faber, and Sam Payne (dimension 44 over F3\F_3; [arXiv:2206.07759]);
  • Dusan Dragutinović (dimension 55 over F2\F_2; [arXiv:2202.07809]);
  • Steve Huang, Kiran Kedlaya and Jun Bo Lau (dimension 66 over F2\F_2; [arXiv:2402.00716]).
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  • Review status: beta
  • Last edited by Kiran S. Kedlaya on 2024-10-08 10:16:36
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