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.

- Howe and Lauter, Improved upper bounds for the number of points on curves over finite fields [MR:2038778]
- Howe, Maisner, Nart, and Ritzenthaler, Principally polarizable isogeny classes of abelian surfaces over finite fields [MR:2367179, 10.4310/MRL.2008.v15.n1.a11]
- Howe, Nart, and Ritzenthaler, Jacobians in isogeny classes of abelian surfaces over finite fields [MR:2514865]
- Korchmáros and Torres, On the genus of a maximal curve [MR:1923698, 10.1023/A1017553432375]
- Serre, Rational points on curves over finite fields, URL
- Stohr and Voloch, Weierstrass points and curves over finite field [MR:0812443, 10.1112/plms/s3-52.1.1]
- Zaytsev, An improvement of the Hasse-Weil-Serre bound for curves over some finite fields [MR:3249822, 10.1016/j.ffa.2014.06.003]
- Zaytsev, Optimal curves of low genus over finite fields [MR:3426586, 10.1016/j.ffa.2015.09.008]

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

**Referred to by:**

**History:**(expand/hide all)

- 2020-11-06 02:10:39 by David Roe
- 2020-07-31 12:39:05 by Kiran S. Kedlaya
- 2019-12-28 16:18:47 by Andrew Sutherland
- 2019-12-15 23:03:18 by Kiran S. Kedlaya
- 2019-11-19 15:51:09 by Kiran S. Kedlaya
- 2019-11-19 15:48:21 by Kiran S. Kedlaya
- 2019-11-19 15:42:37 by Kiran S. Kedlaya
- 2019-11-19 15:37:09 by Kiran S. Kedlaya
- 2019-10-31 21:17:08 by David Roe
- 2019-10-18 12:38:34 by Andrew Sutherland
- 2017-06-21 04:23:24 by Christelle Vincent

**Differences**(show/hide)