Abstract
Schoof’s classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof’s algorithm. While we are currently missing the tools we need to generalize Elkies’ methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this chapter, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.
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Notes
- 1.
We would also like to mention Bisson, Cosset, and Robert’s AVIsogenies software package [1], which provides some functionality in this direction. However, their methods apply to abelian surfaces with a lot of rational 2- and 4-torsion, and applying them to general genus-2 Jacobians (with or without known RM) generally requires a substantial extension of the base field to make that torsion rational. This is counterproductive in the context of point counting.
- 2.
Vanilla is the most common and least complicated flavor of abelian varieties over finite fields. Heuristically, over large finite fields, randomly sampled abelian varieties are vanilla with overwhelming probability. Indeed, being vanilla is invariant in isogeny classes, and Howe and Zhu have shown in [14, Theorem 2] that the fraction of isogeny classes of g-dimensional abelian varieties over \(\mathbb{F}_{q}\) that are ordinary and absolutely simple tends to 1 as q → ∞. All absolutely simple ordinary abelian varieties are vanilla, except those whose endomorphism algebras contain roots of unity; but the number of such isogeny classes for fixed g is asymptotically negligible.
- 3.
For full generality, we should also allow \(\deg f = 6\); the curve \(\mathcal{C}\) then has two points at infinity. This substantially complicates the formulæ without significantly modifying the algorithms or their asymptotic complexity, so we will not treat this case here.
- 4.
With polynomial time estimates like these, who needs enemies?
- 5.
See [24, Chapter 2, Section 2] for details on this subset. For point counting over large finite fields, it is enough to note that since the subset is Zariski open, randomly sampled Jacobians with real multiplication by \(\mathcal{O}_{F}\) have their RM invariants in this subset with overwhelming probability.
- 6.
We emphasize that the subgroup \(\mathcal{A}[\mu ]\) depends on ι, but we have chosen to write \(\mathcal{A}[\mu ]\) instead of the more cumbersome \(\mathcal{A}[\iota (\mu )]\).
- 7.
The polynomials H μ, 3 do not appear there, but only G μ is required to apply our results in §3.5.
- 8.
Available from https://members.loria.fr/EMilio/modular-polynomials/.
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Acknowledgements
This chapter reports on work carried out at the workshop Algebraic Geometry for Coding Theory and Cryptography at the Institute for Pure and Applied Mathematics (IPAM), University of California, Los Angeles, February 22–26, 2016. The authors thank IPAM for its generous support. Chloe Martindale was supported by an ALGANT-doc scholarship in association with Universiteit Leiden and Université de Bordeaux. Maike Massierer was supported by the Australian Research Council (DP150101689).
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Ballentine, S. et al. (2017). Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication. In: Howe, E., Lauter, K., Walker, J. (eds) Algebraic Geometry for Coding Theory and Cryptography. Association for Women in Mathematics Series, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-63931-4_3
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