Quickly constructing curves of genus 4 with many points.

*(English)*Zbl 1417.11122
Kohel, David (ed.) et al., Frobenius distributions: Lang-Trotter and Sato-Tate conjectures. Winter school and workshop on Frobenius distributions on curves, CIRM, Marseille, France, February 17–21, 2014 and February 24–28, 2016. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 663, 149-173 (2016).

Summary: The defect of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing genus-4 double covers of genus-2 curves over finite fields such that the defect of the double cover is not much more than the defect of the genus-2 curve. We give an algorithm that uses this construction to produce genus-4 curves with small defect. Heuristically, for all sufficiently large primes and for almost all prime powers q, the algorithm is expected to produce a genus-4 curve over \(\mathbb F_q\) with defect at most 4 in time \( \widetilde{O}(q^{3/4})\).

As part of the analysis of the algorithm, we present a reinterpretation of results of T. Hayashida [J. Math. Soc. Japan 20, 26–43 (1968; Zbl 0186.26501)] on the number of genus-2 curves whose Jacobians are isomorphic to the square of a given elliptic curve with complex multiplication by a maximal order. We show that a category of principal polarizations on the square of such an elliptic curve is equivalent to a category of right ideals in a certain quaternion order.

For the entire collection see [Zbl 1345.11004].

As part of the analysis of the algorithm, we present a reinterpretation of results of T. Hayashida [J. Math. Soc. Japan 20, 26–43 (1968; Zbl 0186.26501)] on the number of genus-2 curves whose Jacobians are isomorphic to the square of a given elliptic curve with complex multiplication by a maximal order. We show that a category of principal polarizations on the square of such an elliptic curve is equivalent to a category of right ideals in a certain quaternion order.

For the entire collection see [Zbl 1345.11004].