Quickly constructing curves of genus 4 with many points

Abstract

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 F_q with defect at most 4 in time q^{3/4}, up to logarithmic factors. As part of the analysis of the algorithm, we present a reinterpretation of results of Hayashida 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.