Quickly constructing curves of genus 4 with many points

TL;DR

This paper introduces an efficient algorithm to construct genus-4 curves over finite fields with many rational points, leveraging double covers of genus-2 curves and advanced algebraic structures.

Contribution

It presents a novel construction method and an algorithm for generating genus-4 curves with small defect, improving point count bounds over finite fields.

Findings

Algorithm produces genus-4 curves with defect at most 4 for large primes.

Expected runtime is approximately q^{3/4} for large q.

Reinterprets results on Jacobians and principal polarizations in algebraic geometry.

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.