Generalized Howe curves of genus 4, 5, and 6 with completely decomposable Jacobians

TL;DR

This paper develops an efficient computational method to construct superspecial curves of genus 4, 5, and 6 with completely decomposable Jacobians, extending known existence results to larger characteristics.

Contribution

It introduces a new approach focusing on Howe curves with Jacobians decomposing into four elliptic curves, enabling construction of superspecial curves in higher characteristics.

Findings

Confirmed existence of superspecial genus 4 curves for 20000 < p < 10^6.

Constructed superspecial genus 5 and 6 curves from supersingular elliptic curves.

Established existence of such curves in specified characteristic ranges through computational experiments.

Abstract

Superspecial curves are important objects in number theory and algebraic geometry, and the existence in genus $g \geq 4$ remains an open problem for all but finitely many characteristics $p > 0$. As a computational approach to this problem, Kudo-Harashita-Howe (2020) showed that a superspecial curve of genus 4 exists in each characteristic $p$ with $7 < p < 20000$. Their method restricted attention to a specific class of curves, known as Howe curves, for which superspeciality is reduced to those of curves of genus at most 2. In this paper, we focus on a more specific class of curves, namely Howe curves whose Jacobians decompose into a product of four elliptic curves. By restricting our attention to such curves, the superspeciality reduces to the supersingularity of elliptic curves, which enables us to construct a superspecial curve of genus 4 more efficiently than Kudo-Harashita-Howe's method. As our first main result, we confirmed by computer the existence of such superspecial curves of genus 4 in characteristics $p$ with $20000 < p < 10^6$. Using a similar approach, we also propose constructions of superspecial curves of genera 5 and 6 from only supersingular elliptic curves. Furthermore, computational experiments establish the existence of superspecial curves of genus 5 (resp. genus 6) in characteristics $p$ with $13 < p < 10^5$ (resp. $7 < p < 10^5$).