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

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. $7 < p < 10^5$).