Overview
This study investigates the existence of superspecial curves, focusing on generalized Howe curves with completely decomposable Jacobians. The primary objective is to construct superspecial curves of genera 4, 5, and 6 through computational methods, building upon prior work by Kudo-Harashita-Howe (2020).
Research Context
Superspecial curves are significant objects within number theory and algebraic geometry. A standing problem concerns the existence of these curves in genus $g \geq 4$ for infinitely many characteristics $p$. For genus 4, Kudo-Harashita-Howe (2020) demonstrated the existence of a superspecial curve in each characteristic $p$ where $7 < p < 20000$. Their approach utilized Howe curves, a specific class for which the condition of superspeciality reduces to that of curves with genus at most 2.
Approach
This paper adopts a more restricted approach, concentrating on Howe curves whose Jacobians factor into a product of four elliptic curves. This particular restriction simplifies the superspeciality criterion, reducing it to the supersingularity of elliptic curves. This simplification aims to enhance the efficiency of constructing superspecial curves of genus 4 compared to the method described by Kudo-Harashita-Howe. Furthermore, a similar methodology is applied to propose constructions for superspecial curves of genera 5 and 6, using only supersingular elliptic curves as foundational elements.
Findings
- For genus 4, computational experiments confirmed the existence of superspecial Howe curves in characteristics $p$ ranging from $20000$ to $10^6$. These curves belong to the specified class where their Jacobians decompose into a product of four elliptic curves.
- Regarding genus 5, computational experiments established the existence of superspecial curves in characteristics $p$ from $13$ to $10^5$. These constructions are derived from the proposed method involving only supersingular elliptic curves.
- For genus 6, computational experiments indicated the existence of superspecial curves in characteristics $p$ from $7$ to $10^5$. Similar to genus 5, these curves are constructed using the proposed method based on supersingular elliptic curves.