Abstract
In arithmetic and algebraic geometry, superspecial curves have been studied as one of the most important objects, with practical applications to cryptography and coding theory. The enumeration of those curves is a central problem, but if \(g \ge 4\) it is not even known whether a superspecial curve of genus g exists in general characteristic \(p>0\). In this paper, we propose an algorithm with complexity \(O(p^3)\) to enumerate superspecial hyperelliptic curves of genus 4 with automorphism group \(V_4\), where \(V_4\) is the non-cyclic group of order 4. By executing the algorithm over Magma, we enumerate those curves over \(\overline{\mathbb {F}_p}\) for p up to 200. We also succeeded in finding a superspecial hyperelliptic curve of genus 4 in every characteristic p with \(19 \le p \le 6691\).
Keywords
- Hyperelliptic curves
- Superspecial curves
- Genus-4 curves
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https://sites.google.com/view/m-kudo-official-website/english/code/genus4hypv4
Acknowledgements
The authors thank the referees for valuable comments and suggestions. This work was supported by JSPS Grant-in-Aid for Young Scientists 20K14301, and JSPS Grant-in-Aid for Scientific Research (C) 21K03159.
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Ohashi, R., Kudo, M., Harashita, S. (2023). Fast Enumeration of Superspecial Hyperelliptic Curves of Genus 4 with Automorphism Group \(V_4\). In: Mesnager, S., Zhou, Z. (eds) Arithmetic of Finite Fields. WAIFI 2022. Lecture Notes in Computer Science, vol 13638. Springer, Cham. https://doi.org/10.1007/978-3-031-22944-2_6
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