Superspecial Trigonal Curves of Genus 5

Abstract

In this paper, we enumerate superspecial trigonal curves of genus 5 over ${\mathbb{F}}_{p^a}$ for any natural number a if $p\le 7$ and for odd a if $p\le 13.$ This paper provides an algorithm enumerating superspecial trigonal curves of genus 5 over ${\mathbb{F}}_q$ for arbitrary q. We implemented the algorithm over computer algebra system Magma. With help of computational results obtained by our implementation, we prove the nonexistence (resp. existence) of superspecial trigonal curves of genus 5 over ${\mathbb{F}}_q$ for q = 7, 7² and 13 (resp. q = 11). Executing the algorithm also determines the number of isomorphism classes of such curves for q = 11 with explicit defining equations.

Keywords:

• Algebraic curves
• superspecial curves
• maximal curves

2010 Mathematical Subject Classification:

• 14G05
• 14G15
• 14G50
• 14H45
• 14Q05
• 68W30

Acknowledgments

The authors thank the referee for his/her careful reading and for helpful suggestions and comments. This work was supported by JSPS Grant-in-Aid for Scientific Research (C) 17K05196, and JSPS Grant-in-Aid for Research Activity Startup 18H05836 and 19K21026.

Notes

1 We implemented the algorithm on Magma V2.22-7. All the source codes and log files are available at the web page of the first author [26].