Superspecial curves of genus 4 in small characteristic

Abstract

This paper contains a complete study of superspecial curves of genus 4 in characteristic p≤7. We prove that there does not exist a superspecial curve of genus 4 in characteristic 7. This is a negative answer to the genus 4 case of the problem proposed by Ekedahl [9] in 1987. This implies the non-existence of maximal curve of genus 4 over F49, which updates the table at manypoints.org. We give an algorithm to enumerate superspecial nonhyperelliptic curves in arbitrary p≥5, and for p≤7 we execute it with our implementation on a computer algebra system Magma. Our result in p=5 re-proves the uniqueness of maximal curves of genus 4 over F25, see [11] for the original theoretical proof.In Appendix, we present a general method determining Hasse–Witt matrices of curves which are complete intersections.