Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two

Abstract

Let $\mathcal{X}$ be a projective, geometrically irreducible, nonsingular algebraic curve of genus $g\ge 2$ over an algebraically closed field of characteristic two. We classify cyclic subgroups of $\Aut(\mathcal{X})$ of order $N\ge 2g+1$. Besides Kummer extensions of odd degree ramified over three points, precisely two cases occur: $y^2+y=x^m$, with $N=2m=4g+2$, and $y^2+y=c/(x^m+1)$, with $m$ odd, $c\ne 0$, and $N=2m=2g+2$. In particular, $4\nmid N$. We then classify groups $H$ with a cyclic subgroup of index two and $|H|>4g+4$. They are precisely the groups $C_k\times D_{2M}$ on the curves $y^k=x^M+x^{-M}$, where $k,M\ge 3$ are odd and coprime, and $2g=M(k-1)$. Their possible orders are $4g+2M$, where $M$ runs over certain odd divisors of $g$, and $|H|\le 6g$. No curve in this family is ordinary. Hermitian curves occur in the family, and we give two sufficient conditions for maximality over finite fields. For dihedral groups the sharp bound is $4g+4$ in even genus and $4g$ in odd genus. The equality cases are given by explicit Artin--Schreier equations, and in each case the dihedral group is the full automorphism group.