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

Abstract

Let 𝒳 be a projective, geometrically irreducible, nonsingular algebraic curve of genus g≥2 over an algebraically closed field of characteristic two. We classify cyclic subgroups of Aut(𝒳) of order N≥2g+1. Besides Kummer extensions of odd degree ramified over three points, precisely two cases occur: y²+y=x^m, with N=2m=4g+2, and y²+y=c/(x^m+1), with m odd, c≠0, and N=2m=2g+2. In particular, 4nmid N. We then classify groups H with a cyclic subgroup of index two and |H|>4g+4. They are precisely the groups C_k× D_2M on the curves y^k=x^M+x^−M, where k,M≥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|≤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.

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BibTeX RIS

BibTeX

@misc{timpanella2026large,
  title = {{Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two}},
  author = {Marco Timpanella},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34709},
  eprint = {2609.34709},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34709},
}

RIS

TY  - UNPB
TI  - Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two
AU  - Marco Timpanella
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34709
UR  - https://arxiv.org/abs/2609.34709
AB  - Let 𝒳 be a projective, geometrically irreducible, nonsingular algebraic curve of genus g≥2 over an algebraically closed field of characteristic two. We classify cyclic subgroups of Aut(𝒳) of order N≥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≠0, and N=2m=2g+2. In particular, 4nmid N. We then classify groups H with a cyclic subgroup of index two and |H|>4g+4. They are precisely the groups C_k× D_2M on the curves y^k=x^M+x^−M, where k,M≥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|≤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.
ER  -

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