Curves with a large automorphism group admitting a cyclic subgroup of index 2

2026/07/17 by Arianna Dionigi, Massimo Giulietti, Marco Timpanella
#math.AG

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Abstract

The Hurwitz bound on the order of the \mathbb K-automorphism group \rmAut(X) of an algebraic curve X of genus g(X)≥ 2 defined over a field \mathbb K of zero characteristic states that |\rmAut(X)|≤ 84(g(X)-1). Improved bounds are available for the order of certain types of subgroups within automorphism groups. For instance, if a subgroup H of \rmAut(X) is dihedral, then in the complex case, |H| ≤ 4g(X) + 4. More recently it has been shown that a tighter bound holds for H a generalized quasi-dihedral group. In this paper we explore the more general setting of a curve defined over a field of any characteristic, and H a group admitting a cyclic subgroup of index two. We show that the same upper bound for the size of a dihedral group of automorphisms holds for curves defined over an algebraically closed field of characteristic p≠ 2. Then we provide some classification results about (non-dihedral) groups of size larger than 4g(X)+4 admitting a cyclic subgroup of index 2.

Related

• Large automorphism groups of ordinary curves of even genus in odd characteristic
• Large 2-groups of automorphisms of curves with positive 2-rank
• Large automorphism groups of ordinary curves in characteristic 2
• On algebraic curves with many automorphisms in characteristic p
• Algebraic curves with a large cyclic automorphism group