Large 2-groups of automorphisms of algebraic curves over a field of characteristic 2

Let Sbe a 2-subgroup of the K-automorphism group Aut(X)of an algebraic curve Xof genus g(X)defined over an algebraically closed field Kof characteristic 2. It is known that Smay be quite large compared to the classical Hurwitz bound 84(g(X) −1). However, if Sfixes no point, then the size of Sis smaller than or equal to 4(g(X) −1). In this paper, we investigate algebraic curves Xwith a 2-subgroup Sof Aut(X)having the following properties: (I)|S| ≥8and |S| >2(g(X) −1), (II)Sfixes no point on X. Theorem1.2shows that Xis a general curve and that either |S| =4(g(X) −1), or |S| =2g(X) +2, or, for every involution u ∈Z(S), the quotient curve X/uinherits the above properties, that is, it has genus ≥2, and its automorphism group S/ustill has properties (I) and (II). In the first two cases, Sis completely determined. We also give examples illustrating our results. In particular, for every g =2h+1 ≥9, we exhibit a (general bielliptic) curve X of genus g whoseK-automorphism group has a dihedral 2-subgroup Sof order 4(g −1)that fixes no point in X.✩