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

Let S be a 2-subgroup of the K-automorphism group Aut(X) of an algebraic curve X of genus g(X) defined over an algebraically closed field K of characteristic 2. It is known that S may be quite large compared to the classical Hurwitz bound 84(g(X)−1). However, if S fixes no point, then the size of S is smaller than or equal to 4(g(X)−1). In this paper, we investigate algebraic curves X with a 2-subgroup S of Aut(X) having the following properties: (I) |S|≥8 and |S|>2(g(X)−1), (II) S fixes no point on X. Theorem 1.2 shows that X is 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/〈u〉 inherits the above properties, that is, it has genus ≥2, and its automorphism group S/〈u〉 still has properties (I) and (II). In the first two cases, S is 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 whose K-automorphism group has a dihedral 2-subgroup S of order 4(g−1) that fixes no point in X.