Large p-groups of automorphisms of algebraic curves in characteristic p

Abstract

Let S be a p-subgroup of the K-automorphism group Aut(X) of an algebraic curve X of genus g≥2 and p-rank γ defined over an algebraically closed field K of characteristic p≥3. Nakajima [27] proved that if γ≥2 then |S|≤pp−2(g−1). If equality holds, X is a Nakajima extremal curve. We prove that if|S|>p2p2−p−1(g−1) then one of the following cases occurs.(i)γ=0 and the extension K(X)|K(X)S completely ramifies at a unique place, and does not ramify elsewhere.(ii)|S|=p, and X is an ordinary curve of genus g=p−1.(iii)X is an ordinary, Nakajima extremal curve, and K(X) is an unramified Galois extension of a function field of a curve given in (ii).(iii)X is an ordinary, Nakajima extremal curve, and K(X) is an unramified Galois extension of a function field of a curve given in (ii). There are exactly p−1 subgroups M of S such that K(X)|K(X)M is such a Galois extension. Moreover, if some of them is an abelian extension then S has maximal nilpotency class. The full K-automorphism group of any Nakajima extremal curve is determined, and several infinite families of Nakajima extremal curves are constructed by using their pro-p fundamental groups.