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

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 | ≤ p p − 2 ( g − 1 ) . If equality holds, X is a Nakajima extremal curve . We prove that if | S | > p 2 p 2 − 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. [ABSTRACT FROM AUTHOR]