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 [26] proved that if γ ≥ 2 then |S| ≤ p p−2 (g − 1). If equality holds, X is a Nakajima extremal curve. We prove that ifthen 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). There are exactly p − 1 such Galois extensions. 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.

Algebraic curves with many automorphisms

“…Result 2.5 (Nakajima's bound [36]; see also [16] and Theorem 11.84 in [29]). If X has positive p-rank and S is a p-subgroup of Aut(X ) then…”

An 𝔽p2-maximal Wiman sextic and its automorphisms

“…Curves C together with a p-group H of automorphisms such that the bound ( 12) is attained are called Nakajima extremal curves. Giulietti and Korchmáros in [16] showed that the full automorphism group of Nakajima extremal curves has a precise structure. 3 The automorphism group of W Wiman proved that the automorphism group of W over the complex field C is the symmetric group S 5 .…”

Large odd prime power order automorphism groups of algebraic curves in any characteristic

“…Multiply both sides by α. Since α, s 1 and s 2 commute pairwise, Equation (6) gives (7) (αβ) 2 βαs 1 = (αβ) 2 βs 1 α = s 3 1 s 2 α 3 . Furthermore, αβs −1 1 = βα yields (αβ) 2 βαs 1 = (αβ) 3 .…”

“…Let S be a p-subgroup of X . If the p-rank γ(X ) of X is positive then Nakajima's bound yields |S| ≤ 3(g(X ) − 1), [16] see also [10,Theorem 11.84], and this bound is attained by an infinite family of curves; see [7]. If γ(X ) = 0 then G fixes a point of X , see [6] or [10,Lemma 11.129], and Stichtenoth's bound gives |S| ≤ 4p/(p − 1) 2 g, [20]; see also [10,Theorem 11.78].…”