automorphism groups for p-cyclic covers of the affine line

Abstract

Let k be an algebraically closed field of positive characteristic p > 0 and C → P 1 k a p-cyclic cover of the projective line ramified in exactly one point. We are interested in the p-Sylow subgroups of the full automorphism group Aut k C. We prove that for curves C with genus 2 or higher, these groups are exactly the extensions of a p-cyclic group by an elementary abelian p-group. The main tool is an efficient algorithm to compute the p-Sylow subgroups of Aut k C starting from an Artin-Schreier equation for the cover C → P 1 k . We also characterize curves C with genus g C 2 and a p-groupOur methods rely on previous work by Stichtenoth whose approach we have adopted.

Algebraic curves with a large non-tame automorphism group fixing no point

“…As a consequence of Nakajima's results, (1.2) implies that S fixes a point of X . This was observed by Lehr and Matignon [19]. In their investigation on big actions satisfying the condition…”

“…The term of big action was introduced by Lehr and Matignon [19] and found its motivation in earlier work by Stichtenoth [27] and Nakajima [22]. As a consequence of Nakajima's results, (1.2) implies that S fixes a point of X .…”

“…they proved that (1.2) and (1.3) only occur simultaneously when the curve is birationally equivalent over K to an Artin-Schreier curve v(Y q − Y − f (X)) such that f (X) = XP (X) + cX, where P (X) is an additive polynomial of K[X]; see [19]. Matignon and Rocher [20,23,24] continued the work of Lehr and Matignon, especially in order to classify big actions in which |S| g 2 > 4 (p 2 − 1) 2 .…”

On the tangent space of the deformation functor of curves with automorphisms

“…Moreover, according to [Lehr and Matignon 2005] any automorphism σ v corresponding to v ∈ V is given by Observe that w and x have a unique pole of order p m + 1 and p, respectively, at the point above ∞, so we can select the local uniformizer π so that…”

“…Such curves were examined in [van der Geer and van der Vlugt 1992] in connection with coding theory, and their automorphism group was studied in [Lehr and Matignon 2005], is given by…”