Large p-groups actions with |G| /g^2 > 4/ (p^2-1)^2

TL;DR

This paper investigates large p-group actions on algebraic curves, focusing on the ratio of group order to genus squared, and provides a classification for a specific critical ratio value.

Contribution

It offers a classification and parametrization of big p-group actions on curves when the ratio |G|/g^2 equals 4/(p^2-1)^2, a specific threshold.

Findings

Finiteness results on the ratio |G|/g^2 for big actions.

Explicit classification of big actions at the critical ratio.

Parametrization of such big actions.

Abstract

Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. Let (C,G) be a "big action", i.e. a pair (C,G) where G is a p-subgroup of the k-automorphism group of C such that |G|/g > 2p / p-1. We first study finiteness results on the values taken by the quotient |G|/g^2 when (C,G) runs over the big actions satisfying |G|/g^2 >M, for a given positive real M>0. Then, we exhibit a classification and a parametrization of such big actions when M=4/ (p^2-1)^2.