Smooth curves having a large automorphismp-group in characteristicp >0

Abstract

Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g ≥ 2. This paper continues our study of big actions, that is, pairs (C, G) where G is a p-subgroup of the k-automorphism group of C such that |G|/g > 2 p/( p−1). If G 2 denotes the second ramification group of G at the unique ramification point of the cover C → C/G, we display necessary conditions on G 2 for (C, G) to be a big action, which allows us to pursue the classification of big actions.Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J.-P. Serre and continued by Lauter and by Auer. In particular, we obtain explicit examples of big actions with G 2 abelian of large exponent.

Large p-group actions with a p-elementary abelian derived group

“…By the second point, G is included in Z (G), which implies that H i is normal in G. From [MR08] (see Lemma 2.4 and Theorem 2.6), we infer that (C/H i , G/H i ) is a big action whose derived group (G/…”

“…In [MR08], we more specifically concentrate on the properties of G 2 . In particular, we prove that G 2 coincides with the derived group (or commutator subgroup) of G, denoted by G or D(G).…”

“…If (C, G) is a big action and H a normal subgroup of G such that H G , then (C/H, G/H) is still a big action whose derived group is G /H (cf. [MR08,§2]). In particular, when applying this result to H = Fratt(G ) the Frattini subgroup of G , one obtains a big action whose derived group is p-elementary abelian.…”

“…More precisely, we shall parametrize those satisfying |G| g 2 4 (p 2 −1) 2 , knowing that, under this condition, we previously showed that G is a p-elementary abelian group of order dividing p 3 (cf. [MR08,§4]). …”

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

“…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 .…”