Transcendence degree one function fields over a finite field with many automorphisms

Let Kbe the algebraic closure of a finite field Fqof odd characteristic p. For a positive integer mprime to p, let F=K(x, y)be the transcendence degree 1function field defined by yq+y=xm+x−m. Let t =xm(q−1)and H=K(t). The extension F|His a non-Galois extension. Let Kbe the Galois closure of Fwith respect to H. By Stichtenoth [20], Khas genus g(K) =(qm −1)(q−1), p-rank (Hasse–Witt invariant) γ(K) =(q−1)2and a K-automorphism group of order at least 2q2m(q−1). In this paper we prove that this subgroup is the full K-automorphism group of K; more precisely AutK(K) =Δ Dwhere Δis an elementary abelian p-group of order q2and Dhas an index 2cyclic subgroup of order m(q−1). In particular, √m|AutK(K)| >g(K)3/2, and if Kis ordinary (i.e. g(K) =γ(K)) then |AutK(K)| >g3/2. On the other hand, if Gis a solvable subgroup of the K-automorphism group of an ordinary, transcendence degree 1function field Lof genus g(L) ≥2defined over K, then |AutK(K)| ≤34(g(L) +1)3/2<68√2g(L)3/2; see [15]. This shows that Khits this bound up to the constant 68√2. Since AutK(K)has several subgroups, the fixed subfield FNof such a subgroup Nmay happen to have many automorphisms provided that the normalizer of Nin AutK(K)is large enough. This possibility is worked out for subgroups of Δ.